lecture notes introduction to di erential geometry...

Lecture NotesIntroduction to Differential Geometry

MATH 442Instructor: Ivan Avramidi

New Mexico Institute of Mining and Technology

Socorro, NM 87801

August 25, 2005

Author: Ivan Avramidi; File: diffgeom.tex; Date: March 22, 2018; Time: 16:56

Contents

1 Manifolds 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Submanifolds of Euclidean Space . . . . . . . . . . . . . . . . . 4

1.2.1 Submanifolds of Rn . . . . . . . . . . . . . . . . . . . . . 51.2.2 Differential of a Map . . . . . . . . . . . . . . . . . . . . 81.2.3 Main Theorem on Submanifolds of Rm . . . . . . . . . . 9

1.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 Basic Notions of Topology . . . . . . . . . . . . . . . . . 101.3.2 Idea of a Manifold . . . . . . . . . . . . . . . . . . . . . 131.3.3 Rigorous Definition of a Manifold . . . . . . . . . . . . . 151.3.4 Complex Manifolds . . . . . . . . . . . . . . . . . . . . 16

1.4 Tangent Vectors and Mappings . . . . . . . . . . . . . . . . . . . 181.4.1 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . 181.4.2 Vectors as Differential Operators . . . . . . . . . . . . . . 191.4.3 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . 201.4.4 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.5 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 221.4.6 Change of Coordinates . . . . . . . . . . . . . . . . . . . 24

1.5 Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . . . . 251.5.1 Vector Fields in Rn . . . . . . . . . . . . . . . . . . . . . 251.5.2 Vector Fields on Manifolds . . . . . . . . . . . . . . . . . 271.5.3 Straightening Flows . . . . . . . . . . . . . . . . . . . . 27

2 Tensors 292.1 Covectors and Riemannian Metric . . . . . . . . . . . . . . . . . 29

2.1.1 Linear Functionals and Dual Space . . . . . . . . . . . . 292.1.2 Differential of a Function . . . . . . . . . . . . . . . . . . 312.1.3 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . 32

I

II CONTENTS

2.1.4 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . 352.1.5 Curves of Steepest Ascent . . . . . . . . . . . . . . . . . 36

2.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.1 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . 372.2.2 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 422.3.1 Pull-Back of a Covector . . . . . . . . . . . . . . . . . . 432.3.2 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . 442.3.3 The Poincare 1-Form . . . . . . . . . . . . . . . . . . . . 46

2.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4.1 Covariant Tensors . . . . . . . . . . . . . . . . . . . . . 482.4.2 Contravariant Tensors . . . . . . . . . . . . . . . . . . . 502.4.3 General Tensors of Type (p, q) . . . . . . . . . . . . . . . 522.4.4 Linear Transformations and Tensors . . . . . . . . . . . . 542.4.5 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . 552.4.6 Tensor Bundles . . . . . . . . . . . . . . . . . . . . . . . 552.4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.8 Einstein Summation Convention . . . . . . . . . . . . . . 57

3 Differential Forms 593.1 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.1 Permutation Group . . . . . . . . . . . . . . . . . . . . . 593.1.2 Permutations of Tensors . . . . . . . . . . . . . . . . . . 613.1.3 Alternating Tensors . . . . . . . . . . . . . . . . . . . . . 623.1.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . 653.1.5 Exterior p-forms . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Exterior and Interior Products . . . . . . . . . . . . . . . . . . . 693.2.1 Interior Product . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Hodge Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.3.1 Volume Form . . . . . . . . . . . . . . . . . . . . . . . . 733.3.2 Star Operator . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4 Exterior Derivative and Coderivative . . . . . . . . . . . . . . . . 783.4.1 Coderivative . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5 Pullback of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 813.6 Vector Analysis in R3 . . . . . . . . . . . . . . . . . . . . . . . . 83

3.6.1 Vector Algebra in R3 . . . . . . . . . . . . . . . . . . . . 833.6.2 Vector Analysis in R3 . . . . . . . . . . . . . . . . . . . . 84

3.7 Orientation and the Volume Form . . . . . . . . . . . . . . . . . 86

diffgeom.tex; March 22, 2018; 16:56; p. 1

CONTENTS III

3.7.1 Orientation of a Vector Space . . . . . . . . . . . . . . . 863.7.2 Orientation of a Manifold . . . . . . . . . . . . . . . . . 873.7.3 Hypersurfaces in Orientable Manifolds . . . . . . . . . . 883.7.4 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . 893.7.5 Pseudotensors and Tensor Densities . . . . . . . . . . . . 91

4 Integration of Differential Forms 954.1 Integration over a Parametrized Subset . . . . . . . . . . . . . . . 95

4.1.1 Integration of n-Forms in Rn . . . . . . . . . . . . . . . . 954.1.2 Integration over Parametrized Subsets . . . . . . . . . . . 964.1.3 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . 974.1.4 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . 974.1.5 Independence of Parametrization . . . . . . . . . . . . . . 984.1.6 Integrals and Pullbacks . . . . . . . . . . . . . . . . . . . 99

4.2 Integration over Manifolds . . . . . . . . . . . . . . . . . . . . . 1014.2.1 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . 1014.2.2 Integration over Submanifolds . . . . . . . . . . . . . . . 1034.2.3 Manifolds with boundary . . . . . . . . . . . . . . . . . . 104

4.3 Stokes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.1 Orientation of the Boundary . . . . . . . . . . . . . . . . 1054.3.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . 105

4.4 Poincare Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.2 Poincare Lemma . . . . . . . . . . . . . . . . . . . . . . 1124.4.3 Complex Analysis . . . . . . . . . . . . . . . . . . . . . 115

5 Lie Derivative 1175.1 Lie Derivative of a Vector Field . . . . . . . . . . . . . . . . . . . 117

5.1.1 Lie Bracket . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1.2 Flow generated by the Lie Bracket . . . . . . . . . . . . . 120

5.2 Lie Derivative of Forms and Tensors . . . . . . . . . . . . . . . . 1225.2.1 Properties of Lie Derivative . . . . . . . . . . . . . . . . 124

5.3 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1295.3.1 Distributions . . . . . . . . . . . . . . . . . . . . . . . . 1295.3.2 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . 1335.3.3 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.4 Degree of a Map . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4.1 Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . 138


IV CONTENTS

5.4.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4.3 Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . 1435.4.4 Index of a Vector Field . . . . . . . . . . . . . . . . . . . 1475.4.5 Linking Number . . . . . . . . . . . . . . . . . . . . . . 150

6 Connection and Curvature 1536.1 Affine Connection . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.1.1 Covariant Derivative . . . . . . . . . . . . . . . . . . . . 1536.1.2 Curvature, Torsion and Levi-Civita Connection . . . . . . 1566.1.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . 159

6.2 Tensor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.2.1 Covariant Derivative of Tensors . . . . . . . . . . . . . . 1606.2.2 Ricci Identities . . . . . . . . . . . . . . . . . . . . . . . 1616.2.3 Normal Coordinates . . . . . . . . . . . . . . . . . . . . 1626.2.4 Properties of the Curvature Tensor . . . . . . . . . . . . . 1636.2.5 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . 165

6.3 Cartan’s Structural Equations . . . . . . . . . . . . . . . . . . . . 167

7 Homology Theory 1737.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 173

7.1.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.1.2 Finitely Generated and Free Abelian Groups . . . . . . . 1777.1.3 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . 177

7.2 Singular Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1797.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3 Singular Homology Groups . . . . . . . . . . . . . . . . . . . . . 1867.3.1 Cycles, Boundaries and Homology Groups . . . . . . . . 1867.3.2 Simplicial Homology . . . . . . . . . . . . . . . . . . . . 1877.3.3 Betti Numbers and Topological Invariants . . . . . . . . . 1887.3.4 Some Theorems from Algebraic Topology . . . . . . . . . 1907.3.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 194

7.4 de Rham Cohomology Groups . . . . . . . . . . . . . . . . . . . 1967.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.5 Harmonic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 2007.6 Relative Homology and Morse Theory . . . . . . . . . . . . . . . 207

7.6.1 Relative Homology . . . . . . . . . . . . . . . . . . . . . 2077.6.2 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . 211


CONTENTS V

8 Appendix A: Linear Algebra 2178.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . 217

8.1.1 Sets and Maps . . . . . . . . . . . . . . . . . . . . . . . 2178.1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 2188.1.3 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . 219

8.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.3 Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2208.4 Matrix Representations . . . . . . . . . . . . . . . . . . . . . . . 2218.5 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.6 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2238.7 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2238.8 Pullback and Adjoint . . . . . . . . . . . . . . . . . . . . . . . . 226

9 Appendix B: Multivariable Calculus 2299.1 Basic Theorems of Multivariable Calculus . . . . . . . . . . . . . 2299.2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.3 Smooth Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

Bibliography 237

Notation 239


VI CONTENTS


Chapter 1

Manifolds

1.1 Preliminaries• Sets, union, intersection, empty set, subset

∅,R,Rn,C,Cn

• Quantifiers∀,∃

• Maps, domain, codomainF : A→ B

• Image of a set X ⊂ A

F(X) = y ∈ B | y = F(x) for some x ∈ X

• Range (image) of a mapIm(F) = F(A)

• Inverse image of a set

F−1(Y) = x ∈ A | F(x) ∈ Y

• Injections (one-to-one maps)

∀x1, x2 ∈ A, if x1 , x2 then F(x1) , F(x2)

1

2 CHAPTER 1. MANIFOLDS

• Surjections (onto maps)

∀y ∈ B,∃x ∈ A such that y = F(x)

• Bijections

• Invertible mapsF−1 : B→ A

∀y ∈ B, F(F−1(y)) = y, ∀x ∈ A, F(F−1(x)) = x

• Vector spacesRn

• Linear mapsF : V → W

for all a1, a2 ∈ R and x1, x2 ∈ V

F(a1x1 + a2x2) = a1F(x1) + a2F(x2)

• Kernel of a linear map

Ker F = x ∈ V | F(x) = 0

• MatricesA = (Ai j), i = 1, . . . ,m; j = 1, . . . n

Transposition

AT = (A ji), i = 1, . . . ,m; j = 1, . . . n

Product of matrices

(AB)ik =

n∑j=1

Ai jB jk, i = 1, . . . ,m; k = 1, . . . l

• Square matrices


1.1. PRELIMINARIES 3

• Unit matrixI = (δi j), i = 1, . . . , n; j = 1, . . . n

Kronecker symbol

δi j =

1 i = j0 i , j

• Inverse matrix

• Determinantdet(AB) = det A det B

det A−1 = (det A)−1

• Orthogonal matricesAT A = AAT = I

det A = ±1

• Rank of a linear maprank F = dim Im F

Nullitynull F = dim Ker F

Rank-nullity theorem

dim Im F + dim Ker F = dim V

rank F + null F = dim V

• Groups

• Euclidean space Rn

• Maps between Euclidean spaces

F : Rmx → Rr

y

yα = Fα(x j) α = 1, . . . , r; j = 1, . . . ,m

• Jacobian matrix (∂yα

∂xi

)• Jacobian (if m = r)

J = det(∂yα

∂xi

)diffgeom.tex; March 22, 2018; 16:56; p. 7


1.2 Submanifolds of Euclidean Space• Idea: Manifold is a general space that looks locally like a Euclidean space

of the same dimension. This allows to develop the differential and integralcalculus.

• Let n ∈ N be a positive integer. The Euclidean space Rn is a set of points xdescribed by ordered n-tuples (x1, . . . , xn) or real numbers.

• The numbers xi ∈ R, i = 1, . . . , n, are called the Cartesian coordinates ofthe point x.

• The integer n is the dimension of the Euclidean space.

• The distance between two points of the Euclidean space is defined by

d(x, y) =

√√n∑

k=1

(xk − yk)2 .

• The open ball of radius ε centered at x0 is the set of points defined by

Bε(x0) = x ∈ Rn | d(x, x0) < ε .

• A neighborhood of a point x0 is the set of points that contain an open ballaround it.

• Let x0 ∈ Rn be a fixed point with Cartesian coordinates xi, i = 1, . . . , n,

in the Euclidean space and S ⊂ Rn be a neighborhood of x0. An injective(one-to-one) map

f : S → Rn

defined byyi = f i(x1, . . . , xn) , i = 1, . . . , n,

where f i(x) are smooth functions, is called a coordinate system in S .

• Inverse Function Theorem. Let f : S → Rn be a smooth map and x0 ∈ S .Suppose that

det(∂ f i

∂x j

)(x0) , 0 .

Then f is invertible (locally injective) in a neighborhood V of x0, that is,there is a function g : V → Rn such that for any x ∈ V , y ∈ f (V)

g( f (x)) = x, f (g(y)) = y .


1.2. SUBMANIFOLDS OF EUCLIDEAN SPACE 5

1.2.1 Submanifolds of Rn

• Let n, r ∈ N be positive integers. Let M ⊂ Rn+r be a subset of the Euclideanspace Rn+r.

• Then M is a submanifold of Rn+r of dimension n if for any point x0 ∈

M there exists a neighborhood with a coordinate system such that everypoint in this neighborhood has coordinates (x1, . . . , xn, y1, . . . , yr), where thelast r coordinates (y1, . . . , yr) are given by smooth functions of the first ncoordinates (x1, . . . , xn):

yα = f α(x1, . . . , xn) , α = 1, . . . , r .

• The coordinates (x1, . . . , xn) are called the local coordinates for M near x0.

• More generally, letF : Rn+r

x,y → Rrz

be a smooth map described by r equations

zα = Fα(x1, . . . , xn, y1, . . . , yr) , α = 1, . . . , r ,

or, in short,z = F(x, y) .

Let z0 ∈ Rr and

M = F−1(z0) = (x, y) ∈ Rn+r | F(x, y) = z0

be a subset of the Euclidean space Rm described by the locus of r equations

Fα(x, y) = zα0 , α = 1, . . . , r .

Suppose that M is non-empty and let (x0, y0) ∈ M, that is

F(x0, y0) = z0 .

• Then the Implicit Function Theorem says that if

det(∂Fα

∂yβ(x0, y0)

), 0 ,

where α, β = 1, . . . , r, then there is a neighborhood of (x0, y0) such thatthe last r coordinates can be expressed as smooth functions of the first ncoordinates:

yα = f α(x1, . . . , xn) , α = 1, . . . , r .



• If this is true for any point of M, then M is a n-dimensional submanifold ofRn+r.

• The Jacobian matrix has the form

J(x, y) =

(∂Fα(x, y)∂xi ,

∂Fα(x, y)∂yβ

),

where α, β = 1, . . . , r and i = 1, . . . , n.

• The General Implicit Function Theorem says that if at a point (x0, y0) ∈ Mthe Jacobian matrix has the maximal rank equal to r,

rankJ(x0, y0) = r ,

then there exists a coordinate system in a neighborhood of (x0, y0) such thatthe last r coordinates can be expressed as smooth functions of the first ncoordinates.

• If this is true for every point of M, then M is a n-dimensional submanifoldof Rn+r.

• Even more generally, if the Jacobian has a constant rank on M, rank J = k ≤r, (not necessarily the maximal rank equal to r), then M is a submanifold ofRn+r of dimension dim M = n + r − k.

• The number r is called the codimension of M.

• If the codimension r is equal to 1, then M is called a hypersurface.

Examples

• S 1 in R3

Explicitf : [0, 2π]→ R3

x1 = cos t, x2 = sin t, x3 = 1 .

ImplicitF : R3 → R2

S 1 = F−1(1, 1)

(x1)2 + (x2)2 = 1, x3 = 1 .



• T 2 in R3

Explicitf : [0, 2π] × [0, 2π]→ R3

x1 = (2 + sin v) cos u, x2 = (2 + sin v) sin u, x3 = cos v .

ImplicitF : R3 → R

T 2 = F−1(0)[(x1)2 + (x2)2 + (x3)2 + 3

]2− 16

[(x1)2 + (x2)2

]2= 0

• S 2 in R3

F : R3 → R

S 2 = F−1(1)

(x1)2 + (x2)2 + (x3)2 = 1

• S n in Rn+1

F : Rn+1 → R

S n = F−1(1)n∑

i=1

(xi)2 = 1

• S L(n)F : Rn2

→ R

S L(n) = F−1(1)

det X = 1

• S O(n)F : Rn2

→ Rn(n+1)/2

S O(n) = F−1(I)

XT X = I



1.2.2 Differential of a Map

• Let Rn be a Euclidean space and x0 ∈ Rm. Then the tangent space to Rm at

x0 is a vector space Rmx of all vectors in Rm based at x.

• LetF : Rm → Rr

be a smooth map described by r smooth functions

yα = Fα(x1, . . . , xn) , α = 1, . . . , r .

Let x0 ∈ Rm and x = x(t), t ∈ (−ε, ε), be a curve in Rm such that

x(0) = x0 anddxdt

(0) = v ,

where v ∈ Rmx0

is the tangent vector to the curve at x0.

• Let y0 = F(x0) ∈ Rr and y(t) = F(x(t)). Then

y(0) = y0 anddydt

(0) = w ,

where w ∈ Rrx0

is the tangent vector to the image of the curve at y0.

• We compute

wα =

m∑i=1

(∂Fα

∂xi

)(x0)vi .

• Thus there is a linear transformation

F∗ : Rmx0→ Rr

y0,

so thatF∗v = w .

F∗ is called the differential of the map F at x0.



1.2.3 Main Theorem on Submanifolds of Rm

• The matrix of the linear transformation F∗ is exactly the Jacobian matrix.

• Therefore, the differential F∗ at a point x0 is a surjective (onto) map if andonly if m ≥ r and the Jacobian at x0 has the maximal rank equal to r.

• Recall that F∗ : Rmx0→ Rr

y0is surjective if for any w ∈ Rr

y0there is v ∈ Rm

x0

such that F∗v = w.

• Thus, we have the following theorem.

•

Theorem 1.2.1 Let F : Rm → Rr with m > r, y0 ∈ Rr and

M = F−1(y0) = x ∈ Rm | F(x) = y0 .

If M is non-empty and for any x0 ∈ M the differential F∗ : Rmx0→ Rr

y0is

surjective, then M is a n = (m − r)-dimensional submanifold of Rm.



1.3 Manifolds

1.3.1 Basic Notions of Topology

• First we define the basic topological notions in the Euclidean space Rn.

• Let x0 ∈ Rn be a point in Rn and ε > 0 be a positive real number.

• The open ball in Rn of radius ε with the center at x0 is the set

Bε(x0) = x ∈ Rn | d(x, x0) < ε .

• The closed ball in Rn of radius ε with the center at x0 is the set

Bε(x0) = x ∈ Rn | d(x, x0) ≤ ε .

• Let U ⊆ Rn be a subset of Rn. A point x ∈ U is an interior point of Uif there is an open ball Bε(x) of some radius ε > 0 centered at x such thatBε(x) ⊂ U.

• A point x ∈ Rn is a boundary point of U if every open ball Bε(x) of anyradius ε > 0 centered at x contains at least one point from U and one pointfrom its complement (that is not from U).

• The set Uo of all interior points of U is called the interior of U.

• The set ∂U of all boundary points of U is called the boundary of U.

• A set U ⊆ Rn is open if every point of U is an interior point of U, that is,U = Uo.

• A set F ⊆ Rn is closed if its complement Rn\F is open, that is, F = Fo∪∂F.

• The sets ∅ and Rn are both open and closed.

• The union of any collection of open sets is open.

• The intersection of any finite number of open sets is open.


1.3. MANIFOLDS 11

•

Definition 1.3.1 A general topological space is a set M together witha collection of subsets of M, called open sets, that satisfy the followingproperties

1. M and ∅ are open,

2. the intersection of any finite number of open sets is open,

3. the union of any collection of open sets is open.

Such a collection of open sets is called a topology of M.

• A subset of M is closed if its complement M \ F is open.

• The closure S of a subset S ⊆∈ M of a topological space M is the intersec-tion of all closed sets that contain S ; it is equal to S = S ∪ ∂S .

• A subset S ⊆ M of a topological space is dense in M if S = M, that is,every non-empty subset of M contains an element of S .

• A topological space is called separable if it contains a countable densesubset.

• A topology on M naturally induces a topology on any subset of M. LetA ⊆ M be a subset of M. Then the induced topology on A is defined asfollows. A subset V ⊆ A is defined to be open subset of A if there is an opensubset U ⊆ M of M such that V = U ∩ A.

• Let x ∈ M be a point in a topological space M. An open set in M containingthe point x is called a neighborhood of x.

•

Definition 1.3.2 Let M and N be two topological spaces and F : M →N be a map from M into N. The map F is said to be continuous if theinverse image of any open set in N is an open set in M.

That is, if for any open set V ⊂ N the set

F−1(V) = x ∈ M | F(x) ∈ V

is open in M.

• The direct images of open sets do not have to be open!



• Let the map F be bijective, that is, injective (one-to-one) and surjective(onto). Then there exists the inverse map F−1 : N → M.

•Definition 1.3.3 A map F : M → N is called a homeomorphism if itis bijective and both F and F−1 are continuous.

• Homeomorphisms preserve topology, that is, they take open sets to opensets and closed sets to closed sets.

• A topological space M is called Hausdorff if any two points of M havedisjoint neighborhoods.

• A collection of subsets of a topological space M is called an open cover ofM if the union of all subsets in the collection coincides with M.

• A subcollection of subsets that is itself a cover is called a subcover.

•Definition 1.3.4 A topological space M is called compact if everyopen cover of M has a finite subcover.

• A subset M ∈ Rn of a Euclidean space if called bounded if there is an openball BC(0) of some radius C centered at the origin such that M ⊆ BC(0).

•

Theorem 1.3.1 Bolzano-Weierstrass Theorem. Let M ⊆ Rn be asubset of Rn with the induced topology. Then M is compact if and onlyif M is closed and bounded in Rn.

• Properties of Continuous Maps.

•

Theorem 1.3.2 Let M and N be two topological spaces and M becompact. Let F : M → N be a continuous map from M into N. Thenthe image F(M) of M is compact in N.

That is, a continuous image of a compact topological space is compact.

Proof :

1. Let Uαα∈A be an open cover of F(M) in N.

2. Then F−1(Uα)α∈A is an open cover of M.

3. Since M is compact it has a finite subcover F−1(Ui)ni=1.

4. Then Uini=1 is a finite subcover of F(M).


1.3. MANIFOLDS 13

5. Thus F(M) is compact.

•Theorem 1.3.3 A continuous real-valued function f : M → R on acompact topological space M is bounded.

Proof :

1. f (M) is compact in Rn.

2. Thus f (M) is closed and bounded.

1.3.2 Idea of a Manifold• A manifold M of dimension n is a topological space that is locally homeo-

morphic to Rn.

• A manifold M is covered by a family of local coordinate systems Uα; x1α, . . . , x

nαα∈A,

called an atlas, consisting of open sets, called patches (or charts), Uα, andcoordinates xα.

• A point p ∈ Uα ∩ Uβ that lies in two coordinate patches has two sets ofcoordinates xα and xβ related by smooth functions

xiα = f i

αβ(x1β, . . . , x

nβ) , i = 1, . . . , n .

• The coordinates xα and xβ are said to be compatible.

• If all the functions fαβ are smooth, then the manifold M is said to be smooth.If these functions are analytic, then the manifold is said to be real analytic.

• Each patch is homeomorphic to some open subset in Rn.

• Thus, a manifold is locally Euclidean.

• The collection of all patches (charts) is called an atlas.

• The collection of all coordinate systems that are compatible with those usedto define a manifold is called the maximal atlas.



• Let M be an n-dimensional manifold with local coordinate systems Uα; x1α, . . . , x

nαα∈A

and N be an m-dimensional manifold with local coordinate systems Vβ; y1β, . . . , y

mβ β∈B.

The product manifold L = M × N is a manifold

L = M × N = (p, q) | p ∈ M, q ∈ N

with local coordinate systems Wαβ; z1αβ, . . . , z

n+mαβ α∈A,β∈B where

Wαβ = Uα × Vβ

and(z1αβ, . . . , z

nαβ) = (x1

α, . . . , xnα)

and(zn+1αβ , . . . , z

n+mαβ ) = (y1

β, . . . , ymβ ) .

Examples

• Unit Sphere S n.S n = x ∈ Rn+1 | d(x, 0) = 1

Stereographic projection:

Φ : Rn+1 → Rn

Let

R =

√√1 −

n∑j=1

(x j)2

Then

xn+1 = ±R , Φ1,2(x) =

(x1

1 ± R, · · · ,

xn

1 ± R

).

• Torus T n = S 1 × · · · × S 1.

– T n has local coordinates (θ1, · · · , θn) (angles).

– Topologically it is the cube [0, 1]n with the antipodal points identified.

• Real Projective Space RPn.

– RPn is the space of unoriented lines through the origin of Rn+1.


1.3. MANIFOLDS 15

– The set of oriented lines through the origin of Rn+1 is S n.

– Topologically RPn is the sphere S n in Rn+1 with the antipodal pointsidentified, which is the unit ball in Rn with the antipodal points on theboundary (which is a unit sphere S n−1) identified.

– RPn is covered by (n + 1) sets

U j = L ∈ RPn | L with x j , 0 , j = 1, . . . , n + 1 .

– The local coordinates in U j are

v1 =x1

x j , · · · , vn =

xn

x j .

– The (n + 1)-tuple (x1, . . . , xn+1) identified with (λx1, . . . , λxn+1), λ , 0,are called homogeneous coordinates of a point in RPn.

1.3.3 Rigorous Definition of a Manifold• Let M be a set (without topology) and Uαα∈A be a collection of subsets

that is a cover of M, that is, ⋃α∈A

Uα = M .

• Letϕα : Uα → Rn, α ∈ A ,

be injective maps such that ϕα(Uα) are open subsets in Rn.

• The set ϕα(Uα ∩ Uβ) is an open subset in Rn. The maps

fαβ = ϕα ϕ−1β : ϕα(Uα ∩ Uβ)→ R

are called the transition functions (or the overlap functions). We assumethat the transition functions are smooth.

• The pair (Uα, ϕα) is called a coordinate patch (or chart).

• A point p ∈ Uα is assigned coordinates of the point ϕα(p) in Rn. Thus, ϕα iscalled a coordinate map.



• Now, we take the maximal atlas of such coordinate patches.

• The topology in M is defined as follows.

• Let W ⊂ M be a subset of M. A point p ∈ W in W is said to be an interiorpoint if there is a coordinate chart (Uα, ϕα) including p such that U ⊂ W.

• A subset W of M is declared to be open if all of its points are interior points.

• If the resulting topological space is Hausdorff and separable, then M is saidto be an n-dimensional smooth manifold.

• The regularity of the transition functions determines the regularity of themanifold. If the transition functions are only differentiable once, then themanifold is called differentiable. If they are of class Ck, then the manifoldis called a manifold of class Ck. If they are of class C∞, then the manifoldis called smooth. If the transition functions are analytic, then the manifoldis called analytic.

• Let F : M → R be a real-valued function on M. Let (Uα, xα) be a localcoordinate system. Then the function

Fα = F ϕ−1α : ϕα(Uα)→ R

is a function of n real variables Fα(x1α, . . . , x

nα).

• The function F is said to be smooth if the function Fα is smooth in termsof local coordinates xα.

• The process of replacing the map F by the function Fα = F ϕ−1α is usually

omitted, and the functions F and Fα are identified, so that we think of thefunction F directly in terms of local coordinates.

1.3.4 Complex Manifolds• Let M be a set and Uαα∈A be its cover. Let

ϕα : Uα → Cn

be injective maps from Uα to the complex n-space Cn so that ϕα(Uα) areopen subsets of Cn.


1.3. MANIFOLDS 17

• Let the transition functions

fαβ = ϕα ϕ−1β : ϕα(Uα ∩ Uβ)→ Cn ,

defined byzkα = f k

αβ(z1β, . . . , z

nβ) ,

where zkα = xk

α + iykα and zk

β = xkβ + iyk

β, k = 1, . . . , n, be complex analytic.That is, they satisfy Cauchy-Riemann conditions

∂xkα

∂x jβ

=∂yk

α

∂y jβ

,∂xk

α

∂y jβ

= −∂yk

α

∂x jβ

,

or∂zk

α

∂z jβ

= 0 ,

with j, k = 1, . . . , n.

• Then M is called a n-dimensional complex manifold.

• The topological dimension of a n-dimensional complex manifold is 2n.



1.4 Tangent Vectors and Mappings• A tangent vector to a submanifold M of Rn at a point p0 ∈ M is the velocity

vector p at p0 to some parametrized curve p = p(t) lying on M and passingthrough the point p0.

• Whitney theorem: Every n-dimensional manifold can be realized as a sub-manifold of R2n, or as a smooth submanifold of R2n+1.

• So, every manifold is a submanifold of a Euclidean space.

• However, the intrinsic geometry of M does not depend on its embedding ina Euclidean space.

1.4.1 Tangent Vectors

• We fix a point p0 ∈ M on manifold M and consider a curve p : (−ε, ε)→ Msuch that

p(0) = p0 .

• Let (Uα, xjα) be a local chart about p0. The curve is described in local coor-

dinates byxiα = xi

α(t) .

• The velocity vector p(0) is described in (Uα, xα) by an n-tuple(x1α(0), . . . , xn

α(0)).

• Let (Uβ, xjβ) be another local chart containing p0. Then the velocity vector

p(0) is described in (Uβ, xβ) by an n-tuple(x1β(0), . . . , xn

β(0)).

• These n-tuples are related by the chain rule

xiβ(0) =

n∑j=1

∂xiβ

∂x jα

(p0)x jα(0) .


1.4. TANGENT VECTORS AND MAPPINGS 19

•

Definition 1.4.1 A tangent vector at a point p0 ∈ M of a manifoldM is a map that assigns to each coordinate chart (Uα, xα) about p0 anordered n-tuple (X1

α, . . . , Xnα) such that

Xiβ =

n∑j=1

∂xiβ

∂x jα

(p0)X jα .

1.4.2 Vectors as Differential Operators• Let f : M → R be a real-valued function on M.

• Let p ∈ M be a point on M and X be a tangent vector at p.

• Let (Uα, xα) be a coordinate chart about p.

• The (directional) derivative of f with respect to X at p (or along X, or inthe direction of X) is defined by

Xp( f ) = DX( f ) =

n∑j=1

(∂ f

∂x jα

)(p)X j

α .

• Theorem 1.4.1 DX( f ) does not depend on the local coordinate system.

Proof :

1. Chain rule.

• The intrinsic properties are invariant under a change of coordinate system.They should not depend on the choice of the local chart.

• There is a one-to-one correspondence between tangent vectors to M at p andfirst-order differential operators acting on real-valued functions in a localcoordinate chart (Uα, xα) by

Xp =

n∑j=1

X jα

∂

∂x jα

∣∣∣∣∣∣p

.

• Therefore, we can identify tangent vectors and the differential operators.



• Let us fix an index j = 1, . . . , n. The curve

xi(t) = xi0 , i , j, x j(t) = t ,

is called the j-th coordinate curve.

• The velocity vector to this curve is given by

xi(0) = δij ,

where δij is the Kronecker symbol defined by

δij =

1, if i = j ,0, if i , j .

• The differential operator corresponding to this velocity vector is

∂

∂x jα

.

1.4.3 Tangent Space• Let M be a manifold and p ∈ M be a point in M. The tangent space TpM

to M at p is the real vector space of all tangent vectors to M at p.

• Let (U, x) be a local chart about p. Then the vectors

∂

∂x1

∣∣∣∣∣∣p

, · · · ,∂

∂xn

∣∣∣∣∣∣p

form a basis in the tangent space called the coordinate basis, or the coor-dinate frame.

• A vector field X on an open set U ⊂ M in a manifold M is the differentiableassignment of a tangent vector Xp to each point p ∈ U.

• In local coordinates

X =

n∑j=1

X j(x)∂

∂x j .

• Example.



1.4.4 Mappings

• Let M be an n-dimensional manifold and N be an m-dimensional manifoldand let F : M → N be a map from M to N. Let (Uα, ϕα)α∈A be an atlas in Mand (Vβ, ψβ)β∈B be an atlas in N. We define the maps

Fαβ = ψβ F ϕ−1α : ϕα(Uα)→ ψβ(Vβ)

from open sets in Rn to Rm defined by

yaβ = Fa

αβ(x1α, . . . , x

nα) ,

where a = 1, . . . ,m.

• The map F is said to be smooth if Faαβ are smooth functions of local coor-

dinates xiα, i = 1, . . . , n.

• The process of replacing the map F by the functions Fαβ = ψβ F ϕ−1α is

usually omitted, and the maps F and Fαβ are identified, so that we think ofthe map F directly in terms of local coordinates.

• If n = m and the map F : M → N is bijective and both F and F−1 aredifferentiable, then F is called a diffeomorphism.

• That is, a diffeomorphism is a differentiable homeomorphism with differen-tiable inverse.

• If this is only true in a neighborhood of a point p ∈ M, then F is called alocal diffeomorphism.

• Example.



•

Definition 1.4.2 Let M and N be two manifolds and F : M → N bea map from M into N. Let p0 ∈ M be a point in M and X ∈ Tp0 M bea tangent vector to M at p0. Let p = p(t), t ∈ (−ε, ε), be a curve in Msuch that

p(0) = p0, p(0) = X .

Then the differential of F is the map

F∗ : Tp0 M → TF(p0)N

defined by

F∗X =ddt

F(p(t))

∣∣∣∣∣∣t=0

.

• F∗ does not depend on the curve p(t).

• The matrix of the linear transformation F∗ in therms of the coordinate bases∂/∂yα and ∂/∂xi is the Jacobian matrix

(F∗)αi =∂yα

∂xi ,

that is,

(F∗X)αi =

n∑i=1

∂yα

∂xi Xi .

1.4.5 Submanifolds

•

Definition 1.4.3 Let M be an n-dimensional manifold and W ⊂ M bea subset of M. Then W is an r-dimensional embedded submanifold ofM if W is locally described as the common locus of (n−r) differentiableindependent functions

Fα(x1, . . . , xn) = 0 , α = 1, . . . , n − r ,

such that the Jacobian matrix has rank (n− r) at each point of the locus,that is,

rank(∂Fα

∂xi (x))

= n − r , ∀x ∈ W.

• Every embedded submanifold of a manifold is itself a manifold.



•

Theorem 1.4.2 Let n and r be two positive integers such that n > r.Let M be an n-dimensional manifold and N be an r-dimensional man-ifold. Let q ∈ N be a point in N such that the inverse image W =

F−1(q) , ∅ is nonempty. Suppose that for each point p ∈ W the differ-ential F∗ of the map F is surjective, that is, has the maximal rank

rank F∗(p) = r .

Then W is an (n − r) dimensional submanifold of M.

• Example. Morse Map. (Height function F : M → R for trousers surfaceM in R3).

• Let p0 ∈ M and v ∈ Tp0 M be a tangent vector to M at p0. Then F∗ : Tp0 M →R = TF(p0)M is the projection of v to z-axis defined by F∗(v1, v2, v3) = v3.

• Let p0 ∈ M and z = F(p0). F−1(z) is an embedded submanifold if F∗ isonto, that is, Tp0 M is not horizontal. If Tp0 M is horizontal, then F∗ = 0(hence, not onto).

•

Definition 1.4.4 Let M and N be two manifolds and F : M → N be adifferentiable map from M into N.

A point p ∈ M is a regular point if the differential F∗ : TpM → TF(p)Nis surjective.

A point p ∈ M is a critical point if it is not regular.

A point q ∈ N is a regular value of F if the inverse image F−1(q) iseither empty or consists only of regular points.

A point q ∈ N is a critical value of F if it is not regular.

• Thus, we can reformulate the main theorem as follows.

Theorem 1.4.3 Let n and r be two positive integers such that n >r. Let M be an n-dimensional manifold and N be an r-dimensionalmanifold. Let q ∈ N be a regular value of F. Then the inverse imageW = F−1(q) , ∅ is either empty or is an (n−r) dimensional submanifoldof M.



•

Theorem 1.4.4 Sard’s Theorem. Let M and N be two manifolds andF : M → N be a smooth map from M into N. Then the set of criticalvalues of F is a set of measure zero in N, that is, almost all values of Fare regular values.

• The critical values of a map F : M → N cannot fill up any open set in N.

1.4.6 Change of Coordinates

•

Theorem 1.4.5 Inverse Function Theorem. Let M and N be two n-dimensional manifolds and F : M → N be a differentiable map from Minto N. Let p0 ∈ M be such that the differential F∗ : Tp0 M → TF(p0)N isbijective. Then F is a local diffeomorphism in a neighborhood U of p0,that is, F(U) is open in N and F : U → F(U) is a diffeomorphism.

•

Theorem 1.4.6 Let M be a manifold, p0 ∈ M be a point in M, and(U, x) be a local coordinate chart about p0. Let F : U → Rn be adifferentiable map defined by yi = F i(x), i = 1, . . . , n, such that

det(∂F i

∂x j

)(p0) , 0 .

Then there is a neighborhood V of p0 such that the n-tuple (y1, . . . , yn)forms a compatible coordinate system in V.

• Example.


1.5. VECTOR FIELDS AND FLOWS 25

1.5 Vector Fields and Flows

1.5.1 Vector Fields in Rn

• Let x = (x j) ∈ Rn be a point in Rn and v(x) ∈ TxRn be a vector at x given by

v =

n∑j=1

v j(x)∂ j ,

where ∂ j = ∂/∂x j and the components v j(x) are smooth (or just differen-tiable) functions of x. Then v(x) is called a vector field in Rn.

• Let t ∈ (−ε, ε) andϕt : Rn → Rn

be a family of diffeomorphisms such that

ϕ0 = Id

and for any t, t1, t2 ∈ (−ε, ε) such that −t, t1 + t2 ∈ (−ε, ε) there holds

ϕt1 ϕt2 = ϕt1+t2

andϕ−t = ϕ−1

t .

Such a one-parameter group of diffeomorphisms is called a flow on Rn.

• A flow ϕt defines a vector field v by

v(x) =ddtϕt(x)

∣∣∣∣t=0

with the components

v j(x) =dx j

dt.

• The corresponding differential operator

v( f )(x) =ddt

f (ϕt(x))∣∣∣∣t=0

=

n∑j=1

v j(x)∂ f∂x j

is the derivative along the streamline of the flow through the point p.



• Conversely, every vector field v determines a flow, which is determined byrequiring v to be the velocity field of the flow.

• Such a flow is defined as the solution of the system of ordinary first-orderdifferential equations

dx j

dt= v j(x1(t), . . . , xn(t)) , j = 1, . . . , n,

with the initial conditionsx j(0) = x j

0 .

The solution of this system defines the integral curves of the vector field v.

•

Theorem 1.5.1 Fundamental Theorem on Vector Fields in Rn Let ⊂ Rn

be an open set in Rn and v : U → Rn be a smooth vector field on U.Then for any point x0 ∈ U there is ε > 0 and a neighborhood V of x0

such that:

1. there is a unique curve (called the integral curve of v) x(t) =

φt(x0), t ∈ (−ε, ε) such that for any t ∈ (−ε, ε)

x(t) ≡dx(t)

dt= v(x(t))

and x(0) = x0;

2. the mapV × (−ε, ε)→ Rn

defined by (x, t) 7→ ϕt(x) is smooth and for any t1, t2 ∈ (−ε, ε)such that t1 + t2 ∈ (−ε, ε)

ϕt1 ϕt2 = ϕt1+t2

holds in V. The family of maps ϕt is called a local one-parametergroup of diffeomorphisms or a local flow.

• Remarks.

• The local flow is only defined in a small neighborhood of the point x0.

• The one-parameter group is not a group in the strict sense.


1.5. VECTOR FIELDS AND FLOWS 27

• The integral curves exist only for a small time.

• If the vector field is not differentiable, then the integral curve is not unique.

1.5.2 Vector Fields on Manifolds• Let W be an open subset of a mnifold M and v be a smooth vector field on

W.

• Let (Uα, xα) be a local chart in W.

• If W ⊂ Uα, then one can proceed as in Rn.

• If W is not contained in a single chart, then we choose a cover of W andproceed as follows. Let p ∈ W and (Uα, xα) and (Uβ, xβ) be two chartscovering p.

• Then the integral curves in both local coordinate systems have the samemeaning and define a unique integral curve in M. This defines a local flowon W in M. We just need to check that if the flow equations are satisfied inone coordinate system, then they are satisfied in another coordinate system.

1.5.3 Straightening Flows• Let U be an open set in a manifold M and ϕt : U → M be a local flow on

a M such that ϕ0(p) = p. Then ϕt(p) depends smoothly on both the time tand the point p.

• Note that if a vector field does not vanish at a point p, i.e. vp , 0, then itdoes not vanish in a neighborhood of p.

• Let p0 be a fixed point in M and W be a sufficiently small hypersurfacepassing through p0 such that vp is transversal to W at every point of p ∈ W.This just means that v is nowhere tangent to W.

• If W is small enough it can be covered by a single coordinate chart. Let(u1, . . . , un−1) be local coordinates for W such that the the point p0 has coor-dinates (0, . . . , 0).

• Then for a small neighborhood of a point p and sufficiently small t the n-tuple (u1, . . . , un−1, t) gives a local coordinate system near p in M.



• Proof: (follows from Inverse Function Theorem).

• Consider a map F : W × (−ε, ε)→ M given by F(p, t) = ϕt(p).

• The differential of this map at p0 (that is, at u = 0), is

F∗∂

∂u j =∂

∂u j , j = 1, . . . , n − 1 .

Also,

F∗v =ddtϕt(p0) = v .

ThusF∗ = Id ,

and, hence, (u1, . . . , un−1, t) can be used as a coordinate system near p0.

• In these coordinates the flow is

ϕs(u, t) = (u, t + s)

andv =

∂

∂t.

• Thus, near a non-singular point of a vector field v one can introduce localcoordinates (u1, . . . , un−1, un) such that

du j

dt= δ j

n

that is,

du j

dt= 0, if j = 1, . . . , n − 1, and

dun

dt= 1 .

• Near a non-singular point of a vector field all flows are qualitatively thesame.

• Let v be a vector field on a manifold M. A point p ∈ M is called a singularpoint of a vector field v if vp = 0 and a non-singular if vp , 0.


Chapter 2

Tensors

2.1 Covectors and Riemannian Metric

2.1.1 Linear Functionals and Dual Space

• Let E be a real n-dimensional vector space.

• Let B = e1, . . . , en be a basis in E.

• Then for any v ∈ E

v =

n∑j=1

v je j

• The real numbers (v1, . . . , vn) are called the components of v with respectto the basis B.

• Remark. Every real vector space of dimension n is isomorphic to Rn.

•

Definition 2.1.1 Let E be a real vector space. A linear functional onE is a linear real-valued function on E. That is, it is a map

α : E → R

satisfying the linearity conditions: ∀a, b, ∈ R and ∀v,w ∈ E

α(av + bw) = aα(v) + bα(w) .

29

30 CHAPTER 2. TENSORS

• Given a basis e j, j = 1, . . . , n, we define

a j = α(e j) .

Then for any v ∈ E we have

α(v) =

n∑j=1

a jv j .

•Definition 2.1.2 Let E be a real vector space. The set of all linearfunctionals on E is called the dual space and denoted by E∗.

• The dual space is a real vector space under the natural operations of additionand multiplication by scalar defined by: ∀α, β ∈ E∗, c1, c2 ∈ R, v ∈ E,

(c1α + c2β)(v) = c1α(v) + c2β(v) .

• Let e j be a basis in E and σ j, j = 1, . . . , n, be linear functionals on Esuch that

σ j(ei) = δji .

• Thenσ j(v) = v j .

•

Theorem 2.1.1 Let E be a real vector space and e j, j = 1, . . . , n, bea basis in E. Let and σ j, j = 1, . . . , n, be linear functionals on E suchthat

σ j(ei) = δji .

Then the linear functionals σ j for a basis in the dual space E∗, calledthe dual basis to the basis e j, so that for any α ∈ E∗

α =

n∑j=1

a jσj .

The real numbersa j = α(e j)

are called the components of the linear functional α with respect to thebasis σ j.

Proof :


2.1. COVECTORS AND RIEMANNIAN METRIC 31

1.

2.1.2 Differential of a Function

•

Definition 2.1.3 Let M be a manifold and p ∈ M be a point in M. Thespace T ∗pM dual to the tangent space TpM at p is called the cotangentspace.

•

Definition 2.1.4 Let M be a manifold and f : M → R be a real valuedsmooth function on M. Let p ∈ M be a point in M. The differentiald f ∈ T ∗pM of f at p is the linear functional

d f : Tp → R

defined byd f (v) = vp( f ) .

• In local coordinates x j the differential is defined by

d f (v) =

n∑j=1

v j(x)∂ f∂x j

In particular,

d f(∂

∂x j

)=∂ f∂x j .

Thus

dxi

(∂

∂x j

)= δi

j .

anddxi(v) = vi .

• The differentials dxi form a basis for the cotangent space T ∗pM called thecoordinate basis.

• Therefore, every linear functional has the form

α =

n∑j=1

a jdx j .



• That is why, the linear functionals are also called differential forms, or1-forms, or covectors, or covariant vectors.

•

Definition 2.1.5 A covector field α is a differentiable assignment ofa covector αp ∈ T ∗pM to each point p of a manifold. This means thatthe components of the covector field are differentiable functions of localcoordinates.

• Therefore, a covector field has the form

α =

n∑j=1

a j(x)dx j

• Under a change of local coordinates x jα = x j

α(xβ) the differentials transformaccording to

dx jα =

n∑i=1

∂x jα

∂xiβ

dxiβ .

• Therefore, the components of a covector transform as

aαi =

n∑j=1

∂x jα

∂xiβ

aβj .

2.1.3 Inner Product• Let E be a n-dimensional real vector space.

• The inner product (or scalar product) on E is a symmetric bilinear non-degenerate functional on E × E, that is, it is a map 〈 , 〉 : E × E → R suchthat

1. ∀v,w ∈ E,〈v,w〉 = 〈w, v〉

2. ∀v,w,u ∈ E, ∀a, b ∈ R

〈av + bu,w〉 = a〈v,w〉 + b〈u,w〉

3. ∀v ∈ E,

if 〈v,w〉 = 0, ∀w ∈ E, then v = 0 .



4. If, in addition, ∀v ∈ E,〈v, v〉 ≥ 0

and〈v, v〉 = 0 if and only if v = 0 ,

then the inner product is called positive definite.

• For a positive-definite inner product the norm of a vector v is defined by

||v|| =√〈v, v〉

• Let e j be a basis in E.

• Then the matrix gi j defined by

gi j = 〈ei, e j〉

is a metric tensor, more precisely it gives the components of the metrictensor in that basis.

• The matrix gi j is symmetric and nondegenerate, that is,

gi j = g ji, det gi j , 0 .

For a positive definite inner product, this matrix is positive-definite, thatis, it has only positive real eigenvalues. One says, that the metric has thesignature (+ · · ·+). In special relativity one considers metrics wich are notpositive definite but have the signature (− + · · ·+).

• Two vectors v,w ∈ E are orthogonal if

〈v,w〉 = 0 .

• A vector u ∈ E is called unit vector if

||u|| = 1 .

• The basis is called orthonormal if

gi j = δi j .



• The inner product is given then by

〈v,w〉 =

n∑i, j=1

gi jviw j .

• Let v ∈ E. Then we can define a linear functional ν ∈ E∗ by

ν(w) = 〈v,w〉 .

• Therefore, each vector v ∈ E defines a covector ν ∈ E∗ called the covariantversion of the vector v.

• Given a basis e j in E and the dual basis σ j in E∗ we find

νi =

n∑j=1

gi jv j .

• This operation is called lowering the index of a vector.

• Therefore, we can denote the components of the covector ν correspondingto a vector v by the same symbol and call them the covariant componentsof the vector.

• In an orthonormal basis, of course,

vi = vi .

• Similarly, given a covector ν ∈ E∗ we can define a vector v ∈ E such that∀w ∈ E∗

ν(w) = 〈v,w〉 .

• Let gi j represent the matrix inverse to the matrix gi j.

• Then the contravariant components can be computed from the covariantcomponents by

vi =

n∑j=1

gi jv j .

This operation is called raising the index of a covector.



• Thus, the vector spaces E and E∗ are isomorphic. The isomorphism is pro-vided by the inner product (the metric).

• The space E can also be considered as the space of linear functionals on E∗.A vector v ∈ E defines a linear functional v : E∗ → R by, for any α ∈ E∗

v(α) = α(v).

2.1.4 Riemannian Manifolds

•

Definition 2.1.6 Let M be a manifold. A Riemannian metric on M isa differentiable assignment of a positive definite inner product in eachtangent space TpM to the manifold at each point p ∈ M.

If the inner product is non-degenerate but not positive definite, then it isa pseudo-Riemannian metric.

A Riemannian manifold is a pair (M, g) of a manifold with a Rieman-nian metric on it.

• Let p ∈ M be a point in M and (U, xα) be a local coordinate system about p.Let ∂i be the coordinate basis in TpM and gαi j = 〈∂αi , ∂

αj 〉 be the components

of the metric tensor in the coordinate system xα. Let (Uβ, xβ) be anothercoordinate system containing p. Then the components of the metric tensortransform as

gαi j =

n∑k,l=1

∂xkβ

xiα

xlβ

x jα

gβkl .

•

Definition 2.1.7 Let (M, g) be a Riemannian manifold and f : M → Rbe a smooth function. Then the gradient of f is a vector field grad fassociated with the covector field d f . That is, for any p ∈ M and anyv ∈ TpM

〈 grad f , v〉 = d f (v) .

• In local coordinates,

( grad f )i =

n∑j=1

gi j ∂ f∂x j .



2.1.5 Curves of Steepest Ascent• Let (M, g) be a Riemannian manifold.

• Let p ∈ M be a point in M. For positive-definite inner product there is theSchwartz inequality: for any v,w ∈ TpM,

|〈v,w〉| ≤ ||v|| ||w|| .

• Let u ∈ TpM be a unit vector. Then for any f ∈ C∞(M)

u( f ) = 〈 grad f ,u〉 .

• Therefore,|u( f )| = |〈 grad f ,u〉| ≤ || grad f || .

• Thus, f has a maximum rate of change in the direction of the gradient.

•

Definition 2.1.8 Let a ∈ R be a real number. Then the level set off : M → R is the subset of M defined by

S = f −1(a) = p ∈ M | f (p) = a .

• Let f : M → R be a smooth function, a ∈ R and S = f −1(a) be the level setof f . Let p0 ∈ S be a point in the level set S and let p = p(t), t ∈ (−ε, ε), bea curve in S so that p(0) = p0, f (p(t)) = a and p(0) ∈ Tp0S . Then

ddt

f (p(t)) = 〈 grad f , p〉 = d f ( p) = 0 .

• Thus, the gradient of f is orthogonal to the level set of f .

• The flowdpdt

=grad f|| grad f ||

defined by the gradient of a smooth function f is called the Morse defor-mation. It has the property that

d fdt

= 1

and, therefore, in time t it maps the level set f −1(a) into the level set f −1(a +

t).


2.2. TANGENT BUNDLE 37

2.2 Tangent Bundle

2.2.1 Fiber Bundles• Let M and E be smooth manifolds and π : E → M be a smooth map. Then

the triple (E, π,M) is called a bundle.

• The manifold M is called the base manifold of the bundle and the manifoldE is called the bundle space of the bundle (or the bundle space manifold).The map π is called the projection.

• The inverse image π−1(p) of a point p ∈ M is called the fiber over p.

• The projection map is supposed to be surjective, that is, the differential π∗has the maximal rank equal to dim M.

• Let Uαα∈A be an atlas of local charts covering the base manifold M and letUαβ = Uα ∩ Uβ, Uαβγ = Uα ∩ Uβ ∩ Uγ etc.

• A fiber bundle is a bundle all fibers of which, π−1(p), ∀p ∈ M, are diffeo-morphic to a common manifold F called the typical fiber of the bundle (orjust the fiber).

• For a fiber bundle, the inverse images π−1(Uα) are diffeomorphic to Uα × F.That is, there are diffeomorhisms

hα : Uα × F → π−1(Uα) ,

such that for any p ∈ Uα ⊂ M, σ ∈ F

π(hα(p, σ)) = p .

The diffeomorphisms

ϕαβ = h−1β hα : Uαβ × F → Uαβ × F

are called the transition functions of the bundle.

• The transition functions are defined by, ∀p ∈ Uαβ ⊂ M, σ ∈ F,

ϕαβ(p, σ) = (p, (Rαβ(p))(σ)) .

That is, for all p ∈ Uαβ there are diffeomorphisms Rαβ(p) of the fiber

Rαβ(p) : F → F .



• It is required that the set of all transformations Rαβ(p) ∈ G for all α, β andp ∈ Uαβ ⊂ M forms a group G. This group is called the structure group ofthe bundle.

• Thus, the transition functions ϕαβ determine smooth maps

Rαβ : Uαβ → G ,

that assign to each point p ∈ Uαβ an element Rαβ(p) ∈ G of the structuregroup.

• Of course, from the definition of these maps we immediately obtain theconsistency conditions (or compatibility conditions)

Rαβ(p) = (Rβα(p))−1 , ∀p ∈ Uαβ ,

Rαβ(p)Rβγ(p)Rγα(p) = IdM , ∀p ∈ Uαβγ .

• A principal bundle is a fiber bundle (E, π,M) whose fiber F coincides withthe structure group, that is, F = G.

• A fiber bundle with any fiber F is fully determined by the transition func-tions satisfying the consistency conditions. The fiber F does not play muchrole in this construction.

• Let F be a manifold and Diff(F) be the set of all diffeomorphisms F → F.Let G be a group and e ∈ G be the identity element of G. Then a mapT : G → Diff(F) such that

T (e) = IdF ,

T (R−1) = (T (R))−1 , ∀R ∈ G ,

T (R1R2) = T (R1) T (R2) , ∀R1,R2 ∈ G ,

is called a representation of the group G.

• Given a fiber bundle (E, π,M) with a fiber F and a structure group G onecan construct another fiber bundle (E′, π′,M) with a fiber F′ and the samestructure group G as follows. One takes a representation of the structuregroup T : G → Diff(F′) on the fiber F′ and simply replaces the transitionfunctions Rαβ by T (Rαβ). Such a fiber bundle is called a bundle associatedwith the original bundle.



• Thus, every fiber bundle is an associated bundle with some principal bun-dle. So, all bundles can be constructed as associated bundles from principalbundles. All we need is the structure group. The fiber is not important.

• A vector bundle is a fiber bundle whose fiber is a vector space.

• A section of a bundle (E, π,M) is a map s : M → E such that the image ofeach point p ∈ M is in the fiber π−1(p) over this point, that is, s(p) ∈ π−1(p),or

π s = IdM .

2.2.2 Tangent Bundle

•

Definition 2.2.1 Let M be a smooth manifold. The tangent bundleT M to M is the collection of all tangent vectors at all points of M.

T M = (p, v) | p ∈ M, v ∈ TpM

• Let dim M = n.

• Let p ∈ M be a point in the manifold M, (U, x) be a local chart and (xi) bethe local coordinates of the point p.

• Let ∂i = ∂/∂xi be the coordinate basis for TpM.

• Let v =∑n

i=1 vi∂i ∈ TpM. Then the local coordinates of the point (p, v) ∈T M are

(x1, . . . , xn, v1, . . . , vn) .

• Remarks.

• The coordinates (xi) are local; they are restricted to the local chart U, thatis, (xi) ∈ U ⊂ Rn.

• The coordinates vi are not restricted, that is, (vi) ∈ Rn, they take any valuesin Rn.

• The open set U × Rn ⊂ R2n is a local chart in the tangent bundle T M.

• Let (Uα, xα) and (Uα, xα) be two local charts containing the point p.



• Then the local coordinates of the point (p, v) in overlapping local charts arerelated by

xiα = xi

α(xβ)

viα =

n∑j=1

∂xiα

∂x jβ

v jβ

• This is a local diffeomorphism.

• Thus, the tangent bundle T M is a manifold of dimension 2 × dim M.

• A map π : T M → M defined by

π(p, v) = p

is called the projection map. It assigns to a vector tangent to M the pointin M at which the vector sits.

• Locally, if p has coordinates (x1, . . . , xn) and v has components (v1, . . . , vn)in the coordinate basis, then

π(x1, . . . , xn, v1, . . . , vn) = (x1, . . . , xn) .

• Let p ∈ M be a point in M. The set π−1(p) ⊂ T M is called the fiber of thetangent bundle.

• The fiber of the tangent bundle

π−1(p) = TpM

is the tangent space at p.

• Remarks.

• There is no global projection map π′ : T M → Rn defined by π′(p, v) = v.

• In general, T M , M × Rn.

• For any chart U ∈ Mπ−1(U) = U × Rn .

Thus, locally the tangent bundle is a product manifold.



• A vector field is a mapv : M → T M ,

such thatπ v = Id : M → M .

• A vector field is a cross section of the tangent bundle.

• The image of the manifold under a vector field is a n-dimensional subman-ifold of the tangent bundle T M.

• The zero vector field defines the zero section of the tangent bundle.

•

Definition 2.2.2 Let (M, g) be an n-dimensional Riemannian mani-fold. The unite tangent bundle of M is the set T0M of all unit vectorsto M,

T0M = (p, v) | p ∈ M, v ∈ TpM, ||v|| = 1 ,

where, locally, ||v||2 =∑n

i, j=1 gi j(p)viv j.

• The unit tangent bundle is a (2n−1)-dimensional submanifold of the tangentbundle T M.

•

Theorem 2.2.1 Let S 2 be the unit 2-sphere embedded in R3. The unittangent bundle T0S 2 is homeomorphic to the real projective space RP3

and to the special orthogonal group S O(3)

T0S 2 ∼ RP3 ∼ S O(3) .

Proof :

1.



2.3 The Cotangent Bundle

•

Definition 2.3.1 Let M be a smooth manifold. The cotangent bundleT ∗M to M is the collection of all covectors at all points of M

T ∗M = (p, σ) | p ∈ M, σ ∈ T ∗pM

• Let dim M = n.

• Let p ∈ M be a point in the manifold M, (U, x) be a local chart and (xi) bethe local coordinates of the point p.

• Let dxi be the coordinate basis for T ∗pM.

• Let α =∑n

i=1 αidxi ∈ T ∗pM. Then the local coordinates of the point (p, α) ∈T ∗M are

(x1, . . . , xn, α1, . . . , αn) .

• Remarks.

• The open set U × Rn ⊂ R2n is a local chart in the cotangent bundle T ∗M.

• Let (Uα, xα) and (Uα, xα) be two local charts containing the point p.

• Then the local coordinates of the point (p, σ) in overlapping local charts arerelated by

xiα = xi

α(xβ)

σαi =

n∑j=1

∂x jβ

∂xiα

σβj .

• This is a local diffeomorphism.

• Thus, the cotangent bundle T ∗M is a manifold of dimension 2n.

• The projection map π : T ∗M → M is defined by π(p, α) = p.

• A covector field (or a 1-form)is a map

α : M → T ∗M ,

such thatπ α = Id : M → M .

• A covector field is a section of the cotangent bundle.


2.3. THE COTANGENT BUNDLE 43

2.3.1 Pull-Back of a Covector

• Let M and N be two smooth manifolds and n = dim M and m = dim N.

• Let ϕ : M → N be a smooth map.

• The differentialϕ∗ : TpM → Tϕ(p)N

is the linear transformation of the tangent spaces.

• Let xi be a local coordinate system in a local chart about p ∈ M and yα be alocal coordinate system in a local chart about ϕ(p) ∈ N and ∂i and ∂α be thecoordinate bases for TpM and Tϕ(p)N.

• Then the action of the differential ϕ∗ is defined by

ϕ∗

(∂

∂x j

)=

m∑α=1

∂yα

∂x j

∂

∂yα.

• Let v =∑n

i=1 vi∂i. Then

[ϕ∗(v)]α =

n∑j=1

∂yα

∂x j v j .

•

Definition 2.3.2 The pullback ϕ∗ is the linear transformation of thecotangent spaces

ϕ∗ : T ∗ϕ(p)N → T ∗pM

taking covectors at ϕ(p) ∈ N to covectors at p ∈ M, defined as follows.If α ∈ T ∗ϕ(p)N, then ϕ∗(α) ∈ T ∗pM so that

ϕ∗(α) = α ϕ∗ : TpM → R

where α : Tϕ(p)N → R. That is, for any vector v ∈ TpM

(ϕ∗(α)) (v) = α(ϕ∗(v)) .

• Diagram.



• In local coordinates,

[ϕ∗(dyα)] j =

m∑α=1

∂yα

∂x j dx j .

• Let σ =∑mα=1 σαdyα. Then

ϕ∗(σ) =

n∑j=1

m∑α=1

σα

∂yα

∂x j dx j ,

that is, in components,

[ϕ∗(σ)] j =

m∑α=1

σα

∂yα

∂x j .

• Remark.

• In general, for a map ϕ : M → N, the following linear transformationsare well defined: the differential ϕ∗ : TpM → Tϕ(p)N and the pullbackϕ∗ : T ∗ϕ(p)N → T ∗pM.

• The maps T ∗pM → T ∗ϕ(p)N and Tϕ(p)N → TpM are not well defined, ingeneral.

• If dim M = dim N and ϕ : M → N is a diffeomorphism, then all these mapsare well defined.

• Explain.

2.3.2 Phase Space• Let M be a configuration space of a dynamical system with local general-

ized coordinates q1, . . . , qn. Then qi =dqi

dt are the generalized velocities.

• Under a change of local coordinates

qiα = qi

α(qβ)

the velocities transform as components of a vector

q jα =

n∑i=1

∂q jα

∂qiβ

qiβ .



• Therefore, (q1, . . . , qn, q1, . . . , qn) give local coordinates for the tangent bun-dle T M.

• Let L : T M → R be a map. Then L(q, q) is called the Lagrangian.

• The generalized momenta pi are defined by

pi =∂L∂qi .

• The momenta are functions on T M, that is, p : T M → R.

• Under a change of local coordinates qα = qα(qβ) the momenta transform ascomponents of a covector

pαj =

n∑i=1

∂qiβ

∂q jα

pβi ,

• The matrix

Hik(q, q) =∂2L∂qi∂qk

is called the Hessian.

• Suppose that the Hessian is non-degerate

detHik , 0 .

• Then the velocities can be expressed in terms of momenta

qi = qi(q, p) ,

that is, there is a map q : T ∗M → R. More generally, there is a mapT M → T ∗M.

• Thus, (q1, . . . , qn, p1, . . . , pn) give local coordinates for the cotangent bundleT ∗M.

• The cotangent bundle is called the phase space in dynamics.



• The Hamiltonian is a smooth function on the cotangent bundle H : T ∗M →R defined by

H(q, p) =

n∑i=1

∂L∂qi q

i − L(q, q) .

• Example.

• One of the most important examples is the Lagrangian quadratic in veloci-ties

L(q, q) =12

n∑i, j=1

gi j(q)qiq j − V(q) ,

where gi j is a Riemannian metric on M and V is a smooth function on M.

• Then the Hessian is

gik =∂2L∂qi∂qk

and, therefore, nondegenerate.

• The relation between momenta and the velocities is

pi =

n∑j=1

gi j(q)q j , qi =

n∑j=1

gi j(q)p j .

• The Hamiltonian is given by

H(q, p) =12

n∑i, j=1

gi j(q)pi p j + V(q) ,

2.3.3 The Poincare 1-Form• The Poincare 1-form λ is a 1-form on the cotangent bundle T ∗M defined

in local coordinates (q, p) on T ∗M by

λ =

n∑i=1

pidqi .

• Remarks.



• The coordinates p are not functions on M.

• The Poincare form is not a 1-form on M.

• A general 1-form on T ∗M is

α =

n∑i=1

αi(q, p)dqi +

n∑i=1

vi(q, p)dpi .

•Theorem 2.3.1 The Poincare 1-form is well defined globally on thecotangent bundle of any manifold.

Proof :

1. Let (qα, pα) and (qβ, pβ) be two overlapping coordinate patches ofT ∗M.

2. Then

dqiα =

n∑j=1

∂qiα

∂q jβ

dq jβ

andn∑

i=1

pαi dqiα =

n∑j=1

pβjdq jβ .

• We give now an intrinsic definition of the Poincare form.

• Let (q, p) ∈ T ∗M be a point in T ∗M. We want to define a 1-form λ ∈T ∗(q,p)(T

∗M) at this point (q, p) ∈ T ∗M.

• Let π : T ∗M → M be the projection defined for any q ∈ M, p ∈ T ∗q M byπ(q, p) = q.

• Then the pullback is the map π∗ : T ∗q M → T ∗(q,p)(T∗M). For each 1-form

p ∈ T ∗q M it defines a 1-form π∗(p) ∈ T ∗(q,p)(T∗M). This is precisely the

Poincare 1-form λ, that is,λ = π∗(p) .

• Of course, in local coordinates

π∗(p) =

n∑i=1

pidqi .



2.4 Tensors

2.4.1 Covariant Tensors• Let E be a vector space and E∗ be its dual space.

• Let ei be a basis for E and σi be the dual basis for E∗.

• A covariant tensor of rank p (or a tensor of type (0, p)) is a multi-linearreal-valued functional

Q : E × · · · × E︸︷︷︸p

→ R

• Remarks.

• The function Q(v1, . . . , vp) is linear in each argument.

• The functional Q is independent of any basis.

• A covariant vector (covector) is a covariant tensor of rank 1.

• A metric tensor is a covariant tensor of rank 2.

• The components of the tensor Q with respect to the basis ei are definedby

Qi1...ip = Q(ei1 , . . . , eip) .

• Then for any vectors

va =

n∑j=1

v jae j ,

where a = 1, . . . , p, we have

Q(v1, . . . , vp) =

n∑j1,..., jp=1

Q j1... jpvj11 · · · v

jpp .

• The collection of all covariant tensors of rank p forms a vector space de-noted by

Tp = E∗ ⊗ · · · ⊗ E∗︸︷︷︸p

.


2.4. TENSORS 49

• The dimension of the vector space Tp is

dim Tp = np .

• The tensor product of two covectors α, β ∈ E∗ is a covariant tensor α⊗ β ∈E∗ ⊗ E∗ of rank 2 defined by: ∀v,w ∈ E

(α ⊗ β)(v,w) = α(v)β(w) .

• The components of the tensor product α ⊗ β are

(α ⊗ β)i j = αiβ j .

• The tensor product of a covariant tensor Q of rank p and a covariant ten-sor T of rank q is a covariant tensor Q ⊗ T of rank (p + q) defined by:∀v1, . . . , vp,w1, . . . ,wq ∈ E

(Q ⊗ T )(v1, . . . , vp,w1, . . . ,wq) = Q(v1, . . . , vp)T (w1, . . . ,wq) .

• The components of the tensor product Q ⊗ T are

(Q ⊗ T )i1...ip j1... jq = Qi1...ipT j1... jq .

• Thus,⊗ : Tp × Tq → Tp+q .

• Tensor product is associative.

• The basis in the space Tp is

σi1 ⊗ · · · ⊗ σip ,

where 1 ≤ i1, . . . , ip ≤ n.

• A covariant tensor Q of rank p has the form

Q =

n∑i1,...,ip=1

Qi1...ipσi1 ⊗ · · · ⊗ σip .



2.4.2 Contravariant Tensors• A contravariant vector can be considered as a linear real-valued functional

v : E∗ → R .

• A contravariant tensor of rank p (or a tensor of type (p, 0)) is a multi-linear real-valued functional

T : E∗ × · · · × E∗︸︷︷︸p

→ R

• Remarks.

• The function T (α1, · · · , αp) is linear in each argument.

• The functional T is independent of any basis.

• A contravariant vector (covector) is a contravariant tensor of rank 1.

• The components of the tensor T with respect to the basis σi are definedby

T i1...ip = T (σi1 , . . . , σip) .

• Then for any covectors

α(a) =

n∑j=1

α(a) jσj ,

where a = 1, . . . , p, we have

T (α(1), . . . , α(p)) =

n∑j1,..., jp=1

T j1... jpα(1) j1 · · ·α(p) jp .

• The inverse matrix of the components of a metric tensor defines a con-travariant tensor g−1 of rank 2 by

g−1(α, β) =

n∑i, j=1

gi jαiβ j .


2.4. TENSORS 51

• The collection of all contravariant tensors of rank p forms a vector spacedenoted by

T p = E ⊗ · · · ⊗ E︸︷︷︸p

.

• The dimension of the vector space T p is

dim T p = np .

• The tensor product of a contravariant tensor Q of rank p and a contravari-ant tensor T of rank q is a contravariant tensor Q⊗T of rank (p + q) definedby: ∀α1, . . . , αp, β1, . . . , βq ∈ E∗

(Q ⊗ T )(α1, . . . , αp, β1, . . . , βq) = Q(α1, . . . , αp)T (β1, . . . , βq) .


(Q ⊗ T )i1...ip j1... jq = Qi1...ipT j1... jq .

• Thus,⊗ : T p × T q → T p+q .

• Tensor product is associative.

• The basis in the space T p is

ei1 ⊗ · · · ⊗ eip ,

where 1 ≤ i1, . . . , ip ≤ n.

• A contravariant tensor T of rank p has the form

T =

n∑i1,...,ip=1

T i1...ipei1 ⊗ · · · ⊗ eip .

• The set of all tensors of type (p, 0) forms a vector space T p of dimension np

dim T p = np .



2.4.3 General Tensors of Type (p, q)

• A tensor of type (p, q) is a multi-linear real-valued functional

T : E∗ × · · · × E∗︸︷︷︸p

× E × · · · × E︸︷︷︸q

→ R

• The components of the tensor T with respect to the basis ei,σi are definedby

T i1...ip

j1... jq= T (σi1 , . . . , σip , e j1 , . . . , e jq) .

• Then for any covectors

α(a) =

n∑j=1

α(a) jσj ,

where a = 1, . . . , p, and any vectors

vb =

n∑j=1

vkbek ,

where b = 1, . . . , q, we have

T (α(1), . . . , α(p), v1, . . . , vq) =

n∑k1,...,kq=1

n∑j1,..., jp=1

T j1... jp

k1...kqα(1) j1 · · ·α(p) jpv

k1 · · · vkq .

• The inverse matrix of the components of a metric tensor defines a con-travariant tensor g−1 of rank 2 by

g−1(α, β) =

n∑i, j=1

gi jαiβ j .

• The collection of all tensors of type (p, q) forms a vector space denoted by

T pq = E ⊗ · · · ⊗ E︸︷︷︸

p

⊗ E∗ ⊗ · · · ⊗ E∗︸︷︷︸q

.

• The dimension of the vector space Tp is

dim T pq = np+q .


2.4. TENSORS 53

• The tensor product of a tensor Q of type (p, q) and a tensor T of type (r, s)is a tensor Q⊗T of type (p+r, q+s) defined by: ∀α1, . . . , αp, β1, . . . , βr ∈ E∗,v1, . . . , vq,w1, . . . ,ws ∈ E

(Q ⊗ T )(α1, . . . , αp, β1, . . . , βr, v1, . . . , vq,w1, . . . ,ws)= Q(α1, . . . , αp, v1, . . . , vq)T (β1, . . . , βr,w1, . . . ,ws) .


(Q ⊗ T )i1...ip j1... jrk1...kql1...ls

= Qi1...ip

k1...kqT j1... jr

l1...ls.

• Thus,⊗ : T p

q × T rs → T p+r

q+s .

• We stress once again that the tensor product is associative.

• The basis in the space T pq is

ei1 ⊗ · · · ⊗ eip ⊗ σj1 ⊗ · · · ⊗ σ jq ,

where 1 ≤ i1, . . . , ip, j1, . . . , jq ≤ n.

• A tensor T of type (p, q) has the form

T =

n∑j1,..., jq=1

n∑i1,...,ip=1

T i1...ip

j1... jqei1 ⊗ · · · ⊗ eip ⊗ σ

j1 ⊗ · · · ⊗ σ jq .

• Let p, q ≥ 1 and 1 ≤ r ≤ p, 1 ≤ s ≤ q. The (r, s)-contraction of tensors oftype (p, q) is the map

tr rs : T p

q → T p−1q−1

defined by

(tr rs T )i1...ip−1

j1... jq−1=

n∑k=1

T i1...ir−1kir ...ip−1

j1... js−1k js... jq.



2.4.4 Linear Transformations and Tensors• Let A : E → E be a linear transformation.

• Then we can define a tensor A of type (1, 1) by ∀α ∈ E∗, v ∈ E

A(α, v) = α(Av) .

• Let (Aij) be the matrix of the linear transformation A, that is,

Ae j =

n∑i=1

Aijei .

• Then the components of the tensor A are

Aij = A(σi, e j) = σi(Ae j) = Ai

j .

• Thus, one can identify tensors of type (1, 1) and linear transformations onE (and, similarly on E∗ as well).

• Then,

A =

n∑i, j=1

Aijei ⊗ σ

j .

• The identity linear transformation I has the matrix

Iij = δi

j

and defines the tensor of type (1, 1)

I =

n∑i=1

ei ⊗ σi .

• To summarize, a (1, 1) tensor A : E∗ × E → R is identified with the lineartransformation A : E → E.

• The covariant tensor Ai j, the contravariant tensor Ai j and the tensor Aij of

type (1, 1) are related by

Ai j =

n∑k=1

gikAkj , Ai j =

n∑k=1

Aikgk j .


2.4. TENSORS 55

2.4.5 Tensor Fields

•Definition 2.4.1 A tensor field on a manifold M is a smooth assign-ment of a tensor at each point of M.

• Let xiα = xi

α(xβ) be a local diffeomorphism.

• Then

dxiα =

n∑j=1

∂xiα

∂x jβ

dx jβ

and∂

∂xiα

=

n∑j=1

∂x jβ

∂xiα

∂

∂x jβ

• Let T be a tensor of type (p, q). Then

T(α)i1...ip

j1... jq=

n∑k1,...,kp=1

n∑l1,...,lq=1

∂xi1α

∂xk1β

· · ·∂xip

α

∂xkp

β

∂xl1β

∂xlqα

· · ·∂x j1

β

∂x jqα

T(β)k1...kp

l1...lq

2.4.6 Tensor Bundles

•

Definition 2.4.2 Let M be a smooth manifold. The tensor bundle oftype (p, q) T p

q M is the collection of all tensors of type (p, q) at all pointsof M

T pq M = (p,T ) | p ∈ M,T ∈ T p

q, (x)M

• The tensor bundle T pq M is the tensor product of the tangent and cotangent

bundlesT p

q M = T M ⊗ · · · ⊗ T M︸︷︷︸p

⊗T ∗M ⊗ · · · ⊗ T ∗M︸︷︷︸q

.

• Let dim M = n.

• Let x ∈ M be a point in the manifold M, (U, x) be a local chart and (xi) bethe local coordinates of the point p.

• Let ∂i and dxi be the coordinate basis for TxM and T ∗x M.



• Let T be a tensor of type (p, q) and T i1...ip

j1... jqbe the components of the tensor T

in the coordinate basis. Then the local coordinates of the point (p,T ) ∈ T pq M

are(xi,T i1...ip

j1... jq) ,

where 1 ≤ i, i1, . . . , ip, j1, . . . , jq ≤ n.

• Remarks.

• The open set U × Rnp+q⊂ Rn+np+q

is a local chart in the tensor bundle T pq M.

• The bundle T pq M is a manifold of dimension n + np+q.

• The projection map π : T pq M → M is defined by π(x,T ) = x.

• A tensor field of type (p, q) is a map

T : M → T pq M ,

such thatπ T = Id : M → M .

• A tensor field of type (p, q) is a section of the tensor bundle T pq M.

2.4.7 Examples• Riemannian metric gµν.

• Energy-momentum tensor Tµν.

• Stress tensor σi j.

• Riemann curvature tensor Rµαβγ.

• Ricci curvature tensor Rµν.

• Scalar density of weight 1: |g| =√

det gi j.

• Axial vectors (vector product) in R3.

• Strength of the electromagnetic field Fµν.


2.4. TENSORS 57

•

Theorem 2.4.1 Let

A =

n∑i=1

Aidxi

be a covector field (1-form). Let

Fi j = ∂iA j − ∂ jAi .

Then

F =

n∑i, j=1

Fi jdxi ⊗ dx j

is a tensor of type (0, 2).

Proof :

1. Check the transformation law.

• A tensor is called isotropic if it is a tensor product of g, g−1 and I.

• The components of an isotropic tensor are the products of gi j, gi j and δij.

• Every isotropic tensor of type (p, q) has an even rank p + q.

• For example, the most general isotropic tensor of type (2, 2) has the form

Ai jkl = agi jgkl + bδi

kδjl + cδi

lδjk ,

where a, b, c are scalars.

2.4.8 Einstein Summation Convention

• In any expression there are two types of indices: free indices and repeatedindices.

• Free indices appear only once in an expression; they are assumed to take allpossible values from 1 to n.

• The position of all free indices in all terms in an equation must be the same.



• Repeated indices appear twice in an expression. It is assumed that there is asummation over each repeated pair of indices from 1 to n. The summationover a pair of repeated indices in an expression is called the contraction.

• Repeated indices are dummy indices: they can be replaced by any otherletter (not already used in the expression) without changing the meaning ofthe expression.

• Indices cannot be repeated on the same level. That is, in a pair of repeatedindices one index is in upper position and another is in the lower position.

• There cannot be indices occuring three or more times in any expression.


Chapter 3

Differential Forms

3.1 Exterior Algebra

3.1.1 Permutation Group

• A group is a set G with an associative binary operation, · : G×G → G withidentity, called the multiplication, such that each element has an inverse.That is, the following conditions are satisfied

1. for any three elements g, h, k ∈ G, the associativity law holds: (gh)k =

g(hk);

2. there exists an identity element e ∈ G such that for any g ∈ G, ge =

eg = g;

3. each element g ∈ G has an inverse g−1, such that g g−1 = g−1 g = e

• Let X be a set. A transformation of the set X is a bijective map g : X → X.

• The set of all transformations of a set X forms a group Aut(X), with com-position of maps as group multiplication.

• Any subgroup of Aut(X) is a transformation group of the set X.

• The transformations of a finite set X are called permutations.

• The group S p of permutations of the set Zn = 1, . . . , p is called the sym-metric group of order p.

59

60 CHAPTER 3. DIFFERENTIAL FORMS

• Theorem 3.1.1 The order of the symmetric group S p is

|S p| = p! .

• Any subgroup of S p is called a permutation group.

• A permutation ϕ : Zp → Zp can be represented by(1 . . . p

ϕ(1) . . . ϕ(p)

)• The identity permutation is (

1 . . . p1 . . . p

)• The inverse permutation ϕ−1 : Zp → Zp is represented by(

ϕ(1) . . . ϕ(p)1 . . . p

)• The product of permutations is then defined in an obvious manner.

• An elementary permutation is a permutation that exchanges the order ofonly two elements.

• Every permutation can be realized as a product of elementary permutations.

• A permutation that can be realized by an even number of elementary per-mutations is called an even permutation.

• A permutation that can be realized by an odd number of elementary permu-tations is called an odd permutation.

• Proposition 3.1.1 The parity of a permutation does not depend on therepresentation of a permutation by a product of the elementary ones.

• That is, each representation of an even permutation has even number ofelementary permutations, and similarly for odd permutations.

• The sign of a permutation ϕ, denoted by sign(ϕ) (or simply (−1)ϕ), isdefined by

sign(ϕ) = (−1)ϕ =

+1, if ϕ is even,−1, if ϕ is odd


3.1. EXTERIOR ALGEBRA 61

3.1.2 Permutations of Tensors• Let S p be the symmetric group of order p. Then every permutation ϕ ∈ S p

defines a mapϕ : Tp → Tp ,

which assigns to every tensor T of type (0, p) a new tensor ϕ(T ), called apermutation of the tensor T , of type (0, p) by: ∀v1, . . . , vp

ϕ(T )(v1, . . . , vp) = T (vϕ(1), . . . , vϕ(p)) .

• Let (i1, . . . , ip) be a p-tuple of integers. Then a permutation ϕ : Zp → Zp

defines an actionϕ(i1, . . . , ip) = (iϕ(1), . . . , iϕ(p)) .

• The components of the tensor ϕ(T ) are obtained by the action of the permu-tation ϕ on the indices of the tensor T

ϕ(T )i1...ip = Tiϕ(1)...iϕ(p) .

• The symmetrization of the tensor T of the type (0, p) is defined by

Sym(T ) =1p!

∑ϕ∈S p

ϕ(T ) .

• The symmetrization is also denoted by parenthesis. The components of thesymmetrized tensor Sym(T ) are given by

T(i1...ip) =1p!

∑ϕ∈S p

Tiϕ(1)...iϕ(p) .

• The anti-symmetrization of the tensor T of the type (0, p) is defined by

Alt(T ) =1p!

∑ϕ∈S p

sign (ϕ)ϕ(T ) .

• The anti-symmetrization is also denoted by square brackets. The compo-nents of the anti-symmetrized tensor Alt(T ) are given by

T[i1...ip] =1p!

∑ϕ∈S p

sign (ϕ)Tiϕ(1)...iϕ(p) .



• Examples.

• A tensor T of type (0, p) is called symmetric if for any permutation ϕ ∈ S p

ϕ(T ) = T .

• A tensor T of type (0, p) is called anti-symmetric if for any permutationϕ ∈ S p

ϕ(T ) = sign (ϕ)T .

• An anti-symmetric tensor of type (0, p) is called a p-form.

• Remarks.

• Permutation, symmetrization, anti-symmetrization of tensors of type (p, 0).

• Completely symmetric and completely anti-symmetric tensors of type (p, 0).

• An anti-symmetric tensor of type (p, 0) is called a p-vector.

• Partial permutation.

• Examples.

• Notation.

3.1.3 Alternating Tensors• Let (i1, . . . , ip) and ( j1, . . . , jp) be two p-tuples of integers 1 ≤ i1, . . . , ip, j1, . . . , jp ≤

n. The generalized Kronecker symbol is defined by

δi1... ip

j1... jp=

1 if (i1, . . . , ip) is an even permutation of ( j1, . . . , jp)−1 if (i1, . . . , ip) is an odd permutation of ( j1, . . . , jp)

0 otherwise

• One can easily check that

δi1... ip

j1... jp= det

δi1

j1. . . δi1

jp...

. . ....

δip

j1. . . δ

ip

jp



• Also, there holdsδ

i1... ip

j1... jp= p!δi1

[ j1· · · δ

ip

jp] .

• Thus, the Kronecker symbols δi1... ip

j1... jpare the components of the tensors

p!Alt(I ⊗ · · · ⊗ I︸︷︷︸p

)

of type (p, p), which are anti-symmetric separately in upper indices and thelower indices.

• Thus, the anti-symmetrization can also be written as

T[i1...ip] =1p!δ

j1... jp

i1...ipT j1... jp .

• Notation.

• Obviously, the Kronecker symbols vanish for p > n

δi1...ip

j1... jp= 0 if p > n .

• The contraction of Kronecker symbols gives Kronecker symbols with lowerindices, more precisely, we have the theorem.

•

Theorem 3.1.2 For any p, q ∈ N, 1 ≤ p, q ≤ n, there holds

δi1... ipl1...lqj1... jpl1...lq

=(n − p)!

(n − p − q)!δ

i1... ip

j1... jp.

Proof :

1.

•

Corollary 3.1.1For any q ∈ N, 1 ≤ q ≤ n we have

δi1...iqi1...iq

=n!

(n − q)!.

In particular,δi1...in

i1...in= n! .



•

Lemma 3.1.1 There holds

δi1...ip j1 ... jrl1...lp m1...mr

δk1...krj1... jr

= r!δi1...ip k1 ...kr

l1...lp m1...mr

Proof :

1.

• In general, let A be a p-form (an antisymmetric tensor of type (0, p)) and Bbe a p-vector (an anti-symmetric tensor of type (p, 0)). Then

Ai1...ipδi1...ip

j1... jp= p!A j1... jp ,

Bi1...ipδj1... jp

i1 ... ip= p!B j1... jp .

• Let (i1, . . . , in) be an n-tuple of integers 1 ≤ i1, . . . , in ≤ n. The completelyanti-symmetric (alternating) Levi-Civita symbols are defined by

εi1...in = δ1 ... ni1...in , εi1...in = δi1...in

1 ... n ,

so that

εi1...in = εi1...in =

1 if (i1, . . . , in) is an even permutation of (1, . . . , n)−1 if (i1, . . . , in) is an odd permutation of (1, . . . , n)

0 otherwise

• Theorem 3.1.3 There holds the identity

εi1...inε j1... jn =∑ϕ∈S n

sign(ϕ) δi1jϕ(1)· · · δin

jϕ(n)

= n!δi1[ j1· · · δin

jn]

= δi1...inj1... jn

.

The contraction of this identity over k indices gives

εi1...in−km1...mkε j1... jn−km1...mk = k!(n − k)!δi1[ j1· · · δin−k

jn−k]

= k!δi1...in−kj1... jn−k

.

In particular,εm1...mnεm1...mn = n! .

• It is easy to see that there holds also

δi1...in−p

l1...ln−pε j1... jpl1...ln−p = (n − p)!ε j1... jpi1...in−p



3.1.4 Determinants

• The set of all n × n real matrices is denoted by Mat(n,R).

• The determinant is a map det : Mat(n,R) → R that assigns to each matrixA = (Ai

j) a real number det A defined by

det A =∑ϕ∈S n

sign (ϕ)A1ϕ(1) · · · An

ϕ(n) ,

• The most important properties of the determinant are listed below:

Theorem 3.1.4 1. The determinant of the product of matrices is equal tothe product of the determinants:

det(AB) = det A det B .

2. The determinants of a matrix A and of its transpose AT are equal:

det A = det AT .

3. The determinant of the inverse A−1 of an invertible matrix A is equalto the inverse of the determinant of A:

det A−1 = (det A)−1

4. A matrix is invertible if and only if its determinant is non-zero.

• The determinant of a matrix A = (Aij) can be written as

det A = εi1...in A1i1 . . . A

nin

= ε j1... jn A j11 . . . A jn

n

=1n!εi1...inε j1... jn A j1

i1 . . . Ajn

in .

Here, as usual, a summation over all repeated indices is assumed from 1 ton.



3.1.5 Exterior p-forms• An exterior p-form (or simply a p-form) is an anti-symmetric covariant

tensor α ∈ Tp of type (0, p).

• The collection of all p-forms forms a vector space Λp, which is a vectorsubspace of Tp

Λp ⊂ Tp .

• In particular,Λ0 = R and Λ1 = T1 = E∗ .

• In other words, a 0-form is a smooth function, and a 1-form is a covectorfield.

• Let α ∈ Λp be a p-form.

• Let ei be a basis in E and σi be the dual basis in E∗.

• The components of the p-form α are

αi1...ip = α(ei1 , . . . eip) .

• The components are completely anti-symmetric in all indices, that is,

αi1...ip = sign (ϕ)αiϕ(1)...iϕ(p) .

In particular, under a permutation of any two indices the form changes sign

α... i ... j ... = −α... j ... i ... ,

which means that the components vanish if any two indices are equal

α... i ... i ... = 0 (no summation!) .

• Thus, all non-vanishing components have different indices.

• Therefore, the values of all components αi1...ip are completely determined bythe values of the components with the indices i1, . . . , ip reordered in strictlyincreasing order

1 ≤ i1 < · · · < ip ≤ n .



• Notation.

• To deal with forms it is convenient to introduce multi-indices. We willdenote a p-tuple of integers from 1 to n by a capital letter

I = (i1, . . . , ip) ,

where 1 ≤ i1, . . . , ip ≤ n. For a p-tuple of the same integers ordered in anincreasing order we define

I = (i1, . . . , ip) .

where 1 ≤ i1 < i2 < · · · < ip ≤ n. We call I an increasing p-tuple associatedwith I.

• Therefore, a collection of p-forms

p!Alt(σi1 ⊗ · · · ⊗ σip

),

where 1 ≤ i1 < i2 < · · · < ip ≤ n, forms a basis in the space Λp.

• Thus, every p-form α ∈ Λp has the form

α =∑

1≤i1<···<ip≤n

αi1···ip p!Alt(σi1 ⊗ · · · ⊗ σip

).

• Therefore, the dimension of the space Λp is equal to the number of distinctincreasing p-tuples of integers from 1 to n.

•

Theorem 3.1.5 The dimension of the space Λp of p-forms is

dim Λp =

(np

)=

n!p!(n − p)!

.

Proof :

1.

• In particular,dim Λ0 = dim Λn = 1 ,

dim Λ1 = dim Λn−1 = n ,

etc.



• There are no p-forms with p > n.

• Similarly to the norm of vectors and covectors we define the inner productof exterior p-forms α and β in a Riemannian manifold by

(α, β) =1p!

gi1 j1 · · · gi1 jpαi1...ipβ j1... jp .

• This enables one to define also the norm of an exterior p-form α by

||α|| =√

(α, α)


3.2. EXTERIOR AND INTERIOR PRODUCTS 69

3.2 Exterior and Interior Products• Since the tensor product of two skew-symmetric tensors is not a skew-

symmetric tensor to define the algebra of antisymmetric tensors we need todefine the anti-symmetric tensor product called the exterior (or wedge)product.

• If α is an p-form and β is an q-form then the exterior product of α and β isan (p + q)-form α ∧ β defined by

α ∧ β =(p + q)!

p!q!Alt (α ⊗ β) .

• In components

(α ∧ β)i1...ip+q =(p + q)!

p!q!α[i1...ipβip+1...ip+q] .

• Let α ∈ Λp be a p-form. Then

p = deg(α)

is called the degree (or rank) of α.

•

Theorem 3.2.1 The exterior product has the following properties

(α ∧ β) ∧ γ = α ∧ (β ∧ γ) (associativity)

α ∧ β = (−1)deg(α)deg(β)β ∧ α (anticommutativity)

(α + β) ∧ γ = α ∧ γ + β ∧ γ (distributivity) .

Proof :

1.

• The exterior square of any p-form α of odd degree p (in particular, for any1-form) vanishes

α ∧ α = 0 .



• The exterior algebra Λ (or Grassmann algebra) is the set of all forms ofall degrees, that is,

Λ = Λ0 ⊕ · · · ⊕ Λn .

• The dimension of the exterior algebra is

dim Λ =

n∑p=0

(np

)= 2n .

• A basis of the space Λp is

σi1 ∧ · · · ∧ σip , (1 ≤ i1 < · · · < ip ≤ n) .

An p-form α can be represented in one of the following ways

α = αi1...ipσi1 ⊗ · · · ⊗ σip

=1p!αi1...ipσ

i1 ∧ · · · ∧ σip

=∑

i1<···<ip

αi1...ipσi1 ∧ · · · ∧ σip .

• The exterior product of a p-form α and a q-form β can be represented as

α ∧ β =1

p!q!α[i1...ipβip+1...ip+q]σ

i1 ∧ · · · ∧ σip+q .

•

Theorem 3.2.2 Let σ j ∈ Λ1, 1 ≤ j ≤ n, and α j ∈ Λ1, 1 ≤ j ≤ n, betwo collections of n 1-forms related by a linear transformation

α j =

n∑i=1

A jiσ

i , 1 ≤ j ≤ n ,

Thenα1 ∧ · · · ∧ αn = det Ai

j σ1 ∧ · · · ∧ σn .

Proof :

1.


3.2. EXTERIOR AND INTERIOR PRODUCTS 71

•

Theorem 3.2.3 Let α j ∈ Λ1 = E∗, 1 ≤ j ≤ p, be a collections of p1-forms and vi ∈ E, 1 ≤ i ≤ p, be a collection of p vectors. Let

A ji = α j(vi) , 1 ≤ i, j ≤ p .

Then (α1 ∧ · · · ∧ αp

)(v1, . . . , vp) = det Ai

j .

Proof :

1.

•

Theorem 3.2.4 A collections of p 1-forms α j ∈ Λ1 = E∗, 1 ≤ j ≤ p,is linearly dependent if and only if

α1 ∧ · · · ∧ αp = 0 .

•

Corollary 3.2.1 Let xi = xi(x′), i = 1, . . . , n, be a local diffeomor-phism. Then

dx1 ∧ · · · ∧ dxn = det(∂xl

∂x′m

)dx′1 ∧ · · · ∧ dx′n .

3.2.1 Interior Product• The interior product of a vector v and a p-form α is a (p − 1)-form ivα

defined by, for any v1, . . . , vp−1,

ivα(v1, . . . , vp−1) = α(v, v1, . . . , vp−1) .

In particular, if p = 1, then ivα is a scalar

ivα = α(v)

and if p = 0, then by definition

ivα = 0 .

• In components,(ivα)i1...ip−1 = v jα ji1...ip−1 .



• The interior product is a map

iv : Λp → Λp−1 ,

oriv : Λ→ Λ .

• A map L : Λ→ Λ is called an derivation if for any α ∈ Λp, β ∈ Λq,

L(α ∧ β) = (Lα) ∧ β + α ∧ Lβ .

• A map L : Λ→ Λ is called an anti-derivation if for any α ∈ Λp, β ∈ Λq,

L(α ∧ β) = (Lα) ∧ β + (−1)pα ∧ Lβ .

•Theorem 3.2.5 Let v ∈ E be a vector. The interior product iv : Λ→ Λ

is an anti-derivation.


3.3. HODGE DUALITY 73

3.3 Hodge Duality

3.3.1 Volume Form

• Let v(1), . . . , v(n) be an ordered n-tuple of vectors. The volume of the par-allelepiped spanned by the vectors v(1), . . . , v(n) is a real number definedby

|vol (v(1), . . . , v(n))| =√

det((v(i), v( j))) .

• If the vectors v(1), . . . , v(n) are orthonormal, then

|vol (v(1), . . . , v(n))| = 1 .

• If the vectors v(1), . . . , v(n) are linearly dependent, then

vol (v(1), . . . , v(n)) = 0 .

• The volume |vol (v(1), . . . , v(n))| does not depend on the orientation of vec-tors.

• The signed volume of the parallelepiped spanned by an ordered n-tuple ofvectors v(1), . . . , v(n) is

vol (v(1), . . . , v(n)) = sign(v(1), . . . , v(n)) |vol (v(1), . . . , v(n))| ,

where the sign of the signed volume is defined by

sign(v(1), . . . , v(n)) =

+1, if v(1), . . . , v(n) is positively oriented

−1, if v(1), . . . , v(n) is negatively oriented



•

Theorem 3.3.1 Let ei be a basis in a vector space E, σi be the dual ba-sis of 1-forms in E∗, g = (gi j) be a Riemannian metric and v(1), . . . , v(n)

be a set of n vectors. Let vi( j) = σi(v( j)) be the contravariant components

of the vectors v( j) and vi ( j) = (ei, v( j)) = gikvk( j) be the covariant compo-

nents of these vectors. Then:

vol (v(1), . . . , v(n)) =√|g| det(vi

( j)))

= Ei1...in vi1(1) · · · vin

(n)

and

vol (v(1), . . . , v(n)) =1√|g|

det(vi ( j))

= Ei1...in vi1 (1) · · · vin (n)

In particular,vol (e1, . . . , en) = o(ei)

√|g| ,

where o(ei) is equal to +1 or −1 for positively oriented and negativelyoriented basis ei.

• The n-formvol =

√|g|σ1 ∧ · · · ∧ σn

is called the Riemannian volume element (or volume form).

• We have (σi1 ∧ · · · ∧ σip

)(e j1 , . . . , e jp) = δ

i1...ip

j1··· jp.

• The components of the volume form are

vol (ei1 , . . . , ein) =√|g|εi1...in = Ei1...in .

3.3.2 Star Operator

• The volume form allows one to define the duality of p-forms and (n − p)-vectors.



• For each p-form Ai1...ip one assigns the dual (n − p)-vector by

A j1... jn−p =1p!

E j1... jn−pi1...ip Ai1...ip .

• Similarly, for each p-vector Ai1...ip one assigns the dual (n − p)-form by

A j1... jn−p =1p!

E j1... jn−pi1...ip Ai1...ip .

• By lowering and raising the indices of the dual forms we can define theduality of forms and poly-vectors separately.

• The Hodge star operator

∗ : Λp → Λn−p

maps any p-form α to a (n − p)-form ∗α dual to α defined as follows.

• For each p-form α the form ∗α is the unique (n − p)-form such that

α ∧ ∗α = (α, α)vol .

• In particular,∗1 = vol , ∗vol = 1 .

• In components, this means that

(∗α)ip+1... in =1p!εi1... ipip+1...in

√|g|gi1 j1 · · · gip jpα j1... jp

=1p!

1√|g|

gip+1 jp+1 · · · gin jnεj1... jp jp+1... jnα j1... jp .

•

Theorem 3.3.2 For any p-form α there holds

∗2α = (−1)p(n−p)α .

In particular, if n is odd, then for any p

∗2 = Id .



•

Theorem 3.3.3 Let α be a 1-form and v be the corresponding vector,that is, vi = gi jα j. Then

∗α = ivvol .

• A collection ω(1), . . . , ω(n−1) of (n − 1) 1-forms defines a 1-form α by

α = ∗[ω(1) ∧ · · · ∧ ω(n−1)

].

• Then∗α = (−1)n−1ω(1) ∧ · · · ∧ ω(n−1)

andω(1) ∧ · · · ∧ ω(n−1) ∧ α = (α, α)vol .

• In components,α j = g jkEi1...in−1kωi1 (1) · · ·ωin−1 (n−1) .

• If the 1-forms ω(1), . . . , ω(n−1) are linearly dependent, then α = 0.

• If the collection of 1-forms ω(1), . . . , ω(n−1) is linearly independent, thenω(1), . . . , ω(n−1), α are linearly independent and form a basis in E∗.

•

Theorem 3.3.4 Let σ1, . . . , σn be an orthonormal basis of 1-forms.Then

σ j = (−1)n− j ∗[σ1 ∧ · · · ∧ σ j−1 ∧ σ j+1 ∧ · · · ∧ σn

],

In particular,σn = ∗ [σ1 ∧ · · · ∧ σn−1] .

• Similarly, a collection v(1), . . . , v(n−1) of (n − 1) vectors defines a covectorα by: for any vector v

α(v) = vol (v(1), . . . , v(n−1), v)

or, in components,

α j = Ei1...in−1 jvi1(1) · · · vin−1

(n−1) .



•

Theorem 3.3.5 Let v(1), . . . , v(n−1) be an ordered (n − 1)-tuple of lin-early independent vectors and ω(1), . . . , ω(n−1) be the corresponding1-forms. Let α be 1-form defined by

α = ∗[ω(1) ∧ · · · ∧ ω(n−1)

],

and N be the corresponding vector, that is,

N i = gik√|g|εi1...in−1kvi1

(1) · · · vin−1(n−1)

Then:

1. The vector N is orthogonal to all vectors v(1), . . . , v(n−1), that is,

(N, v( j)) = gikN ivk( j) = 0 , ( j = 1, . . . , n − 1) .

2. The n-tuple v1, . . . , vn−1,N forms a positively oriented basis.

3. The volume of the parallelepiped spanned by the vectorsv(1), . . . , v(n−1),N is determined by the norm of N

vol (v1, . . . , vn−1,N) = (N,N) .



3.4 Exterior Derivative and Coderivative• From now on, if not specified otherwise, we will denote the derivatives by

∂i =∂

∂xi .

• The exterior derivative of a 0-form (that is, a function) f is a 1-form d fdefined by: for any vector v

(d f )(v) = v( f ) .

• In local cordinatesd f = ∂ j f dx j .

• The exterior derivative of a 1-form (that is, a covector) A is a 2-form dAdefined by: for any vectors v,w

(dA)(v,w) = v(A(w)) − w(A(v)) − A([v,w]) .

• In local cordinates

dA =12

(∂iA j − ∂ jAi

)dxi ∧ dx j .

• Let α be a p-form

α =1p!αi1...ipdxi1 ∧ · · · ∧ dxip .

• The exterior derivative of α is a (p + 1)-form dα defined by

dα =1p!

dαi1...ipdxi1 ∧ · · · ∧ dxip

=1p!∂i1αi2...ip+1dxi1 ∧ dxi2 ∧ · · · ∧ dxip+1 .

• In components

(dα)i1i2...ip+1 = (p + 1)∂[i1αi2...ip+1]

=

p+1∑k=1

(−1)k−1∂ikαi1...ik−1ik+1...ip+1


3.4. EXTERIOR DERIVATIVE AND CODERIVATIVE 79

•

Theorem 3.4.1 The exterior derivative is a linear map

d : Λp → Λp+1 .

Proof : Show that dα is a form whose value does not depend on the coordi-nate system.

•

Theorem 3.4.2 Let α ∈ Λp be a p-form and v1, . . . , vp+1 be a collec-tion of (p + 1) vectors. Then

(dα)(v1, . . . , vp+1) =

p+1∑k=1

(−1)k−1vk(α(v1, . . . , vk−1, vk+1, . . . , vp+1))

−

p+1∑k=1

k−1∑i=1

(−1)i+k−1α([vi, vk], v1, . . . , vi, vi+1, . . . , vk−1, vk+1, . . . , vp+1))

Proof : Calculation.

• Remark. This formula can be taken as the intrinsic definition of the exteriorderivative.

•Theorem 3.4.3 For any p-form

d2 = 0 .

Proof : Easy.

•

Theorem 3.4.4 The exterior derivative d : Λ → Λ is an anti-derivation on the exterior algebra.

That is, for any p-form α ∈ Λp and any q-form β ∈ Λq there holds

d(α ∧ β) = (dα) ∧ β + (−1)pα ∧ (dβ) .

Proof : Calculation.

• Examples.



3.4.1 Coderivative• Given a Riemannian metric gµν we also define the co-derivative of p-forms.

• The coderivative is a linear map

δ : Λp → Λp−1

defined byδ = ∗−1d∗ = (−1)(n−p+1)(p−1) ∗ d∗

• That is the coderivative of a p-form α is the (p − 1)-form δα defined by

(δα)i1...ip−1 =1

(n − p + 1)!εi1...ip−1ip...in

√|g|g jipg jp+1ip+1 · · · g jnin

(n − p + 1)∂ j

(1p!ε j1... jp jp+1... jn

√|g|g j1k1 · · · g jpkpαk1...kp

)

•Theorem 3.4.5 For any p-form

δ2 = 0 .

Proof : Follows from ∗2 = ±1 and d2 = 0.

• From this definition, we can also see that, for any 0-form f (a function) ∗ fis an n-form and, therefore, d ∗ f = 0, i.e. a coderivative of any 0-form iszero

δ f = 0 .

• For a 1-form α, δα is a 0-form

δα =1√|g|∂i

( √|g|gi jα j

).

• More generally, one can prove that for a p-form α

(δα)i1...ip−1 = gi1 j1 . . . gip−1 jp−1

1√|g|∂ j

( √|g|g jkg j1k1 · · · g jp−1kp−1αkk1...kp−1

).

• Examples.


3.5. PULLBACK OF FORMS 81

3.5 Pullback of Forms• Let M be a n-dimensional manifold and W be a r-dimensional manifold.

• Let F : M → W be a smooth map of a manifold M to a manifold W.

• Let p ∈ M be a point in M and q = F(p) ∈ W be the image of p in W.

• Let xi, (i = 1, . . . , n), be a local coordinate system about p and yµ, (µ =

1, . . . , r), be a local coordinate system about q so that

yµ = yµ(x) .

• Let f : W → R be a smooth function on W.

• The pullback of f to M is a function F∗ f : M → R on M defined by

F∗ f = f F ,

that is, for any x(F∗ f )(x) = f (y(x)) .

• Suppose that n = r and the map F is bijective.

• Let h : M → R be a smooth function on M.

• The pushforward of h to W is a function F∗h : W → R on W defined by

F∗h = h F−1 ,

that is, for any y(F∗h)(y) = h(x(y)) .

• Remarks. The pullback is well defined for an arbitrary map F.

• The pushforward is only defined for bijections!

• The pullback is the map

F∗ : ΛpW → ΛpM

defined as follows.

Let α ∈ ΛpW be a p-form on W. The pullback of α is a p-form F∗α on Mdefined by: for any vectors v1, . . . , vp

(F∗α)(v1, . . . , vp) = α(F∗v1, . . . , F∗vp) .



• In local cordinates

F∗α =1p!αµ1...µp(y(x))dyµ1 ∧ · · · ∧ dyµp

=1p!∂yµ1

∂xi1· · ·

∂yµp

∂xipαµ1...µp(y(x))dxi1 ∧ · · · ∧ dxip

• In components

(F∗α)i1...ip =∂yµ1

∂xi1· · ·

∂yµp

∂xipαµ1...µp(y(x))

• Remark. The pullback is well-defined only for covariant tensors and thepushforward is well defined only for contravariant tensors.

•

Theorem 3.5.1 Let F : M → W. The pullback F∗ : ΛpW → ΛpMhas the properties:

1. F∗ is linear.

2. For any two forms α and β

F∗(α ∧ β) = (F∗α) ∧ (F∗β)

3. F∗ commutes with exterior derivative. That is, for any p-form α

F∗(dα) = d(F∗α) .

Proof : Direct calculation.


3.6. VECTOR ANALYSIS IN R3 83

3.6 Vector Analysis in R3

3.6.1 Vector Algebra in R3

• In the case of three-dimensional Euclidean space the metric in Cartesiancoordinates is gi j = δi j.

• The bases of p-forms are:

1, dx, dy, dz, dx ∧ dy, dx ∧ dz, dy ∧ dz, dx ∧ dy ∧ dz .

• The star operator acts on this forms by

∗1 = dx ∧ dy ∧ dz,

∗dx = dy ∧ dz, ∗dy = −dx ∧ dz, ∗dz = dx ∧ dy,

∗(dx ∧ dy) = dz, ∗(dy ∧ dz) = dx, ∗(dx ∧ dz) = −dy,

∗(dx ∧ dy ∧ dz) = 1 .

• So, any 2-form

α = α12dx ∧ dy + α13dx ∧ dz + α23dy ∧ dz

is represented by the dual 1-form

∗α = α12dz − α13dy + α23dx ,

• That is(∗α)i =

12εi jkα

jk

(∗α)1 = α23 , (∗α)2 = α31 , (∗α)3 = α12 ,

• Any 3-form α

α = α123dx ∧ dy ∧ dz

is represented by the dual 0-form

∗α =13!εi jkα

i jk = α123 .



• Now, let α and β be two 1-forms

α = α1dx + α2dy + α3dz , β = β1dx + β2dy + β3dz .

Then∗β = β1dy ∧ dz + β2dz ∧ dx + β3dx ∧ dz

and

α∧β = (α1β2−α2β1)dx∧dy + (α1β3−α3β1)dx∧dz + (α2β3−α3β2)dy∧dz ,

α ∧ (∗β) = (α1β1 + α2β2 + α3β3)dx ∧ dy ∧ dz .

Therefore,

∗(α ∧ β) = (α1β2 − α2β1)dz − (α1β3 − α3β1)dy + (α2β3 − α3β2)dx ,

∗[α ∧ (∗β)] = α1β1 + α2β2 + α3β3 ,

or∗(α ∧ β) = α × β ,

∗[α ∧ (∗β)] = α · β .

3.6.2 Vector Analysis in R3

• Zero-Forms.For a 0-form f we have

(d f )i = ∂i f ,

so thatd f = grad f .

• One-Forms.For a 1-form

α = α1dx + α2dy + α3dz

we have(dα)i j = ∂iα j − ∂ jαi

that is,

dα = (∂1α2 − ∂2α1)dx∧ dy + (∂2α3 − ∂3α2)dy∧ dz + (∂3α1 − ∂1α3)dz∧ dx .


3.6. VECTOR ANALYSIS IN R3 85

• Therefore(∗dα)i = εi jk∂ jαk ,

so that

∗dα = (∂2α3 − ∂3α2)dx + (∂3α1 − ∂1α3)dy + (∂1α2 − ∂2α1)dz .

• We see that∗dα = curlα .

• Two-Forms.For a 2-form β there holds

(dβ)i jk = ∂iβ jk + ∂ jβki + ∂kβi j ,

ordβ = (∂1β23 + ∂2β31 + ∂3β12)dx ∧ dy ∧ dz .

• Hence,

∗dβ =12εi jk∂iβ jk = ∂1β23 + ∂2β31 + ∂3β12 .

• Now let α be a 1-form

α = α1dx + α2dy + α3dz .

Then∗α = α1dy ∧ dz − α2dx ∧ dz + α3dx ∧ dy ,

andd ∗ α = (∂1α1 + ∂2α2 + ∂3α3)dx ∧ dy ∧ dz ,

or∗d ∗ α = ∂1α1 + ∂2α2 + ∂3α3 .

So,∗d ∗ α = divα .



3.7 Orientation and the Volume Form

3.7.1 Orientation of a Vector Space

• Let E be a vector space. Let ei = e1, . . . , en and e′j = e′1, . . . , e′n be two

different bases in E related by

ei = Λ jie′j ,

where Λ = (Λij) is a transformation matrix.

• Note that the transformation matrix is non-degenerate

det Λ , 0 .

• Since the transformation matrix Λ is invertible, then the determinant det Λ

is either positive or negative.

• If det Λ > 0 then we say that the bases ei and e′i have the same orien-tation, and if det Λ < 0 then we say that the bases ei and e′i have theopposite orientation.

• If the basis ei is continuously deformed into the basis e′j, then both baseshave the same orientation.

• Since det I = 1 > 0 and the function det : GL(n,R)→ R is continuous, thena one-parameter continuous transformation matrix Λ(t) such that Λ(0) = Ipreserves the orientation.

• This defines an equivalence relation on the set of all bases on E called theorientation of the vector space E.

• This equivalence relation divides the set of all bases in two equivalenceclasses, called the positively oriented and negatively oriented bases.

• A vector space together with a choice of what equivalence class is positivelyoriented is called an oriented vector space.


3.7. ORIENTATION AND THE VOLUME FORM 87

3.7.2 Orientation of a Manifold• Let M be a manifold and let TpM be the tangent space at a point p ∈ M.

• Let (U, x) be a local coordinate patch about a point p ∈ M.

• Then the vectors∂

∂xi , i = 1, . . . , n

form a basis in TpM.

• Let (U′, x′) be another local coordinate system about a point p, that is, thereis a local diffeomorphism xi = xi(x′).

• Then the vectors∂

∂x′i=∂x j

∂x′i∂

∂x j

form another basis in TpM.

• The orientation of the bases ∂i and ∂′j is the same (or consistent) if

det(∂xi

∂x′ j

)> 0 .

• If it is possible to choose an orientation of all tangent spaces TpM at allpoints in a continuous fashion, then the orientation of all tangent spaces isconsistent.

• A manifold M is called orientable if there is an atlas such that the ori-entation of all charts of this atlas can be chosen consistently, that is, theJacobians of all transition functions have positive determinant.

• Each connected orientable manifold has exactly two possible orientations.One orientation can be declared positive, then the other orientation is neg-ative.

• An orientable manifold with a chosen orientation is called oriented.

• Remarks.

• If a manifold can be covered by a single coordinate chart then it is ori-entable.



• Not all manifolds are orientable.

• Transport of the orientation.

• Let p, q ∈ M be two points in a manifold M and C(t) be a curve in Mconnecting p and q, i.e.

C(0) = p, C(1) = q .

• Let ei(t) be a basis in TC(t)M that continuously depends on t ∈ [0, 1].

• Then the orientation of the basis ei(1) is uniquely determined by the orien-tation of the basis ei(0).

• Thus, the orientation is transported along a curve in a unique way.

• Note that the transportation of the basis is not unique, in general. Only thetransportation of the orientation is!

• Given a point p and another point q, the orientation at the point q does, ingeneral, depend on the curve C(t) connecting the points p and q.

• If a manifold is orientable, then the transportation of the orientation fromone point to another does not depend on the curve connecting the points.

•

Corollary 3.7.1 Let M be a manifold. If there exist two points p andq in M and two curves C1(t) and C2(t) joining the points p and q suchthat the orientation at q transported from p along the curves C1 and C2

are different, then M is nonorientable.

That is, if there exists a closed curve C(t) in M such that the transportof the orientation along C leads to a reversal of orientation, then M isnonorientable.

• Example. Mobius Band.

3.7.3 Hypersurfaces in Orientable Manifolds• Let V ⊂ W be a submanifold of a manifold W.

• A vector field N on W is transverse to V if N is nowhere tangent to V .



• That is, N is transverse to V if for any point p ∈ V , Np < TpV .

• If N is transverse to V , then N , 0 on V .

• Let W be an n-diemsnional manifold. An (n − 1)-diemsnional submanifoldM of W is called a hypersurface in W.

• A hypersurface M in W is two-sided in W if there is a continuous vector Nin W transverse to M.

• Examples. Normals to surfaces in R3.

•Theorem 3.7.1 A two-sided hypersurface in an orientable manifold isorientable.

• Remarks.

• Orientability of a manifold is an intrinsic property of a manifold.

• Two-sidedness of a manifold depends on the embedding of the manifold asa hypersurface in a higher-dimensional manifold.

• Example.

• Every manifold (even a nonorientable one) M is a two-sided hypersurfacein a manifold M × R.

3.7.4 Projective Spaces• The real projective space RP2 is the sphere S 2 with antipodal points identi-

fied.

• The sphere S 2 is two-sided in R3, so it is orientable.

• Let e1, e2 be a basis in TpS 2 and N be an outward pointing normal vector toS 2.

• Then N, e1, e2 is a basis in R3.

• We say that the basis e1, e2 is positively oriented in S 2 if the basis N, e1, e2

is positive in R3.



• Letr : R3 → R3

be the reflection map defined by

r(x) = −x .

• Leta : S 2 → S 2

be the antipodal map in S 2 defined as the restriction of the reflection map toS 2

a = r∣∣∣S 2

• The reflection map reverses the orientation of the basis in R3!

• Let p ∈ S 2 be a point on S 2 and N, e1, e2 ∈ TpS 2 be a positively orientedbasis. Then −e2,−e1,−N is negatively oriented at Ta(p)S 2, where a(p) isthe antipodal point.

• The vector −N is the outward normal at the antipodal point a(p).

• Thus the basis −e1,−e2 is negatively oriented at a(p).

• Thus, if the basis e1, e2 at p ∈ S 2 is transported along a curve C on S 2 toa(p), then the orientation of the transported basis e′1, e

′2 is the opposite of

the orientation of the basis −e1,−e2 at a(p).

• Thus, the antipodal map reverses the orientation on S 2.

• In RP2 the basis −e1,−e2 at a(p) represents the same basis e1, e2 at p.

• Thus, the curve C on S 2 is a closed curve C′ on RP2 going through p.

• Thus, the transportation of the basis along the closed curve C′ in RP2 re-verses the orientation.

• Thus, RP2 is not orientable!

• More generally, the even-dimensional projective spaces RP2n are not ori-entable.

• The odd-dimensional projective spaces RP2n are orientable.



3.7.5 Pseudotensors and Tensor Densities

• Let E be an oriented vector space and B be the set of all bases on E. Thenthe orientation is a function

o : B → R

defined by

o(ei) =

+1 if ei is positively oriented

−1 if ei is negatively oriented

• A pseudo-tensor T on a vector space E assigns, for each orientation o ofE a tensor To such that when the orientation is reversed the tensor changessign, i.e.

T−o = −To .

• That is, a pseudo-tensor is a collection of two tensors T+ and T−, one foreach orientation.

• A pseudo-tensor field on a manifold is a smooth assignment of a pseudo-tensor to each point of the manifold.

• Let xiα = xi

α(xβ) be a local diffeomorphism and

J(xβ) = det

∂xiα

∂x jβ

.• Since this transformation is a diffeomorphism J , 0. Thus, there are two

cases J > 0 and J < 0.

• A pseudo-tensor of type (p, q) is a geometric object T whose componentsT i1...ip

j1... jqin the coordinate basis ∂i and dxi transform according to

T(α)i1...ip

j1... jq(xα) = sign (J)

n∑k1,...,kp=1

n∑l1,...,lq=1

∂xi1α

∂xk1β

· · ·∂xip

α

∂xkp

β

∂xl1β

∂xlqα

· · ·∂x j1

β

∂x jqα

T(β)k1...kp

l1...lq(xβ)



• A tensor density of weight w of type (p, q) is a geometric object T whosecomponents T i1...ip

j1... jqin the coordinate basis ∂i and dxi transform under a dif-

feomorphism xiα = xi

α(xβ) with J > 0 according to

T(α)i1...ip

j1... jq(xα) = J−w(xβ)

n∑k1,...,kp=1

n∑l1,...,lq=1

∂xi1α

∂xk1β

· · ·∂xip

α

∂xkp

β

∂xl1β

∂xlqα

· · ·∂x j1

β

∂x jqα

T(β)k1...kp

l1...lq(xβ)

•

Theorem 3.7.2 Let g = (gi j) be a Riemannian metric and

|g| = det(gi j) .

Then√|g| is a scalar density of weight 1.

Proof :

1. Let x = x(x′) be a local diffeomorphism.

2. Then

g′i j =∂xk

∂x′i∂xl

∂x′ jgkl ,

3. Then

|g′| = det(∂x∂x′

)2

|g| .

4. Thus √|g′| = det

(∂x∂x′

) √|g| .

5. Therefore,√|g| is a scalar density of weight 1.

• The Levi-civita symbol are not tensors!



•

Theorem 3.7.3 Let gi j be the components of a Riemannian metric, |g| =det(gi j), and εi1...in and εi1...in be the Levi-Civita symbols and Ei1...in andEi1...in be defined by

Ei1...in =√|g| εi1...in , Ei1...in =

1√|g|

εi1...in

Then

1. εi1...in represents the components of a pseudo-n-form (that is, apseudo-tensor density of type (0, n)) of weight (−1).

2. εi1...in represents the components of a pseudo-n-vector (that is, apseudo-tensor density of type (n, 0)) of weight 1.

3. Ei1...in represents the components of a pseudo-n-form.

4. Ei1...in represents the components of a pseudo-n-vector.

Proof :

1. Check the transformation law.


Chapter 4

Integration of Differential Forms

4.1 Integration over a Parametrized Subset

4.1.1 Integration of n-Forms in Rn

• Let U ⊂ Rn be a closed ball in Rn and ui, i = 1, . . . , n be the Cartesiancoordinates in Rn.

• Let f : U → R be a continuous real-valued function on U.

• Then the integral of f over U is the multiple integral∫U

f =

∫U

f (u) du1 · · · dun .

• Let o be an orientation of U so that o(u) = +1 if the coordinate basis of cov-ectors (du1, . . . , dun) has the same orientation as o and o(u) = −1 otherwise.

• Letα = f (u) du1 ∧ · · · ∧ dun

be an n-form.

• Then the integral of α over U is defined by∫Uα = o(u)

∫U

f (u) du1 . . . dun .

• The integral of the form α over U reverses sign if the orientation of U isreversed.

95

96 CHAPTER 4. INTEGRATION OF DIFFERENTIAL FORMS

4.1.2 Integration over Parametrized Subsets• Let M be an n-dimensional manifold with local coordinates xi, i = 1, . . . , n.

• Let 0 ≤ p ≤ n and U be an oriented region in Rp with orientation o andcoordinates uµ, µ = 1, . . . , p.

• Let F : U → M be a smooth map given locally by

xi = F i(u) .

• Then the image F(U) ⊂ M of the set U is called a p-subset of M and thecollection (U, o, F) is called an oriented parametrized p-subset of M.

• A parametrized 1-subset is called a curve in M.

• A parametrized 2-subset is called a surface in M.

• Remarks.

• A p-subset is not a submanifold, in general.

• A p-subset could have different dimensions at different points.

• Usually, the differential F∗ has rank p (that is, F(U) is a submanifold) al-most everywhere.

• Let α ∈ Λp be a p-form on M

α =1p!αi1...ip dxi1 ∧ · · · ∧ dxip .

• The integral of α over an oriented parametrized p-subset F(U) is definedby ∫

F(U)α =

∫U

F∗α .

• In more detail,∫F(U)

α = o(u)∫

U(F∗α)

(∂

∂u1 , . . . ,∂

∂up

)du1 · · · dup

= o(u)∫

Uα

(F∗

∂

∂u1 , . . . , F∗∂

∂up

)du1 · · · dup

=1p!

o(u)∫

Uαi1...ip(x(u))

∂xi1

∂u1 · · ·∂xip

∂up du1 · · · dup .


4.1. INTEGRATION OVER A PARAMETRIZED SUBSET 97

4.1.3 Line Integrals• Let U = [a, b] ⊂ R be an interval.

• Then a map F : U → M defines an oriented curve C = F(U) in M

xi = F i(t) .

• Let α be a 1-form in Mα = αi(x) dxi .

• Then the integral of α over C is called the line integral.

• In more detail ∫Cα =

∫ b

aα

[F∗

(ddt

)]=

∫ b

aαi(x(t))

dxi

dtdt .

4.1.4 Surface Integrals• Let U ⊂ R2 be an oriented region in the plane, for example, U = [a, b] ×

[c, d].

• Then a map F : U → M defines an oriented parametrized surface S = F(U)in M

xi = F i(u1, u2) .

• Let α be a 2-form in M

α =12αi j(x) dxi ∧ dx j .

• Then the integral of α over C is called the surface integral.

• In more detail∫Sα =

∫Uα

[F∗

(d

du1

), F∗

(d

du2

)]du1 du2 =

∫U

12αi j(x(u))

dxi

du1

dx j

du2 du1 du2 .

• Example in R3.



4.1.5 Independence of Parametrization

• Let U,V ⊂ Rp be regions in Rp with coordinates uµ, µ = 1, . . . , p, and vν,ν = 1, . . . , p, respectively.

• Let H : U → V be an diffeomorphism so that V = H(U) defined by

vµ = Hµ(u) .

• Then for any function f : V → R there holds a formula for the change ofvariables in multiple integrals∫

Vf (v) dv1 . . . dvp =

∫U

f [v(u)]

∣∣∣∣∣∣ ∂(v1, . . . , vp)∂(u1, . . . , up)

∣∣∣∣∣∣ du1 . . . dup

• Let M be an n-dimensional manifold with n ≥ p.

• Let U and V be oriented regions, uµ and vν be positively-oriented coordi-nates on U and V , and the diffeomorphism H be orientation-preserving, thatis, the Jacobian

∂(v1, . . . , vp)∂(u1, . . . , up)

> 0

is positive.

• Let F : U → M and G : V → M be a smooth maps so that

F = G H .

• Then F(U) is an oriented parametrized p-subset of M and G is a reparametriza-tion of this subset

xi = F i(u) = Gi(H(u)) .

• Let α ∈ Λp be a p-form on M

α =1p!αi1...ip(x) dxi1 ∧ · · · ∧ dxip


4.1. INTEGRATION OVER A PARAMETRIZED SUBSET 99

• Then∫G(V)

α =

∫V

G∗α

=1p!

∫Vαi1...ip(G(v))

∂xi1

∂v1 · · ·∂xip

∂vp dv1 · · · dvp

=1p!

∫Uαi1...ip(G(H(u)))

∂xi1

∂v1 · · ·∂xip

∂vp

∂(v1, . . . , vp)∂(u1, . . . , up)

du1 · · · dup

=1p!

∫Uαi1...ip(F(u))

∂xi1

∂u1 · · ·∂xip

∂up du1 · · · dup

=

∫Uα

• Thus, the integral is independent of the parametrization of a p-subset.

4.1.6 Integrals and Pullbacks• Let M be an n-dimensional manifold and W be an r-dimensional manifold.

• Let ϕ : M → W be a smooth map.

• Let U ⊂ Rp be an oriented region in Rn and F : U → M be an orientedparametrized p-subset of M.

• Then ψ = ϕ F : U → W is an oriented parametrized p-subset of W.

• Let α ∈ ΛpW be a p-form on W.

• Then ∫ψ(U)

α =

∫Uψ∗α =

∫U

(F∗ ϕ∗)α =

∫U

F∗(ϕ∗α) =

∫F(U)

ϕ∗α

• Let S = F(U) be an oriented subset of M. Then ψ(U) = ϕ(F(U)) = ϕ(S ) isan oriented subset of W.

• The general pullback formula takes the form∫ϕ(S )

α =

∫Sϕ∗α .



• Remarks.

• We have only defined integrals of forms over subsets of M covered by asingle coordinate chart.

• We need to define the integrals over general submanifolds, covered by mul-tiple charts.


4.2. INTEGRATION OVER MANIFOLDS 101

4.2 Integration over Manifolds

4.2.1 Partition of Unity• Let M be a manifold, p0 ∈ M be a point in M and (U, x) be a local coordinate

chart about p0.

• Let xi = xi(p) be the local coordinates of the point p and xi0 = xi(p0) be the

local coordinates of the point p0.

• Let

||x − x0|| =

√√n∑

i=1

(xi − xi0)2

• A neighborhood of p0 is a subset of M defined by

Bε(p0) = p ∈ M | ||x − x0|| < ε .

• Every neighborhood of a point in M is an open set in M.

• Let A ⊂ M be a subset of M. A point p ∈ M is called an accumulationpoint (or a limit point) of A if every neighborhood of p contains at leastone point in A other than p.

• A subset of M is closed if and only if it contains all of its limit points.

• A closure of A, denoted by A, is a set obtained by adding to A all itsaccumulation points.

• A closure of any set is a closed set.

• A function f : M → R is continuous if the inverse image of every open setin R is open in M.

• Obviously, the set R \ 0 of non-zero real numbers is open.

• Thus, the subset of M where f is not equal to zero, that is, the set f −1(R\0),is open.

• The support of f is the closure of the set f −1(R \ 0),

supp f = f −1(R \ 0) .



• Thus, f vanishes outside its support supp f .

• There could be points in supp f where f vanishes.

• A bump function is a smooth function ϕε : R→ R such that

0 ≤ ϕε(t) ≤ 1 ∀t ∈ R

and

ϕε(t) =

1, if |t| < ε4

0, if |t| > ε2

• Similarly, we define support of any tensor field.

• For example, an n-form ω ∈ Λn on M defined by

ω = ϕε(||x − x0||) dx1 ∧ · · · ∧ dxn

has a support inside the neighborhood Bε(p0),

suppω ⊂ Bε(p0) .

Such an n-form is called a bump form.

• Let UαNα=1 be a (finite) atlas for the manifold M.

• A partition of unity is a set ϕαNα=1 of functions ϕα : M → R with theproperties

0 ≤ ϕα(p) ≤ 1 ∀α, p ∈ Msuppϕα ⊂ Uα ∀α

N∑α=1

ϕα = 1

• Notice that for any α, suppϕα is closed and ϕα vanishes outside Uα.

• A general theorem from analysis says that every manifold has a partition ofunity.

• Example.


4.2. INTEGRATION OVER MANIFOLDS 103

4.2.2 Integration over Submanifolds

• A manifold M is compact if every open cover of M has a finite subcover.

• Thus, very compact manifold has a finite atlas.

• A subset of Rn is compact if it is closed and bounded.

• Let V be a p-dimensional compact oriented manifold.

• Let UαNα=1 be a finite atlas of V .

• Let each Uα be positively oriented.

• Let ϕα be a partition of unity on V .

• Let β be a p-form over V .

• The integral of β over V is defined by

∫Vβ =

N∑α=1

∫Uα

ϕαβ

• This integral does not depend on the atlas and the partition of unity.

• Now, let V be a p-dimensional compact oriented submanifold of an n-dimensional manifold M described by the inclusion map

i : V → M .

• Let β ∈ ΛpM be a p-form on M.

• The integral of a p-form β on M over V ⊂ M is defined by∫Vβ =

∫V

i∗β .



4.2.3 Manifolds with boundary• Recall that an open ball in Rn is the set

Bε(x0) = x ∈ Rn | ||x − x0|| < ε

• Let us consider also sets

Hε(x0) = x ∈ Rn | ||x − x0|| < ε, xn − xn0 ≥ 0 .

Such sets are called half-open.

• A n-dimensional manifold with boundary consists of the the interior Mo

and the boundary ∂M.

• The interior Mo is a genuine n-dimensional manifold such that all its pointshave neighborhoods diffeomorphic to open balls in Rn.

• The boundary ∂M is a subset of M such that all its points have neighbor-hoods diffeomorphic to half-open sets.

• Usually, the boundary ∂M is itself an (n − 1)-dimensional submanifold ofM (without boundary).

• Boundary may be disconnected. It can also be not smooth.

• Local coordinates (x1, . . . , xn−1, xn) in M in the neighborhoods of points onthe boundary can always be chosen in such a way that (x1, . . . , xn−1) are thecoordinates along the boundary and 0 ≤ xn < δ with some δ.

• A compact manifold is a manifold which is closed and bounded (say, as asubmanifold of some RN).

• A closed manifold is a manifold which is compact and does not have aboundary.


4.3. STOKES’S THEOREM 105

4.3 Stokes’s Theorem

4.3.1 Orientation of the Boundary• Let M be an n-dimensional orientable manifold with boundary ∂M, which

is an (n − 1)-dimensional manifold without boundary.

• Let M be oriented.

• Then an orientation on M naturally induces an orientation on ∂M.

• Let p ∈ ∂M and e2, . . . , en be a basis in Tp∂M.

• Let N ∈ TpM be a tangent vector at p that is transverse to ∂M and pointsout of M.

• Then N, e2, . . . , en forms a basis in TpM.

• Then, by definition, the basis e2, . . . , en has the same orientation as thebasis N, e2, . . . , en. That is, e2, . . . , en is positively oriented in ∂M ifN, e2, . . . , en is positively oriented in M.

4.3.2 Stokes’ Theorem

•

Theorem 4.3.1 Let M be an n-dimensional manifold and V be a p-dimensional compact oriented submanifold with boundary ∂V in M. Letω ∈ Λp−1M be a smooth (p − 1)-form in M. Then∫

Vdω =

∫∂Vω .

Proof :

1. Leti : V → M

be the inclusion map of the submanifold V .

2. Then ∫V

dω =

∫V

i∗(dω) =

∫d(i∗ω)

and ∫∂Vω =

∫∂V

i∗ω



3. Letβ = i∗ω

4. Then we need to show ∫V

dβ =

∫∂Vβ .

5. Let Vα be a finite cover of V and ϕα be the corresponding partitionof unity.

6. We have ∫V

dβ =∑α

∫Vα

d(ϕαβ)

and ∫∂Vβ =

∑α

∫∂Vϕαβ .

7. Thus, we need to show that∫Vα

d(ϕαβ) =

∫∂Vϕαβ .

8. The charts Vα can be of two different kinds: I) those that are fully inthe interior of V , that is, Vα ∩ ∂V = ∅, which are diffeomorphic to theopen balls in Rp, and II) those which contain points of the boundary,which are diffeomorphic to half-open balls in Rp.

9. Case I. Let Vα be open sets in V fully in the interior of V .10. Let Uα ⊂ Rp be the open sets in Rp such that fα : Uα → Vα be the

local coordinate diffeomorphisms.11. Then ∫

Vαd(ϕαβ) =

∫Uα

f ∗α (d(ϕαβ)) =

∫Uα

d( f ∗α (ϕαβ))

12. Letγα = f ∗α (ϕαβ) .

13. Then

γα =

p∑i=1

(−1)i−1γα,idx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxp

and

dγα =

p∑i=1

∂γα,i

∂xi dx1 ∧ · · · ∧ dxp



14. Therefore,∫Uα

dγα =

p∑i=1

∫Uα

∂γα,i

∂xi dx1∧· · ·∧dxp =

p∑i=1

∫Rp

∂γα,i

∂xi dx1∧· · ·∧dxp = 0 .

15. Hence ∫Vα

d(ϕαβ) = 0 .

16. Also, since Uα is disjoint from the boundary∫∂Vϕαβ = 0 .

17. Thus, for each chart disjoint from the boundary∫Vα

d(ϕαβ) =

∫∂Vϕαβ = 0 .

18. Case II. Now, let us consider the half-open charts Vα at the boundary.

19. Let Uα ⊂ Rp be the half-open sets in Rp such that fα : Uα → Vα be the

local coordinate diffeomorphisms.

20. LetWα = Vα ∩ ∂V

andYα = f −1

α (Wα) .

21. Notice that for any point on Yα, xp = 0.

22. Then ∫Vα

d(ϕαβ) =

p∑i=1

∫Rp

∂γα,i

∂xi dx1 ∧ · · · ∧ dxp

23. We have ∫ ∞

−∞

∂γα,i

∂xi dxi = 0

for any i , p, and ∫ ∞

0

∂γα,p

∂xp dxp = −γα,p

∣∣∣∣xp=0

.



24. Therefore,∫Vα

d(ϕαβ) = −

∫Rp−1

γα,p(x1, . . . , xp−1, 0) dx1 · · · dxp

25. Further, ∫∂Vϕαβ =

∫Wα

ϕαβ =

∫Yα

f ∗α (ϕαβ) =

∫Yαγα

26. Since on Yα, xp = 0, then dxp = 0. Therefore,∫Yαγα = (−1)p−1γα,p(x1, . . . , xp−1, 0)dx1 ∧ · · · ∧ dxp−1

27. We have that ∂1, . . . , ∂p is positively oriented on V and −∂p is theoutward normal to ∂V .

28. Thus, ∂1, . . . , ∂p−1 has orientation (−1)p on ∂V .

29. Thus ∫∂Vϕαβ = −

∫Yαγα,p(x1, . . . , xp−1, 0)dx1 · · · dxp−1

30. So, ∫Vα

d(ϕαβ) =

∫∂Vϕαβ

which establishes the theorem.

• Examples.

• Newton formula. Let C be an oriented curve, ∂C = Q − P be its boundaryand f be a smooth real-valued function. Then∫

Cd f =

∫∂C

f = f (Q) − f (P)

• Green formula. Let S be an oriented surface in a 2-dimensional manifoldM with parameters (u1, u2) and ∂S be its boundary with the induced ori-entation. Let U be the preimage of S on the plane R2 with the boundary∂U = [a, b]. Let A be a one-form in M. Then∫

U

(∂A2

∂x1 −∂A1

∂x2

)∂(x1, x2)∂(u1, u2)

du1 ∧ du2 =

∫ b

a

(A1

dx1

dt+ A2

dx2

dt

)dt



• Gauss formula. Let D be a region in a 3-dimensional manifold M and Ube the parameter preimage of D in R3. Let ∂D be the boundary of D and ∂Ube the boundary of U. Let F be a 2-form in M. Then∫

U

(∂F12

∂x3 +∂F23

∂x1 +∂F31

∂x2

)∂(x1, x2, x3)∂(u1, u2, u3)

du1 ∧ du2 ∧ du3

=

∫∂U

(F12

∂(x1, x2)∂(z1, z2)

+ F23∂(x2, x3)∂(z1, z2)

+ F31∂(x3, x1)∂(z1, z2)

)dz1 ∧ dz2

• Stokes’ formula. Let S be a surface in a 3-dimensional manifold M and∂S be its boundary. Let U be the preimage of S in the parameter plane R2

and ∂U = [a, b] be its boundary. Let A be a 1-form in M. Then∫U

(∂A2

∂x1 −∂A1

∂x2

)∂(x1, x2)∂(u1, u2)

+

(∂A3

∂x1 −∂A1

∂x3

)∂(x1, x3)∂(u1, u2)

+

(∂A3

∂x2 −∂A2

∂x3

)∂(x2, x3)∂(u1, u2)

du1 ∧ du2

=

∫ b

a

(A1

dx1

dt+ A2

dx2

dt+ A3

dx3

dt

)dt



4.4 Poincare Lemma

4.4.1 Review• Manifolds.

• Vector fields

• Differential forms.

• Maps

• Differential of a map

• Pullback of a map

• Flow of a vector field.

• Exterior derivative.

• Submanifolds.

• Let M be an n-dimensional manifold and W be an r-dimensional submani-fold of M. It can be described explicitly or implicitly.

• Implicit description. Let x = (x1, . . . , xn) be the local coordinates in aneighborhood of a point p ∈ M. Then the submanifold W can be describedby a map F : M → W as the common locus of (n − r) differentiable inde-pendent functions

Fα(x) = 0 , α = 1, . . . , n − r ,

such that the Jacobian matrix has the maximal rank equal to (n − r) at eachpoint of the locus, that is,

rank(∂Fα

∂xi (x))

= n − r , ∀x ∈ W.

• Explicit description. Let y = (y1, . . . , yr) be local coordinates in a neigh-borhood of a point q ∈ W. Then the submanifold W can be described by amap f : W → M such that p = f (q) and

xi = f i(y), i = 1, . . . , n.


4.4. POINCARE LEMMA 111

• The number (n − r) is called the codimension of the submanifold W.

• The r vectors eµ, µ = 1, . . . , r with the components

eiµ =

∂xi

∂yµ

are tangent vector to the submanifold W and form the basis of the tangentspace TqW.

• Manifolds with boundary.

• The boundary of a manifold M with local coordinates xi is a co-dimension1 submanifold.

• Orientation of the boundary.

• Let M be an n-dimensional orientable manifold with boundary ∂M, whichis an (n − 1)-dimensional manifold without boundary.

• Let v(1), . . . , v(n−1) be an ordered (n− 1)-tuple of linearly independent vec-tors and

N i = gik√|g|εki1...in−1v

i1(1) · · · vin−1

(n−1)

Then the vector N is orthogonal to all vectors v(1), . . . , v(n−1), and the n-tuple N, v1, . . . , vn−1 forms a positively oriented basis.

• Let M be oriented.

• Then an orientation on M naturally induces an orientation on ∂M.

• Let p ∈ ∂M and e2, . . . , en be a basis in Tp∂M.

• Let N ∈ TpM be a tangent vector at p that is transverse to ∂M and pointsout of M.

• Then N, e2, . . . , en forms a basis in TpM.

• Then, by definition, the basis e2, . . . , en has the same orientation as thebasis N, e2, . . . , en. That is, e2, . . . , en is positively oriented in ∂M ifN, e2, . . . , en is positively oriented in M.



• Let V be a p-dimensional compact oriented manifold with boundary ∂V andω ∈ Λp−1V be a smooth (p − 1)-form in V . Then∫

Vdω =

∫∂Vω .

4.4.2 Poincare Lemma

•

Definition 4.4.1 Let M be a manifold.A p-form α on M is called closed if dα = 0.

A p-form α on M is called exact if there is a (p − 1)-form β such thatα = dβ. The form β is called a potential of α.

•

Theorem 4.4.1

1. Every exact form is closed (Poincare Lemma).That is, if α = dβ, then dα = 0.

2. The exterior product of closed forms is closed.That is, if dα = 0 and dβ = 0, then d(α ∧ β) = 0.

3. The exterior product of a closed form and an exact form is exact.That is, if dα = 0 and β = dγ, then there isσ such that α∧β = dσ.

4. Let M be an n-dimensional orientable compact manifold withoutboundary and α be an exact n-form on M, that is, α = dβ forsome (n − 1)-form β. Then ∫

Mα = 0 .

5. Let M be an n-dimensional oriented compact manifold withboundary ∂M and α be a closed (n−1)-form on M, that is, dα = 0.Then ∫

∂Mα = 0 .

Proof : Use Stokes theorem.



• Example. On R2 − 0

β =xdy − ydx

x2 + y2

Why is β , dθ?

Thendβ = 0∮

Cdβ = 2π

if 0 is inside C and 0 if 0 is outside C.

•

Definition 4.4.2 Let M be a manifold. Suppose that for every closedoriented smooth curve C there is a smooth oriented 2-dimensional sur-face S and a map F : S → M such that ∂F(S ) = C, that is, the curve Cis the boundary of the surface S . Then the manifold M is said to havefirst Betti number equal to zero, B1 = 0.

• Example. For T 2

B1 , 0.

•

Theorem 4.4.2 Let M be a manifold with first Betti number equal tozero. Then every closed 1-form on M is exact. That is, if α is a 1-formsuch that dα = 0, then there is a function f such that α = d f .

Proof :

1. Let α be a closed one-form on M.

2. Let x and y be two points in M and Cxy be an oriented curve with theinitial point y and the final point x.

3. Let f be defined by

f (x) =

∫Cxy

α .

4. Then f is independent on the curve Cxy and so is well defined.

5. Finally, we show thatd f = α .



•

Theorem 4.4.3 Let α ∈ λ1M. Suppose that for any closed curve C∮Cα = 0.

Then α is exact.

•

Theorem 4.4.4 Let α be a closed p-form in Rn. Then there is(p − 1)-form β in Rn such that α = dβ.

That is, every closed form in Rn is exact.

Proof :

1. Let α be a closed p-form in Rn.

2. We define a (p − 1)-form β by

βi1...ip−1(x) =

∫ 1

0dτ τp−1 x jα ji1...ip−1(τx)

3. We can show thatdβ = α .

•

Corollary 4.4.1 Let M be a manifold and α be a closed p-form onM. Then for every point x in M there is a neighborhood U of x and a(p − 1)-form β on M such that α = dβ in U.

Proof :

1. Use the fact that a sufficiently small neighborhood of a point in M isdiffeomorphic to an open ball in Rn.

2. Pullback the form α from M to Rn by the pullback F∗ of the diffeo-morphism F : V → U, where U ⊂ M and V ⊂ Rn.

3. Use the previous theorem.



4.4.3 Complex Analysis

• Let M = C2 andz = x + iy,

Thendz = dx + idy, dz = dx − idy

∂

∂z=

12

(∂

∂x− i

∂

∂y

)∂

∂z=

12

(∂

∂x+ i

∂

∂y

)and

dz ∧ dz = −2idx ∧ dy.

• Let f = u + iv be a function. Then

f dz = udx − vdy + i(udy + vdx)

and

d( f dz) = (−uy − vx)dx ∧ dy + i(ux − vy)dx ∧ dy =∂ f∂z

dz ∧ dz

• So, ∫C

f dz =

∫C

[udx − vdy + i(udy + vdx)

]=

∫S

d( f dz)

• Thus, f dz is closed if and only if f is holomorphic (satisfies cauchy-Riemannequations)

ux = vy, uy = −vx

or∂ f∂z

= 0

• The system of pde

∂iA j − ∂ jAi = Fi j (dA = F)

can be solved if and only if

∂[kFi j] = 0 (dF = 0) .


Chapter 5

Lie Derivative

5.1 Lie Derivative of a Vector Field

5.1.1 Lie Bracket• Let M be a manifold.

• Let X be a vector field on M.

• Let ϕt : M → M be the flow generated by X.

• Let x ∈ M. Then ϕt(x) is the point on the integral curve of the vector fieldX going through x and such that

ϕ0(x) = x

anddϕt(x)

dt= Xϕt(x) .

• Let ϕt∗ : TxM → Tϕt(x)M be the differential of the diffeomorphism ϕt.

• Notice thatϕ0 ∗ = Id

is the identity andϕ−t∗ = (ϕt∗)−1

is the inverse transformation.

117

118 CHAPTER 5. LIE DERIVATIVE

• In local coordinates for small t we have

ϕit(x) = xi + tXi(x) + O(t2) ,

so that

(ϕt∗)ij =

∂ϕit(x)∂x j = δi

j + t∂Xi(x)∂x j + O(t2)

• Thus (ddtϕt∗

)i

j

∣∣∣∣∣∣t=0

=∂Xi(x)∂x j .

• Let f be a smooth function on M.

• The flow ϕt naturally defines a new function

(ϕt∗ f )(x) = ( f ϕt)(x) = f (ϕt(x)) .

• Then for small tf ϕ = f + tX( f ) + O(t2) .

• Thusddt

( f ϕt)∣∣∣∣t=0

= X( f ) .

• Let Y be another vector field on M.

• Thenddt

Y(( f ϕt))∣∣∣∣t=0

= Y(X( f )) .

• Then at the point ϕt(x) we have two different well defined vectors, Yϕt(x) andϕt∗Yx.

• Diagram.

•

Definition 5.1.1 A vector field Y is invariant under the flow ϕt gener-ated by a vector field X if

Yϕt(x) = ϕt∗Yx .

An invariant vector field Y is also called a Jacobi field.


5.1. LIE DERIVATIVE OF A VECTOR FIELD 119

•

Definition 5.1.2 The Lie derivative of the vector field Y with respectto the vector field X is the vector field LXY defined at a point x by

(LXY)x = limt→0

1t

(Yϕt(x) − ϕt∗Yx

)• Remark. Notice that this can also be written as

(LXY)x = limt→0

1t

(ϕ−t∗Yϕt(x) − Yx

)or

(LXY)x =ddt

(ϕ−t∗Yϕt(x)

) ∣∣∣∣t=0.

•Proposition 5.1.1

LXY = [X,Y] .

Proof :

1. We compute in local coordinates

(LXY)i =ddt

(ϕ−t∗Yϕt(x)

)i ∣∣∣∣t=0

=ddt

[(ϕ−t∗))i

jY j(ϕt(x))] ∣∣∣∣

t=0

=ddt

(ϕ−t∗))ij

∣∣∣∣∣∣t=0

Y j(x) + δijddt

Y j(ϕt(x))

∣∣∣∣∣∣t=0

= −∂Xi

∂x j Y j(x) +∂Y i

∂x j X j(x)

= [X,Y]i .

•

Definition 5.1.3 The Lie bracket of two vector fields X and Y is avector field [X,Y] such that for any smooth function f on M

[X,Y] = X(Y( f )) − Y(X( f )) .

• Notice that[X,Y] = −[Y,X] .



• In particular,LXX = 0 .

• In local coordinates the Lie bracket is given by

[X,Y]i = X j∂ jY i − Y j∂ jXi .

• The linear ordinary differential equation

[X,Y]i =dY i(t)

dt− (∂ jXi)(ϕt(x))Y j(t) = 0

for Y i(t) is called the Jacobi equation. For a given vector field X and giveninitial conditions for Y i it defines a unique Jacobi field along the flow ϕt.

• In particular,L∂i∂ j = 0 .

5.1.2 Flow generated by the Lie Bracket

•

Theorem 5.1.1 Let M be a manifold. Let X and Y be vector fields onM and ϕX

t and ϕYt be the flows generated by X and Y respectively. Let

σt : M → M be a diffeomorphism defined by

σt = ϕY−t ϕ

X−t ϕ

Yt ϕ

Xt .

Let f be a smooth function on M. Then

[X,Y]x( f ) = limt→0

1t2

[f (σt(x)) − f (x)

]=

ddt

f (σ√t)

∣∣∣∣∣∣t=0

.

which means

[X,Y] =ddtσ√t

∣∣∣∣∣∣t=0

.

Proof :

1. Diagram.


5.1. LIE DERIVATIVE OF A VECTOR FIELD 121

2. Use Taylor expansion in local coordinates.

•

Corollary 5.1.1 Let M be a manifold and W be a submanifold of M.Let X and Y be vector fields on M tangent to W. Then the Lie bracket[X,Y] is also tangent to W.



5.2 Lie Derivative of Forms and Tensors• Let X be a vector field on a manifold M.

• Let ϕt : M → M be the flow generated by X and ϕ∗t : Tϕt(x)M → TxM bethe corresponding pullback.

•

Definition 5.2.1 Let f be a function (0-form) on M. Then the Liederivative of f with respect to X is a function LX f defined by

(LX f )x =ddt

(ϕ∗t f )x

∣∣∣∣t=0

=ddt

f (ϕt(x))∣∣∣∣t=0.

•

Proposition 5.2.1 The Lie derivative of a function f with respect to avector field X is equal to

LX f = X( f ) .

• In local coordinatesLX f = Xi∂i f .

•

Definition 5.2.2 Let α be a 1-form on M. The Lie derivative of α withrespect to X is a 1-form LXα defined by

(LXα)x = limt→0

1t

(ϕ∗tαϕt(x) − αx

)=

ddt

(ϕ∗tα

)x

∣∣∣∣t=0

• We can immediately generalize this to p-forms.

•

Definition 5.2.3 Let α be a p-form on M. The Lie derivative of α withrespect to X is a p-form LXα defined by

(LXα)x = limt→0

1t

(ϕ∗tαϕt(x) − αx

)=

ddt

(ϕ∗tα

)x

∣∣∣∣t=0


5.2. LIE DERIVATIVE OF FORMS AND TENSORS 123

•

Proposition 5.2.2 The Lie derivative of a p-form α with respect to avector field X is given by

(LXα)i1...ip = X j∂ jαi1...ip + α ji2...ip∂i1 X j + · · · + αi1...ip−1 j∂ip X j

Proof :

1. Use that

(ϕ∗tα)i1...ip(x) =∂ϕ

j1t (x)∂xi1

· · ·∂ϕ

jpt (x)∂xip

α j1... jp(ϕt(x))

• In particular, for a 1-form α we have

(LXα)i = X j∂ jαi + α j∂iX j .

• More generally, since the flow ϕt : M → M is a diffeomorphism it naturallyacts on general tensor bundles of type (p, q), that is,

ϕ∗t : (T pq )ϕt(x)M → (T p

q )xM

For a general tensor field T of type (p, q) on M, ϕ∗t T is a tensor field oftype (p, q) defined by

(ϕ∗t T )k1...kp

i1...iq(x) =

∂ϕj1t (x)∂xi1

· · ·∂ϕ

jqt (x)∂xiq

∂xk1

∂ϕm1t (x)

· · ·∂xkp

∂ϕmpt (x)

T m1...mp

j1... jq(ϕt(x))

•

Definition 5.2.4 Let T be a tensor field of type (p, q) on M. The Liederivative of T with respect to X is a tensor field LXT of type (p, q)defined by

(LXT )x = limt→0

1t

(ϕ∗t Tϕt(x) − Tx

)=

ddt

(ϕ∗t T

)x

∣∣∣∣t=0

•

Proposition 5.2.3 The Lie derivative of a tensor field T of type (p, q)with respect to a vector field X is given in local coordinates by

(LXT )k1...kp

i1...iq= X j∂ jT

k1...kp

i1...iq+ T k1...kp

ji2...iq∂i1 X j + · · · + T k1...kp

i1...iq−1 j∂iq X j

−T jk2...kp

i1i2...iq∂ jXk1 − · · · − T k1...kp−1 j

i1...iq∂ jXkp



Proof :

1. Use the definition of the action ϕ∗t .

• In particular, for a tensor gi j of type (0, 2) we obtain

(LXg)i j = Xk∂kgi j + gik∂ jXk + gk j∂iXk .

If gi j is a Riemannian metric, then this tensor is called the strain tensor.

5.2.1 Properties of Lie Derivative

•

Theorem 5.2.1 For any two tensors T and R and a vector field X theLeibnitz rule holds

LX(T ⊗ R) = (LXT ) ⊗ R + T ⊗ (LXR)

Proof :

1. Use the Leibnitz rule for

ddt

[ϕ∗t (T ⊗ R)

]=

ddt

[(ϕ∗t T ) ⊗ (ϕ∗t R)

]

•

Theorem 5.2.2 Let α be a p-form, β be a q-form and X be a vectorfield on M. Then the Leibnitz rule holds

LX(α ∧ β) = (LXα) ∧ β + α ∧ (LXβ)

Proof :

1. Follows from the previous theorem and the definition of the wedgeproduct.

•

Theorem 5.2.3 For any 1-form ω and vector fields X and Y the Leib-nitz rule holds

LX(ω(Y)) = (LXω)(Y) + ω(LXY)

Proof :



1. Computation in local coordinates.

• The Lie derivative of a 1-form ω with respect to a vector field X can bedefined in an intrinsic way. LXω is a 1-form whose value on any vector fieldY is

(LXω)(Y) = X(ω(Y)) − ω([X,Y]) .

• More generally, we have

Theorem 5.2.4 Let X and Y1, . . . ,Yp be vector fields on a manifoldM and α ∈ Λp be a p-form. Then

LX(α(Y1, . . . ,Yp)

)= (LXα)(Y1, . . . ,Yp)

+

p∑i=1

α(Y1, . . . , LXYi, . . . ,Yp) .

Proof : By induction or direct calculation.

• Remark. This can be used as an intrinsic definition of LXα.

•

Theorem 5.2.5 Let X and Y be any two vector fields and c ∈ R. Then

1. LX+Y = LX + LY,

2. LcX = cLX,

3. LXY = −LYX,

4. [LX, LY] = L[X,Y].

Proof :

1.



•

Theorem 5.2.6 Let gi j be a Riemannian metric on an n-dimensionalmanifold M and

vol =√|g|dx1 ∧ · · · ∧ dxn

be the Riemannian volume form. Let X be a vector field on M. Then

LXvol = ( div X)vol ,

where div X is a scalar function defined by

div X = ∗LXvol =1√|g|∂i

( √|g|Xi

).

Proof :

1. Direct calculation. Use LXgi j.

• The scalar div X is called the divergence of the vector field X.

• Remark. Let Y1, . . . ,Yn be vector fields invariant under the vector field X.Then

div X =(LXvol )(Y1, . . . ,Yn)

vol (Y1, . . . ,Yn)

=ddt

log vol (Y1, . . . ,Yn)

∣∣∣∣∣∣t=0

.

• Thus, div X is the logarithmic rate of change of the volume along the flow.

• Remark. Let β ∈ Λn−1 be an (n − 1)-form defined by

β = iXvol .

In componentsβi1...in−1 = X j

√|g|ε ji1...in−1

Thendβ = ( div X)vol .

To prove this compute in local coordinates

(dβ)i1...in = n∂[i1

(X j

√|g|ε ji2...in

)diffgeom.tex; March 22, 2018; 16:56; p. 126


• Recall that a linear mapA : Λp → Λp+r

is a derivation if r is even and for any α ∈ Λp and β ∈ Λq

A(α ∧ β) = (Aα) ∧ β + α ∧ Aβ .

and an anti-derivation if r is odd and

A(α ∧ β) = (Aα) ∧ β + (−1)pα ∧ Aβ .

• Examples.

• The Lie derivative LXΛp → Λp is a derivation.

• The exterior derivative d : Λp → Λp+1 and the interior product iX : Λp →

Λp−1 are anti-derivations.

• Recall the following theorem about exterior derivative

Theorem 5.2.7 Let Y1, . . . ,Yp+1 be vector fields on a manifold M andα ∈ Λp be a p-form. Then

(dα)(Y1, . . . ,Yp+1) =

p+1∑i=1

(−1)i+1Yi

(α(Y1, . . . , Yi, . . . ,Yp+1)

)+

∑1≤i< j≤p+1

(−1)i+ jα([Yi,Y j], . . . , Yi, . . . , Y j, . . . ,Yp+1) .

• In particular, for p = 1 this takes the form

Theorem 5.2.8 Let X and Y be vector fields on a manifold M andα ∈ Λ1 be a 1-form. Then

(dα)(X,Y) = X(α(Y)) − Y(α(X)) − α([X,Y]) .

•

Theorem 5.2.9 The Lie derivative commutes with the exterior deriva-tive, that is,

LXd = dLX .

In other words, for any p-form α and a vector field X there holds

LXdα = dLXα

Proof :



1. Check for p = 0 and p = 1 explicitly.

2. Generalize for p > 1.

•

Theorem 5.2.10 (Cartan Formula) Let X be a vector field on a mani-fold M and α ∈ Λp be a p-form. Then

LXα = iXdα + diXα .

That is,LX = iXd + diX

Proof :

1. Verify it for p = 0 and p = 1.

•

Theorem 5.2.11 Let X and Y be vector fields on a manifold M andα ∈ Λp be a p-form. Then

[LX, iY]α = LXiYα − iYLXα = i[X,Y]α .

That is,[LX, iY] = i[X,Y] .

Proof : Exercise (Use Cartan formula).

•

Theorem 5.2.12 Let M be a n-dimensional manifold, X be a vectorfield on M and ϕt : M → M be the corresponding flow of X. Let W bea p-dimensional oriented compact submanifold of M and Wt = ϕtW bethe image of W under the flow ϕt. Let α be a p-form on M. Then

ddt

∫Wt

α =

∫Wt

LXα


5.3. FROBENIUS THEOREM 129

5.3 Frobenius Theorem

5.3.1 Distributions

•

Definition 5.3.1 Let M be a n-dimensional manifold. A k-dimensionaldistribution (or a tangent subbundle) ∆ : M → ∆x ⊂ TxM is a smoothassignment to each point x ∈ M a k dimensional subspace ∆x of thetangent space TxM.

An submanifold V of M that is everywhere tangent to the distribution iscalled an integral manifold of the distribution.

A k-didmensional distribution ∆ is called integrable if at each pointx ∈ M there is a k-dimensional integral submanifold of ∆.

In other words, the distribution ∆ is integrable if everywhere in M thereexist local coordinates (x1, . . . , xk, y1, . . . , yn−k) such that the coordinatesurfaces ya = ca, a = 1, . . . , n − k, ca being some constants, are integralmanifolds of the distribution ∆. Such a coordinate system is called aFrobenius chart.

• Examples in R3.

• A one-dimensional distribution in R3 is a family of lines at every point inR3 (that is, a vector field). It is always integrable. Integral manifolds of thedistribution are families of integral curves of the vector field.

• A two-dimensional distribution is a smooth family of planes at every pointin R3. It is integrable if there exist nonintersecting surfaces that are every-where tangent to the planes (and fill up a region in R3). Not every two-dimensional distribution is integrable!

• Let α be a 1-form in R3

α = αi(x)dxi ,

where i = 1, 2, 3. A two-dimensional distribution can be described by

α = 0 .

• It is integrable if there exists a diffeomorphism xi = xi(t, u), u = (uµ), µ =

1, 2, such that for a fixed t, xi = xi(t, u) describes a smooth one-parameter



family of surfaces S t and

(F∗α)µ = αi(x(t, u))∂xi

∂uµ= 0

and for fixed u = (u1, u2), xi = xi(t, u) describes a smooth two-parameterfamily of curves Cu transversal to the surfaces S t.

• Let t = t(x) and uµ = uµ(x) be the inverse diffeomorphism. Then the levelsurfaces t(x) = C are the integral surfaces S t of the distribution, so that thecoordinate system (t, u1, u2) is the Frobenius chart.

• Therefore, we have

α = αi∂xi

∂uµduµ + αi

∂xi

∂tdt

= αi∂xi

∂tdt

= f dt ,

where

f = αi∂xi

∂t.

• Therefore,dα = d f ∧ dt .

andα ∧ dα = 0 .

• Thus the distribution is integrable if

α ∧ dα = 0 .

This is called the Euler’s integrability condition.

• In local coordinatesα[i∂ jαk] = 0 .

In R3 this meansα · curlα = 0 .



•

Definition 5.3.2 Let ∆ be a distribution on M. It is said to be ininvolution if it is closed under Lie brackets, that is, for any two vectorfield X and Y in ∆ the Lie bracket [X,Y] is also in the distribution, or

[∆,∆] ⊂ ∆ .

• Let ∆ be an integrable distribution. Let X and Y be two vector fields in ∆.Then X and Y are tangent to the integral manifold of ∆. Therefore, the Liebracket [X,Y] is also tangent to the integral manifold and is in ∆.

• Proposition 5.3.1 Every integral distribution is in involution.

•

Definition 5.3.3 Let α be a 1-form on M. Let x ∈ M be a point suchthat αx , 0. The null space of the form α at x is the (n−1)-dimensionalsubspace of TxM spanned by the vectors X ∈ TxM such that

α(X) = 0 .

• Remark. A 1-form is also called a Pfaffian.

• Let α1, . . . , αn−k be (n − k) linearly independent 1-forms such that

α1 ∧ · · · ∧ αn−k , 0

• Let N1, . . .Nn−k be their null spaces. Then the intersection of the null spacesforms a k-dimensional distribution ∆

∆ =

n−k⋂µ=1

Nµ .

• Locally this distribution is described by (n − k) Pfaffian equations

α1 = · · · = αn−k = 0 .

• Remarks.

• Let ∆ be in involution. Then dαµ|∆ = 0, µ = 1, . . . , (n − k), that is, for anyX,Y ∈ ∆

(dαµ)(X,Y) = 0 .



• Now, suppose that dαµ|∆ = 0, µ = 1, . . . , (n − k). Then the distribution ∆ isin involution.

•

Theorem 5.3.1 Let M be a n-dimensional manifold and α1, . . . , αn−k

be (n − k) one-forms and ω be a (n − k)-form defined by

ω = α1 ∧ · · · ∧ αn−k .

Suppose that the forms α1, . . . , αn−k are linearly independent and ω , 0.Let ∆ be the k-dimensional distribution defined by the intersection of thenull spaces of the 1-forms α1, . . . , αn−k. Then the following conditionsare locally equivalent

1. the distribution ∆ is in involution,

2. dαµ|∆ = 0, that is, for any X,Y ∈ ∆

(dαµ)(X,Y) = 0 .

3. dαµ ∧ ω = 0,

4. there exist 1-forms γµν such that

dαµ =

n−k∑ν=1

γµν ∧ αν .

Proof :

1. (1)⇔ (2). Use the formula

dα(X,Y) = X(α(Y)) − Y(α(X)) − α([X,Y]) .

2. (4) =⇒ (2) and (4) =⇒ (3). Trivial.

3. (2) =⇒ (4). Suppose dαµ|∆ = 0.

4. Let β1, . . . , βk be 1-forms such that α1, . . . , αn−k, β1, . . . , βk is a basis inT ∗x M.

5. Let e1, . . . en−k, v1, . . . , vk be the dual basis in TxM.

6. Then for µ = 1, . . . , (n − k), i = 1, . . . , k

αµ(vi) = 0 .



Thus, span vi = ∆.

7. We have

dαµ =∑µ<ν

Cµλναλ ∧ αν +∑i,ν

Aµλiβi ∧ αν +∑i< j

Bµi jβi ∧ β j

=∑ν

γµν ∧ αν +∑i< j

Bµi jβi ∧ β j .

8. Thus,Bµi j = (dαµ)(vi, v j) = 0 .

9. (3) =⇒ (4). Suppose that dαµ ∧ ω = 0.

10. Then we have

0 = dαµ ∧ ω =∑ν

γµν ∧ αν ∧ ω +∑i< j

Bµi jβi ∧ β j ∧ ω .

=∑i< j

Bµi jβi ∧ β j ∧ α1 ∧ · · · ∧ αn−k .

11. Thus, Bµi j = 0.

• Remarks.

• A k-dimensional distribution ∆ can be described locally by either k linearlyindependent vector fields that span ∆ or by (n − k) linearly independent1-forms whose common null space is ∆.

• If dαµ =∑n−kν=1 γµν ∧ αν for some 1-forms γµν, then we write

dαµ = 0 (modα) .

5.3.2 Frobenius Theorem

•

Definition 5.3.4 Let M be an n-dimensional manifold, W be a k-dimensional manifold and F : W → M be a smooth map.

Then F is an immersion if for each x ∈ W the differential F∗ : TxW →TF(x)M is injective, that is, Ker F∗ = 0.

The image F(W) of the manifold W is called an immersed submani-fold.



• Let M be an n-dimensional manifold.

• Let ∆ be a k-dimensional distribution on M.

• Let X1, . . . ,Xk be vector fields that span ∆ and ϕ1t , . . . , ϕ

kt be the correspond-

ing flows.

• Let x ∈ M be a point in M.

• Let B ∈ Rk be a sufficiently small open ball in Rk around the origin.

• We defineF : B→ M

for any t = (t1, . . . , tk) ∈ B by

F(t) =(ϕk

tk · · · ϕ1t1

)(x) .

• Note that F(0) = x.

• Then for the differential at the origin

F∗T0B = Rk → TxM

we have

F∗∂

∂tµ

∣∣∣∣t=0

= Xµ

∣∣∣x,

where µ = 1, . . . , k.

• Thus,F∗T0B = ∆x .

• So, F∗ is injective at the origin and, hence, in some neighborhood of theorigin.

• Therefore, F(B) is tangent to ∆ at the single point x.

• Thus, F(B) is an immersed submanifold of M.



•

Theorem 5.3.2 Frobenius Theorem. Let M be an n-dimensional man-ifold, ∆ be a k-dimensional distribution in involution on M, X1, . . . ,Xk

be vector fields that span ∆ and ϕ1t , . . . , ϕ

kt be the corresponding flows.

Let x ∈ M, B be a sufficiently small ball around the origin in Rk andF : B→ M be defined by

F(t) =(ϕk

tk · · · ϕ1t1

)(x) .

Then:

1. F(B) is an immersed submanifold of M,

2. F(B) is an integral manifold of ∆,

3. the distribution ∆ is integrable.

Proof :

1. (I) done earlier.

2. (II). We need to show that ∆ is tangent to F(B) at each point of F(B).

3. We have for µ = 1, . . . , k

F∗∂

∂tµ=

(ϕk

tk∗ · · · ϕµtµ∗

)Xµ

[(ϕk−1

tk−1 · · · ϕ1

t1

)(x)

],

4. Thus, the tangent space TF(t)F(B) has a basis(ϕk

tk∗ · · · ϕ2t2∗

)X1

(ϕ1

t1(x))(

ϕktk∗ · · · ϕ

3t3∗

)X2

[(ϕ2

t2 ϕ1t1

)(x)

]. . .

ϕktk∗Xk−1

[(ϕk−1

tk−1 · · · ϕ1

t1

)(x)

]Xk

[(ϕk

tk · · · ϕ1t1

)(x)

].

5. Therefore, we need to show that for each µ = 1, . . . , k, the differentialsϕµt∗ map the distribution ∆ into itself, that is,

ϕµt∗(∆) ⊂ ∆ .

6. Claim: This follows from the fact that ∆ is in involution.



7. Let y ∈ F(B) and Y ∈ ∆y.

8. Let Yµt = ϕµt∗Y, µ = 1, . . . , k.

9. We will show that Yµt ∈ ∆ϕµt (y) for µ = 1, . . . , k.

10. Let ∆ be defined by the 1-forms

α1 = · · · = αn−k = 0 .

11. The vector field Yµt is invariant under the flow of Xµ, so along theorbits ϕµt (y) we have

LXµYµt = 0 .

12. Let f µ1 , . . . , f µn−k be real valued functions defined by

f µi (t) = αi(Yµt) , (i = 1, . . . , n − k) .

13. Then at t = 0 we have the initial conditions

fi(0) = αi(Y) = 0 .

14. Further,ddt

f µi (t) = iYµt iXµdαi .

15. Since ∆ is in involution, by using dαi =∑

k βik ∧ αk, we obtain

ddt

f µi (t) =

n−k∑j=1

γi j(Xµ) f µj (t) .

16. Thus, the above system of the differential equations has the uniquesolution

f µi (t) = 0 .

17. Thus, Yµt is in ∆ for all t and ∆ is tangent to the immersed ball F(B)at each point of F(B).

18. (III). By constructing Frobenius chart.

19. That is, we construct coordinates x1, . . . , xk, y1, . . . , yn−k so that the im-mersed balls F(B) are described locally by

y1 = c1, . . . , yn−k = cn−k ,

where ci are constants.



20. We define a transversal to F(B) (n − k)-dimensional submanifold Wwith local coordinates y1, . . . , yn−k.

21. If the integral balls are sufficiently small, then for different points ofW the integral balls at those points are disjoint.

5.3.3 Foliations• Let M be an n-dimensional manifold and ∆ be a k-dimensional distribution

on M.

• Let ∆ be in involution, and, therefore, integrable.

• Then, at each point x of M there exists an integral manifold of ∆.

• The integral manifold may return to the Frobenius coordinate patch aroundthe point x infinitely many times.

•

Definition 5.3.5 The integral manifolds of an integrable distributiondefine a foliation of M.

Each connected integral manifold is called a leaf of the foliation.

A leaf that is not properly contained in another leaf is called a maximalleaf.

• A maximal leaf is not necessarily an embedded submanifold.

• An immersed submanifold does not have to be an embedded submanifold.

•

Theorem 5.3.3 A maximal leaf of a foliated manifold is an immersedsubmanifold.

More precisely, for each maximal leaf V there is an injective immersionF : V → M that realizes V globally.

• Examples.



5.4 Degree of a Map

5.4.1 Gauss-Bonnet Theorem• Let us consider a compact oriented two-dimensional manifold M (a surface

embedded in R3).

• Let xi, i = 1, 2, 3, be the Cartesian coordinates in R3 and uµ, µ = 1, 2, be thelocal coordinates on M.

• Let F : M → R3 be the embedding map defined locally by

xi = F i(u) .

• The differential of the map F is given by the matrix

(F∗)iµ =

∂xi

∂uµ.

• We assume that F∗ is onto, that is, rank F∗ = 2.

• The tangent space TxM is spanned by the vectors eµ, µ = 1, 2, with compo-nents

eiµ =

∂xi

∂uµ.

• Let δi j be the Euclidean metric in R3.

• Then the induced metric on M is defined by

gµν = (eµ, eν) = δi jeiµe

jν .

This matrix is also called the first fundamental form. It describes theintrinsic geometry of the surface.

• Let N be the vector in R3

N = e1 × e2

with components

Ni = εi jkej1ek

2 = εi jk∂x j

∂u1

∂xk

∂u2 .

• Then N is a vector field that is everywhere normal to M.


5.4. DEGREE OF A MAP 139

• Notice that the norm of the normal vector is

||N||2 = ||e1||2 ||e2||

2 − (e1, e2)2 .

• The second fundamental form, or the extrinsic curvature is defined bythe matrix

bµν =1||N||

(∂eµ∂uν

,N)

=1||N||

∂2xi

∂uµ∂uνNi .

• The second fundamental form describes the extrinsic geometry of the sur-face M.

• The mean curvature of M is defined by

H = gµνbµν .

• The Gauss curvature of M is defined by

K =det bµνdet gαβ

.

• Gauss has shown that the K is an intrinsic invariant. In fact,

K = R1212 =

12

R ,

where R1212 is the only non-vanishing components of the Riemann curvature

of the metric g and R is the scalar curvature. This will be discussed later.

• The Gauss map is the map ϕ : M → S 2 from M to S 2 defined by

ϕ(x) =N(x)||N(x)||

,

that is, it associates to every point x in M the unit normal vector at thatpoint.



•

Theorem 5.4.1 Let M be a closed (compact without boundary) ori-ented 2-dimensional surface embedded in R3. Let gµν be the inducedRiemannian metric, dvol =

√|g|dx be the Riemannian volume element

on M, K be the Gaussian curvature of M and ϕ : M → S 2 be the Gaussnormal map. Then

14π

∫M

dvol K = deg(ϕ)

is an integer that does not depend on the metric and does not changeunder smooth deformations of the surface.

Proof : Later.

• Remark. For a 2-surface of genus g (with g holes)

deg(ϕ) = 1 − g .

• The Euler characteristic of the surface M is a topological invariant equalto

χ = 1 − g .

5.4.2 Laplacian• Let gi j be a Riemannian metric on a manifold V . Let h be a function and X

be its gradient vector field defined by

Xi = gi j∂ jh .

• The Laplacian operator ∆ on the scalar function h is defined by

∆h = div X .

• In local coordinates∆h = g−1/2∂i(g1/2gi j∂ jh) ,

where g = det gi j.

• The operator (−∆) : C∞(V) → C∞(V) acting on smooth functions on acompact manifold V without boundary can be extended to a self-adjointoperator on the Hilbert space L2(V).



• It has a discrete non-negative real spectrum λ∞k=0 bounded from below byzero

0 = λ0 ≤ λ1 ≤ λ2 · · ·

with finite multiplicities.

• The eigenfunctions hk∞k=0 form an orthonormal basis in L2(V), that is,

(hk, hl) =

∫V

dvol hkhl = δkl .

• For each f ∈ L2(V) there is a Fourier series

f =

∞∑k=0

akhk ,

whereak = (hk, f ) =

∫V

f hk .

• The lowest eigenvalue is 0. It is simple (has multiplicity 1). The corre-sponding eigenfunction is the constant h0 = [vol (V)]−1/2.

•

Lemma 5.4.1 Let f be a function on a closed manifold M such that∫M

f = 0. Let λk, hk∞k=1 be the spectral resolution of the operator (−∆).

Then the equation∆h = f

has a unique solution given by the Fourier series

h = −

∞∑k=1

1λk

akhk ,

whereak = (hk, f ) =

∫M

f hk .

Proof :

1.



•

Lemma 5.4.2 Let V be a closed (compact without boundary) orientedn-dimensional manifold and ω be an n-form on V. Then ω is exact ifand only if ∫

Vω = 0 .

Proof :

1. (I). If ω is exact, then ω = dσ for some (n − 1)-form σ.

2. Then by Stokes theorem ∫Vω = 0 .

3. (II). Suppose that ∫Vω = 0 .

4. Let us fix a Riemannian metric on V and let vol be a Riemannianvolume element.

5. Thenω = f vol

for some function f which satisfies∫V

f = 0 .

6. Let h be the solution of the equation

∆h = f .

7. Let X be the gradient of the function h.

8. We define an (n − 1)-form σ by

σ = iXvol .

9. Thendσ = ω .



5.4.3 Brouwer Degree• Let M and V be two closed (compact without boundary) oriented n-dimensional

manifolds.

• Let ϕ : M → V be a smooth map.

• Let ω ∈ ΛnV be an n-form on V .

• Suppose that ∫Vω , 0 .

Then we can normalize it so that∫Vω = 1 .

• Then we can consider the quantity∫Mϕ∗ω∫

Vω

,

which can also be written as ∫ϕ(M)

ω∫Vω

.

• This quantity counts how many times the image of M wraps around V .

•

Corollary 5.4.1 Let M and V be n-dimensional manifolds and ϕ :M → V be a smooth map. Let α and β be n-forms on V such that∫

Vα , 0 and

∫Vβ , 0. Then∫

Mϕ∗α∫

Vα

=

∫Mϕ∗β∫

Vβ

.

Proof :

1. Let

ω =

α∫Vα−

β∫Vβ

.diffgeom.tex; March 22, 2018; 16:56; p. 143


2. We have ∫Vω = 0 .

3. Therefore, the form ω is exact.

4. Hence, the form ϕ∗ω is exact.

5. Thus, ∫Mϕ∗ω = 0 .

• The quantity

deg(ϕ) =

∫Mϕ∗ω∫

Vϕ

does not depend on the choice of the form ω but only on the map ϕ. It iscalled the Brouwer degree of the map ϕ.

• Example.

• In the case of one-dimensional manifolds the degree of the map ϕ : M → S 1

is called the winding number.

• Picture.

•

Theorem 5.4.2 Let V and M be n-dimensional compact oriented man-ifolds without boundary. Let ϕ : M → V be a smooth map. Let y ∈ Vbe a regular value of ϕ so that the differential ϕ∗ : TxM → TyV at anypoint x ∈ ϕ−1(y) is bijective (isomorphism). Then

deg(ϕ) =∑

x∈ϕ−1(y)

sign (ϕ(x)) ,

wheresign (ϕ(x)) = sign (det(ϕ∗)) .

Proof :

1. (I). Claim: the preimage ϕ−1(y) of a regular value is a finite set, that is,

ϕ−1(y) = xi ∈ M | ϕ(xi) = y, i = 1, 2, . . . ,N .



2. Suppose ϕ−1(y) is infinite.

3. Then by compactness argument ϕ−1(y) has a limit point x0 ∈ M, whichis a regular point.Indeed, every sequence has a convergent subsequence. Thus, there is asequence (xk), such that xk ∈ ϕ

−1(y), converging to some x0, xk → x0,so that ϕ(x0) = y. Thus x0 is a regular point of M.

4. Since ϕ∗ : Tx0 M → TyV is bijective at x0, the map ϕ : M → V is adiffeomorphism in a neighborhood of x0.That is, detϕ∗|x0 , 0.

5. This contradicts the fact that there is sequence of points xk → x0 suchthat ϕ(xk) = y.Since, otherwise there exist infinitely many points xk ∈ M such thatϕ(xk) = ϕ(x0) = y contradicting the fact that ϕ is bijective.

6. (II). Claim: The point y ∈ V has a neighborhood whose inverse imageis a disjoint union of neighborhoods of the preimages of y, each ofwhich is diffeomorphic to Vy.

7. Let Wi be disjoint neighborhoods of xi ∈ M such that ϕ : Wi → Vi =

ϕ(Wi) are diffeomorphisms.

8. Let S = ϕ(M − ∪Ni=1Wi).

9. Since M − ∪Ni=1Wi is compact, then S is compact (and closed in V).

10. Let O = V − S .

11. Then O is open and is a neighborhood of y.

12. Now, we defineVy = O ∩ ∩N

i=1Vi .

13. ThenVy ⊂ O, Vy ⊂ Vi .

14. LetUi = ϕ−1(O) ∩Wi .

15. ThenUi ⊂ Wi ⊂ M .



16. Thenϕ−1(Vy) = ∪N

i=1Ui ,

and ϕ : Ui → Vy are diffeomorphisms.

17. (III). Let ω be an n-form on V with support in Vy such that∫

Vω = 1.

18. Let yi, i = 1, . . . , n, be local coordinates in Vy.

19. Since each ϕ : Ui → Vy are diffeomorphisms, one can use yi as localcoordinates on Ui.

20. In such coordinates the diffeomorphism ϕ : Ui → Vy is the identitymap.

21. We notice that the orientation of Ui is described by signϕ(xi).

22. Thus

deg(ϕ) =

∫Mϕ∗ω =

N∑i=1

∫Ui

ϕ∗ω =

N∑i=1

∫ϕ(Ui)

ω

=

N∑i=1

signϕ(xi)∫

Vy

ω =

N∑i=1

sign (ϕ(xi)) .

•

Corollary 5.4.2

1. The Brouwer degree deg(ϕ) of any map ϕ : M → V is an integer.

2. The sum ∑x∈ϕ−1(y)

sign (ϕ(x))

is independent on the regular value y ∈ V.

3. The Brouwer degree deg(ϕ) remains constant under continuousdeformations of the map ϕ.

• Problem. Let ω be the volume form on S n in Rn+1 given by

ω =

n+1∑k=1

(−1)kdx1 ∧ · · · ∧ dxk ∧ · · · ∧ dxn+1 .

Show that the antipodal map ϕ : S n → S n has degree deg(ϕ) = (−1)n+1.



• Problem. Let M be a closed oriented n-dimensional manifold and ϕ : M →S n be a smooth map. We identify ϕ(x) with a unit vector field v on S n inRn+1. Let vol be the volume (n+1)-form in Rn+1. Let vol (S n) be the volumeof the unit sphere S n. Let uµ, µ = 1, . . . , n, be local coordinates on M. Showthat

deg(ϕ) =1

vol (S n)

∫M

vol(v,∂v∂u1 , · · · ,

∂v∂un

)du1 ∧ · · · ∧ dun

5.4.4 Index of a Vector Field

• Let M be a closed (compact without boundary) n-dimensional submanifoldof Rn+1 that is a boundary of a compact region U ⊂ Rn+1, that is,

M = ∂U .

• Let N be an outward-pointing normal to M.

• Then N induces an orientation on M.

• Let v be a unit vector field on M.

• Let S n be the unit n-sphere embedded in Rn centered at the origin.

• We identify the unit vectors in Rn+1 with points in S n.

• Let ϕ : M → S n be a map defined for every x ∈ M by

ϕ(x) = v(x) .

• The Kronecker index of the vector field v on M is defined as the degree ofthe map ϕ:

Index(v) = deg(ϕ) .

• In general, if v is a nonvanishing vector field on M, then its index is definedas the Kronecker index of the unit vector field v/||v||.

• Example. M = S 1 in R2.



• Problem. Let M = ∂U, where U ⊂ Rn+1 is a compact region as above.Show that if a vector field v on M can be extended to a nonvanishing vectorfield on all of the interior region U, then Index(v) = 0.

In other words, a vector field on M with a non-zero index has a singularity(that is, it vanishes) at least at one point inside M.

• Problem. Suppose that a unit vector field v in Rn+1 is such that it is smoothin U except for a finite number of points xa, a = 1, . . . ,N. Let Ba be suf-ficiently small balls around the points xa so that they are in U. Then v isnon-vanishing in U \ ∪N

a=1Ba. Then ∂Ba are small spheres with outward-poiting normals. Let Index(v|∂Ba) be the indices of v on ∂Ba. Show that

Index(v) =

N∑a=1

Index(v|∂Ba) .

That is, the index of a vector field on a closed manifold M is equal to thesum of indices inside M.

•

Theorem 5.4.3 Let vt, t ∈ [0, 1], be a smooth family of non-vanishingvector field on a closed manifold M. Then

Index(v0) = Index(v1) .

Proof :

1. Follows from the fact that the index is an integer.

• Problem. Show that if a vector field v on a sphere S n in Rn+1 never pointsto the origin, then

Index(v) = 1 .

• Corollary. The index of every non-vanishing vector field tangent to S n isequal to

Index(v) = 1 .

•

Theorem 5.4.4 Brouwer Fixed Point Theorem. Let B be a closedunit ball in Rn+1 centered at the origin. Let ϕ : B → B be a smoothmap. Then ϕ has a fixed point.

Proof :



1. Let v be a vector field in Rn+1 defined by

v(x) = ϕ(x) − x .

2. Then v never points to the origin. So, Index(v) = 1, and, therefore, ϕhas a fixed point in B.

3. Alternatively, suppose that ϕ does not have a fixed point.

4. Then v is non-vanishing.

5. Let ψ : B → ∂B = S n be a map defined as follows. The point ψ(x) isthe point on the sphere S n where the line from the ϕ(x) to the point xintersects S n.

6. Then ψ(x) = x for any x ∈ S n.

7. Let α be an n-form on S n normalized by∫

S n α = 1.

8. Since ψ is an identity map on S n, the form ψ∗α is an n-form on Bwhose restriction to S n is equal to α.

9. Since also S n = ∂B and dα = 0, we have

1 =

∫S nα =

∫∂Bψ∗α =

∫B

d(ψ∗α) = 0 ,

which is a contradiction.

• Problem. Let M be a closed n-dimensional submanifold of Rn+1 Let v bea unit vector field on M. Let vol be the volume (n + 1)-form in Rn+1. Letvol (S n) be the volume of the unit sphere S n. Let uµ, µ = 1, . . . , n, be localcoordinates on M. Show that

Index(v) =1

vol (S n)

∫M

vol(v,∂v∂u1 , · · · ,

∂v∂un

)du1 ∧ · · · ∧ dun

• Problem. If v is a non-vanishing vector field, not necessarily unit, then

Index(v) =1

vol (S n)

∫M

1||v||n+1 vol

(v,∂v∂u1 , · · · ,

∂v∂un

)du1 ∧ · · · ∧ dun



•

Corollary 5.4.3 Let M be a closed n-dimensional submanifold of Rn+1

such that M is the boundary of a compact region U ⊂ Rn+1. Let fi,i = 1, . . . , (n + 1), be smooth functions on U. Let

|| f ||2 =

n+1∑i=1

| fi|2 .

Suppose the functions fi do not have common zeros on M, that is, || f || ,0 on M. Let uµ, µ = 1, . . . , n, be local coordinates on M. Let

det( f , d f ) = εi1i2...inin+1 fi1∂ fi2

∂u1 · · ·∂ fin+1

∂un

= det

f1

∂ f1∂u1 · · ·

∂ f1∂un

......

. . ....

fn∂ fn∂u1 · · ·

∂ fn∂un

Suppose that ∫

M

1|| f ||n+1 det( f , d f )du1 ∧ · · · ∧ dun , 0 .

Then the functions fi have a common zero in U, that is, the system of(n + 1) equations

f1 = · · · = fn+1 = 0

has a solution in U.

Proof : Follows from above.

5.4.5 Linking Number

• Let Cµ : S 1 → R3, µ = 1, 2, be two nonintersecting smooth closed curves(loops) in R3 described by

x = x1(θ1) , x = x2(θ2) .

• Let T 2 = S 1 × S 1 be the torus with the local coordinates θ1, θ2.



• Let ϕ : T → S 2 be a smooth map defined by

ϕ(θ) =x1(θ1) − x2(θ2)||x1(θ1) − x2(θ2)||

.

• The Gauss linking number of the loops C1 and C2 is defined by

Link(C1,C2) = deg(ϕ) .

• Problem. Letx12 = x2 − x1 .

Show that

Link(C1,C2) =1

4π

∫C1

(∫C2

1||x12||

3 x12 × dx12

)· dx1

=1

4π

∫ 2π

0

∫ 2π

0dθ1 dθ2

[x2(θ2) − x1(θ1)] × x2(θ2) · x1(θ1)||x2(θ2) − x1(θ1)||3

• Let V be an orientable surface in R3 such that ∂V = C1.

• Let N be the normal vector to V consistent with the orientation of C1.

• Let the curve C2 intersect V transversally.

• The intersection number V C2 of the curve C2 and the surface V is thesigned number of intersections of C2 and V , with an intersection being pos-itive if the tangent vector to C2 at the point of the intersection has the samedirection as N, that is,

V C2 =∑xi∈V

sign (N, x)∣∣∣xi,

where the sum goes over all intersection points xi.

• Problem. Show thatLink(C1,C2) = V C2 .


Chapter 6

Connection and Curvature

6.1 Affine Connection

6.1.1 Covariant Derivative

•

Definition 6.1.1 Let M be an n-dimensional manifold. An affine con-nection is an operator

∇ : C∞(T M) ×C∞(T M)→ C∞(T M)

that assigns to two vector fields X and Y a new vector field ∇XY, thatis linear in both variables, that is, for any a, b ∈ R and any vector fieldsX, Y and Z,

∇X(aY + bZ) = a∇XY + b∇XZ∇aX+bYZ = a∇XZ + b∇YZ

and satisfies the Leibnitz rule, that is, for any smooth function f ∈C∞(M) and any two vector fields X and Y,

∇X( f Y) = X( f )Y + f∇XY= (d f )(X)Y + f∇XY

• Let xµ, µ = 1, . . . , n, be local coordinates and ∂µ be the basis of vector fields.It will be called a coordinate frame for the tangent bundle.

• A basis of vector fields ei = eµi ∂µ, i = 1, . . . , n, for the tangent bundle T M is

153

154 CHAPTER 6. CONNECTION AND CURVATURE

called a frame.

• We will denote partial derivatives by

∂i f = f,i

and the action of frame vector fields on functions by

ei( f ) = eµi ∂µ = f|i .

• For any frame the commutator of the frame vector fields defines the com-mutation coefficients

[ei, e j] = Cki jek .

• The commutation coefficients are scalar functions, in general.

•

Theorem 6.1.1 A frame ei is a coordinate frame if and only if for anyi, j = 1, . . . , n,

[ei, e j] = 0 .

Proof :

1. Need to show that there is a local coordinate system xi such that ei =

∂i.

2. Use the dual basis of 1-forms to prove that they are exact.

•

Definition 6.1.2 Let ei be a frame of vector fields and σ j be the dualframe of 1-forms. The symbols ωi

jk defined by

ωik j = σi(∇e jek)

are called the coefficients of the affine connection.

• We denote∇i = ∇ei .

• Then∇ie j = ωk

jiek ,


6.1. AFFINE CONNECTION 155

• Then, if X = Xiei is a vector field, then

∇X = Xi∇i .

• If Y = Y je j is another vector field then

∇XY = Xiei(Yk) + ωk

jiY j

ek

=[

dYk + ωkjiY jσi

](X)

ek .

• That is,

∇XY = Xi(∇iY)kek

where

(∇iY)k = ei(Yk) + ωkjiY j

• We will often write simply ∇iYk meaning (∇eiY)k. This should not be con-fused with the covariant derivative of the scalar functions Yk.

• The tensor field ∇Y of type (1, 1) with components ∇iYk, that is,

∇Y = σi ⊗ ∇iY = ∇iYkσi ⊗ ek .

is called the covariant derivative of the vector field Y.

• The components of the covariant derivative are also denoted by Yk;i, in con-

trast to partial derivatives ∂iYk, which are also denoted by Yk,i.

• In the coordinate frame ei = ∂i the covariant derivative takes the form

∇iYk = ∂iYk + ωkjiY j .



6.1.2 Curvature, Torsion and Levi-Civita Connection

•

Definition 6.1.3 Let X and Y be vector fields on a manifold M. Thenthe vector field

T (X,Y) = ∇XY − ∇YX − [X,Y]

defines a tensor field T of type (1, 2), called the torsion, so that for any1-form σ

T (σ,X,Y) = σ(T (X,Y)) .

The affine connection is called torsion-free (or symmetric) if the tor-sion vanishes, that is, for any X, Y,

∇XY − ∇YX = [X,Y]

• The components of the torsion tensor are defined by

T ijk = σi(T (e j, ek)) .

• In the coordinate frame the components of the torsion tensor are given by

T ijk = ωi

k j − ωijk .

•

Definition 6.1.4 Let X, Y and Z be vector fields on a manifold M.Then the vector field

R(X,Y)Z =[∇X,∇Y] − ∇[X,Y]

Z

defines a tensor field R of type (1, 3), called the Riemann curvature, sothat for any 1-form σ

R(σ,Z,X,Y) = σ(R(X,Y)Z) .

The affine connection is called flat if the curvature vanishes, that is, forany X, Y, Z,

[∇X,∇Y]Z = ∇[X,Y]Z .

• The components of the curvature tensor are defined by

Rijkl = σi(R(ek, el)e j)) .



•

Theorem 6.1.2 The components of the curvature tensor have the form

Rijkl = ωi

jl|k − ωijk|l + ωi

mkωm

jl − ωimlω

mjk −Cm

klωijm .

Proof :

1.

• In the coordinate frame the components of the curvature tensor are given by

Rijkl = ∂kω

ijl − ∂lω

ijk + ωi

mkωm

jl − ωimlω

mjk .

• For a Riemannian manifold (M, g) the metric tensor g has the components

gi j = g(ei, e j) = gµνeµi eνj .

• This metric is used to lower and raise the frame indices.

•

Definition 6.1.5 Let (M, g) be a Riemannian manifold and ∇ be anaffine connection on M. Then the connection ∇ is called compatiblewith the metric g if for any vector fields X, Y and Z it satisfies thecondition

Z(g(X,Y)) = g(∇ZX,Y) + g(X,∇ZY) .

An affine connection that is torsion-free and compatible with the metricis called the Levi-Civita connection.

• Wed defineωi jk = gimω

mjk Ci jk = gimCm

jk .

• The coefficients of the Levi-Civita connection satisfy the equation

gi j|k = ωi jk + ω jik .

•

Theorem 6.1.3 Then

ωi jk =12

(gi j|k + gik| j − g jk|i + Cki j + C jik −Ci jk

).

Proof :



1. Direct calculation.

• The coefficients of the Levi-Civita connection in a coordinate frame arecalled Christoffel symbols and denoted by Γi

jk.

•

Corollary 6.1.1 The coefficients of the Levi-Civita connection in acoordinate frame (Christoffel symbols) have the form

Γijk =

12

gim(∂ jgmk + ∂kg jm − ∂mg jk

).

Christoffel symbols have the following symmetry property

Γijk = Γi

k j .

Proof :


•

Corollary 6.1.2 The coefficients of the Levi-Civita connection in anorthonormal frame have the form

ωi jk =12

(Cki j + C jik −Ci jk

).

They have the following symmetry properties

ωi jk = −ω jik .

Proof :


•Theorem 6.1.4 Each Riemannian manifold has a unique Levi-Civitaconnection.

Proof :

1. By construction.



6.1.3 Parallel Transport• Let x0 and x1 be two points on a manifold M and C be a smooth curve

connecting these points described locally by xi = xi(t), where t ∈ [0, 1] andx(0) = x0 and x(1) = x1. The tangent vector to C is defined by

X = x(t),

where the dot denotes the derivative with respect to t.

• Let Y be a vector field on M. We say that Y is parallel transported along Cif

∇XY = 0 .

• The vector field Y is parallel transported along C if its components satisfythe linear ordinary differential equation

ddt

Y i(x(t)) + ωijk(x(t))xk(t)Y j(x(t)) = 0 .

• Problem. Solve the equation of parallel transport in terms of a Taylor seriesup to terms qubic in the connection coefficients.

• A curve C such that the tangent vector to C is trasported parallel along C,that is,

∇x x = 0,

is called the geodesics.

• The coordinates of the geodesics x = x(t) satisfy the non-linear second-order ordinary differential equation

xi + ωijk(x(t))xk x j = 0 .



6.2 Tensor Analysis

6.2.1 Covariant Derivative of Tensors• We extend the definition of the affine connection from the tangent bundle

T M to arbitrary tensor bundles T pq M = ⊗pT M ⊗q T ∗M.

• The affine connection on a tensor bundle T pq M is an operator

∇ : C∞(T M) ×C∞(T pq M)→ C∞(T p

q M)

that assigns to a vector field X and a tensor field T of type (p, q) a newtensor field ∇XT of type (p, q).

• The covariant derivative of a tensor field T of type (p, q) is a linear opera-tor

∇ : C∞(T pq M)→ C∞(T p

q+1M)

that assigns to a tensor field T of type (p, q) a new tensor field ∇T of type(p, q + 1).

• First of all, the covariant derivative of a 1-form α on a manifold M is atensor ∇α of type (0, 2) such that for any two vector fields X and Y

(∇α)(X,Y) = (∇Xα)(Y) = X[α(Y)] − α(∇XY) .

• Then the covariant derivative of a tensor T of type (p, q) is a tensor ∇Tof type (p, q + 1) such that for any vector fields X,Y1, . . . ,Yq and 1-formsω1, ωp

(∇T )(X,Y1, . . . ,Yq, ω1, . . . , ωp) = (∇XT )(X,Y1, . . . ,Yq, ω1, . . . , ωp)= X[T (Y1, . . . ,Yq, ω1, . . . , ωp)]

−

q∑i=1

T (Y1, . . . ,∇XYi, . . .Yq, ω1, . . . , ωp)

−

p∑j=1

T (Y1, . . . ,Yq, ω1, . . . ,∇Xω j, . . . , ωp) .

• Let ei be a basis of vector fields and σi be the dual basis of 1-forms.


6.2. TENSOR ANALYSIS 161

• The covariant derivative of a 1-form has the form

(∇iα)k = ei(αk) − ωlkiαl

In the coordinate frame this simplifies to

(∇iα)k = ∂iαk − ωlkiαl .

• The covariant derivative of a tensor of type (p, q) in a coordinate basis hasthe form

∇iTj1... jp

k1...kq= ∂iT

j1... jp

k1...kq+

p∑m=1

ω jmliT

j1... jm−1l jm+1... jp

k1...kq−

q∑n=1

ωlkniT

j1... jp

k1...kn−1lknkq

• The parallel transport of tensor fields is defined similarly to vector fields.Let C be a smooth curve on a manifold M described locally by xi = xi(t),where t ∈ [0, 1], with the tangent vector X = x(t).

• Let T be a tensor field on M. We say that T is parallel transported alongC if

∇XT = 0 .

6.2.2 Ricci Identities

• The commutators of covariant derivatives of tensors are expressed in termsof the curvature and the torsion.

• In a coordinate basis for a torsion-free connection we have the followingidentities (called the Ricci identities):

[∇i,∇ j]Yk = Rkli jY l

[∇i,∇ j]αk = −Rlki jαl

[∇i,∇ j]Tj1... jp

k1...kq=

p∑m=1

R jmli jT

j1... jm−1l jm+1... jp

k1...kq−

q∑n=1

Rlkni jT

j1... jp

k1...kn−1lknkq



6.2.3 Normal Coordinates• Let (M, g) be a Riemannian manifold and Γi

jk be the Christoffel coefficientsdefining the Levi-Civita connection in a coordinate basis.

• Let x0 be a point in M and xi be a local coordinate system in a coordinatepatch about x0.

• We expand the metric in a Taylor series at the point x0

gi j(x) = gi j(x0)+[∂kgi j](x0)(xk−xk0)+

12

[∂k∂lgi j](x0)(xk−xk0)(xl−xl

0)+O((x−x0)3)

• The matrix gi j(x0) is a constant real symmetric matrix with real eigenvalues.

•

Theorem 6.2.1 There exists a coordinate system such that

gi j(x0) = δi j , [∂kgi j](x0) = 0 ,

so that the Taylor series has the form

gi j(x) = δi j +12

[∂k∂lgi j](x0)(xk − xk0)(xl − xl

0) + O((x − x0)3) .

Such coordinates are called Riemann normal coordinates.Proof :

1.

•

Corollary 6.2.1 In Riemann normal coordinates the Christoffel sym-bols vanish at x0

Γijk(x0) = 0

and the curvature of the Levi-Civita connection at x0 is expressed interms of second derivatives of the metric at x0,

Ri jkl(x0) =12

∂k∂ jgil − ∂l∂ jgik − ∂i∂lgk j + ∂k∂ig jl

∣∣∣∣x0,

so that the Taylor series of the metric at x0 has the form

gi j(x) = δi j −13

Rik jl(x0)(xk − xk0)(xl − xl

0) + O((x − x0)3) .



Proof :

1.

6.2.4 Properties of the Curvature Tensor

• Let (M, g) be an n-dimensional Riemannian manifold. We will restrict our-selves to the Levi-Civita connection below. We define some new curvaturetensors.

• The Ricci tensor

Ri j = Rkik j .

• The scalar curvature

R = gi jRi j = gi jRkik j .

• The Einstein tensor

Gi j = Ri j −12

gi jR .

• The trace-free Ricci tensor

Ei j = Ri j −1n

gi jR .

• The Weyl tensor (for n > 2)

Ci jkl = Ri j

kl −4

n − 2R[i

[kδj]

l] +2

(n − 1)(n − 2)Rδ[i

[kδj]

l]

= Ri jkl −

4n − 2

E[i[kδ

j]l] −

2n(n − 1)

Rδ[i[kδ

j]l]



•

Theorem 6.2.2 The Riemann curvature tensor of the Levi-Civita con-nection has the following symmetry properties

1. Ri jkl = −Ri jlk

2. Ri jkl = −R jikl

3. Ri jkl = Rkli j

4. Ri[ jkl] = Ri

jkl + Rikl j + Ri

l jk = 0

5. Ri j = R ji

Proof :

1.

•

Theorem 6.2.3 The Weyl tensor has the same symmetry properties asthe Riemann tensor and all its contractions vanish, that is,

Cijik = 0 .

Proof :

1.

•

Theorem 6.2.4 The number of algebraically independent componentsof the Riemann tensor of the Levi-Civita connection is equal to

n2(n2 − 1)12

.

Proof :

1.



•

Corollary 6.2.2 In dimension n = 2 the Riemann tensor has only oneindependent component determined by the scalar curvature

R1212 =

12

R .

The trace-free Ricci tensor Ei j vanishes, that is

Ri jkl = Rδ[i

[kδj]

l]

Ri j =12

Rgi j .

Proof :

1.

•

Corollary 6.2.3 In dimension n = 3 the Riemann tensor has six in-dependent components determined by the Ricci tensor Ri j. The Weyltensor Ci jkl vanishes, that is,

Ri jkl = 4R[i

[kδj]

l] + Rδ[i[kδ

j]l] .

Proof :

1.

6.2.5 Bianchi Identities• Let (M, g) be an n-dimensional Riemannian manifold. We will restrict our-

selves to the Levi-Civita connection below.

•

Theorem 6.2.5 The Riemann tensor satisfies the following identities

∇[mRi jkl] = ∇mRi j

kl + ∇kRi jlm + ∇lRi j

mk = 0 .

These identities are called the Bianchi identities.Proof :

1.



•

Corollary 6.2.4 The divergences of the Riemann tensor and the Riccitensor have the form

∇iRi jkl = ∇kR

jl − ∇lR

jk ,

∇iRij =

12∇ jR .

The divergence of the Einstein tensor vanishes

∇iGij = 0 .

Proof :

1.

• Problem. By using the Bianchi identities simplify the Laplacian of theRiemann tensor, ∆Ri j

kl = ∇m∇mRi jkl.


6.3. CARTAN’S STRUCTURAL EQUATIONS 167

6.3 Cartan’s Structural Equations• Let ∂µ be a coordinate basis for the tangent bundle T M and ei = eµi ∂µ be an

orthonormal frame of vector fields.

• Thengi j = g(ei, ek) = gµνe

µi eνj = δi j .

• We use this metric to lower and raise the frame indices.

• Let dxµ be a coordinate basis for the cotangent bundle T M and σi = σiµdxµ

be an orthonormal frame of 1-forms dual to ei.

• Thenσi(e j) = σi

µeµj = δi

j

andgµνσi

µσjν = δi j

σiµe

νi = δνµ .

• The commutators of the frame vector fields define the commutation coeffi-cients Ci

jk by[ei, e j] = Ck

i jek ,

or, in components,

eνj|i − eνi| j = eµi ∂µeνj − eµj∂µe

νi = Ck

i jeνk .

That is,Ck

i j = σk([ei, e j])

orCk

i j = σkν

(eµi ∂µe

νj − eµj∂µe

νi

).

•

Proposition 6.3.1 There holds

dσi = −12

Cijkσ

j ∧ σk .

or(dσi)(e j, ek) = −

12

Cijk .

Proof :



1. Direct calculation using the duality condition.

• Let ωijk be the coefficients of the Levi-Civita connection in the orthonormal

frame.

• Then∇ie j = ωk

jiek

and∇iσ

j = −ω jkiσ

k ,

where ∇i = ∇ei .

• This can also be written as

ωkji = σk

µeνi eµj;ν = −σk

µ;νeνi eµj .

• Thus

ωkji − ω

ki j =

(σkν;µ − σ

kµ;ν

)eνi e

µj

=(σkν,µ − σ

kµ,ν

)eνi e

µj

= (dσk)(e j, ei) .

•

Proposition 6.3.2 There holds

dσi = ωijkσ

j ∧ σk .

Proof :

1. Follows from above.

• Since the torsion of the Levi-Civita connection is zero, we have

∇ie j − ∇ jei = [ei, e j] .

• Therefore,ωk

ji − ωk

i j = Cki j .



• Since the Levi-Civita connection is compatible with the metric, we have

0 = ∇ig(e j, ek) = g(∇ie j, ek) + g(e j,∇iek) .

• Thus,ωk ji + ω jki = 0 .

• Finally, we obtain,

•

Proposition 6.3.3 The coefficients of the Levi-Civita connection in anorthonormal frame are given in terms of the commutation coefficientsby

ωi jk =12

(Cki j + C jik −Ci jk

).

Proof :

1. Use the equations

ωki j + ωik j = 0ωi jk + ω jik = 0ω jki + ωk ji = 0

andωk

ji − ωk

i j = Cki j .

• Now we define the connection 1-forms

Aij = ωi

jkσk

and the curvature 2-forms

F ij =

12

Rijklσ

k ∧ σl .

• Then the equationdσi = ωi

jkσj ∧ σk

can be written asdσi +Ai

j ∧ σj = 0 .

This is called Cartan’s first structural equation.



• The curvature 2-forms are obtained from the connection 1-forms by Car-tan’s second structural equation

F ij = dAi

j +Aik ∧A

kj .

This equation is equivalent to the expression for the curvature componentsand can be obtained from that.

• Cartan’s third structural equation

dF ij +Ai

k ∧ Fk

j − Fik ∧A

kj = 0 .

is equivalent to Bianchi identities.

• Cartan structural equations can be written in a very compact way by in-troducing the covariant exterior derivative acting on vector valued andmatrix valued forms.

• Let α be a p-form valued in a vector space V (in our case V = Rn). Sucha form is called a twisted form. Let αi be the components of this form ina fixed basis in V . That is, αi is a p-form for each i = 1, . . . , n. Then thecovariant exterior derivative

D : Λp ⊗ V → Λp+1 ⊗ V

is defined by(Dα)i = dαi +Ai

j ∧ αj

or, in matrix form,Dα = dα +A∧ α

with obvious notation.

• Let V∗ be the dual vector space to V (in our case it is again Rn). We considerp-forms valued in V∗ (covectors) and naturally extend the operator D tosuch forms

D : Λp ⊗ V∗ → Λp+1 ⊗ V∗

by(Dα)i = dαi − (−1)pα j ∧A

ji

or, in matrix form,Dα = dα − (−1)pα ∧A .



• Finally, we consider matrix-valued p-forms valued in V ⊗V∗ and extend theoperatorD to such forms

D : Λp ⊗ V ⊗ V∗ → Λp+1 ⊗ V ⊗ V∗

by(Dα)i

j = dαij +Ai

k ∧ αkj − (−1)pαi

k ∧Ak

j

or, in matrix form,

Dα = dα +A∧ α − (−1)pα ∧A .

• Now, let σ = (σi), F = (F ij) and α = (αi) be an arbitrary vector-valued

1-form. Then

Dσ = 0D2α = F ∧ α

DF = 0 .

• Problem. Let the dimension n = 2k of the manifold M be even. We definethe following 2l-forms

Ω(l) = tr F ∧ · · · ∧ F︸︷︷︸l

and the n-formΩ = εi1...i2kFi1i2 ∧ · · · ∧ Fi2k−1i2k

1. Prove that these forms are independent of the orthonormal basis andare closed, that is,

dΩ(l) = dΩ = 0 .

2. Find the expressions in local coordinates for these forms.

• These forms define so called characteristic classes, which are closed in-variant forms whose integrals over the manifold do not depend on the metricand, therefore, are topological invariants of the manifold.


Chapter 7

Homology Theory

7.1 Algebraic Preliminaries

7.1.1 Groups• A group is a set G with a binary operation, ·, called the group multiplication,

that is,

1. associative,

2. has an identity element,

3. every element has an inverse.

• A group is Abelian if the group operation is commutative.

• For an Abelian group the group operation is called addition and is denotedby +. The identity element is called zero and denoted by 0. The inverseelement of an element g ∈ G is denoted by (−g).

• Let G and E be Abelian groups. A map F : G → E is called a homomor-phism if for any g, g′ ∈ G,

F(g +G g′) = F(g) +E F(g′) ,

where +G and +E are the group operations in G and E respectively.

• In particular,

F(0G) = 0E , and F(−g) = −F(g) .

173

174 CHAPTER 7. HOMOLOGY THEORY

• Let F : G → E be a homomorphism of an Abelian group G into an Abeliangroup E. The set of elements of G mapped to the identity element of E isdenoted by

Ker F = g ∈ G | F(g) = 0E ,

where 0E is the identity element of E, and called the kernel of the homo-morphism F.

• The image of the homomorphism F : G → E is the set

Im F = h ∈ E | h = F(g) for some g ∈ G .

• A homomorphism F : G → E of a group G into a group E is called anisomorphism if it is a bijection.

• A subset H of a group G is called a subgroup of G if it contains the identityelement and is closed under the group operation.

• For any homomorphism F : G → E the kernel Ker F is a subgroup of G.

• Let G be an Abelian group and H be a subgroup of G. We say that twoelements of G are equivalent if they differ by an element of H.

• Let g ∈ G be an element of G. Then the set of all elements of G equivalentto g, denoted by [g] = g + H, is an equivalence class of g called a coset.

• The set of cosets is denoted by G/H.

• An element g used to describe a coset [g] is called a representative.

• The set of cosets G/H is an Abelian group, called the quotient group, withthe addition defined by

[g] + [g′] = [g + g′] .

• The projection map π : G → G/H defined by

π(g) = [g]

is a homomorphism.


7.1. ALGEBRAIC PRELIMINARIES 175

•

Theorem 7.1.1 Fundamental Theorem of Homomorphisms. Let Gand F be groups. Let F : G → F be a homomorphism. Then

G/Ker F Im F .

Proof :

1. SinceF(g + Ker F) = F(g) .

•

Theorem 7.1.2 Let G and E be Abelian groups, and H ⊂ G and N ⊂ Ebe their subgroups so that G/H and E/N are the quotient groups. LetF : G → E be a homomorphism such that the image of the subgroup Hof G is the subgroup N of E, that is,

F(H) = N .

Then the homomorphism F induces a homomorphism of the quotientgroups

F∗ : G/H → E/N .

Proof :

1. Letπ : E → E/N

be the projection homomorphism defined by, for any x ∈ E

π(x) = [x] = x + N .

2. ThenF∗ = π F : G/H → E/N

is a homomorphism.

• A field is a set K with two binary operations, addition, +, and multiplication,·, that satisfy the following conditions:

1. both addition and multiplication are associative,

2. both addition and multiplication are commutative,



3. both addition and multiplication have identity elements, the additiveidentity 0 and the multiplicative identity 1, such that 0 , 1,

4. the multiplication is distributive with respect to addition,

5. every element has an additive inverse,

6. every nonzero element has a multiplicative inverse.

• In particular, any field is an Abelian group with respect to addition.

• A vector space over a field K consists of a set E, whose elements are calledvectors, and the field K, whose elements are called scalars, with two oper-ations: vector addition, + : E × E → E, and multiplication by scalars,· : K × E → E, that satisfy the following conditions:

1. the vector addition is associative and commutative,

2. there is an additive identity, called the zero vector,

3. every vector has an additive inverse, called the opposite vector,

4. for any v ∈ E, a, b ∈ K,

(a) a(bv) = (ab)v,(b) (a + b)v = av + bv,(c) a(u + v) = au + av,(d) 1 v = v .

• Let E be a vector space and F be a vector subspace of E. Then E/F is avector space.

• Let E be an inner product vector space and F be its subspace. Then

E/F = F⊥

is the orthogonal complement of F in E, and

π : E → F⊥

is the orthogonal projection to F⊥.


7.1. ALGEBRAIC PRELIMINARIES 177

7.1.2 Finitely Generated and Free Abelian Groups• Let G be an Abelian group. Let g1, . . . , gr ∈ G be some elements of G and

H =

r∑k=1

nkgk | gk ∈ G, nk ∈ Z

be a set of linear combinations of gk.

• Then H is a subgroup of G. The elements gk are called the generators of Hand H is said to be generated by gk.

• If a group G is generated by finitely many elements of G, then G is called afinitely generated group.

• The elements g1, . . . , gr are linearly independent if for any integer coeffi-cients n1, . . . , nr the linear combination

r∑k=1

nkgk , 0

is not equal to zero.

• A finitely generated group G is called a free Abelian group of rank r if itis generated by r linearly independent elements.

7.1.3 Cyclic Groups• An Abelian group generated by one element is called a cyclic group.

• Infinite cyclic groups.

• Finite cyclic group.

•

Theorem 7.1.3 Fundamental Theorem of Finitely GeneratedAbelian Groups. Let G be a finitely generated Abelian group with mgenerators. Then G is isomorphic to the direct sum of cyclic groups,

G Z ⊕ · · · ⊕ Z︸︷︷︸r

⊕Zk1 ⊕ · · · ⊕ Zkp ,

where m = r + p. The number r is called the rank of G.

Proof :



1.


7.2. SINGULAR CHAINS 179

7.2 Singular Chains• The standard Euclidean p-simplex in Rp is the convex set ∆p ⊂ R

p gener-ated by an ordered (p + 1)-tuple (P0, P1, . . . , Pp) of points in Rp

P0 = (0, . . . , 0), Pi = (0, . . . , 0, 1, 0, . . . , 0), i = 1, . . . , p ,

all of whose components are 0 except for the i-th component, which is equalto 1.

• We use the following notation

∆p = (P0, P1, . . . , Pp) .

• Let M be an n-dimensional manifold. A singular p-simplex in M is adifferentiable map

σp : ∆p → M .

• By slightly abusing notation we denote the image σp(∆p) of ∆p in M underthe map σp just by σp.

• Let α be a p-form on M and σp be a p-simplex in M. We define the integralof α over σp by ∫

σp

α =

∫∆p

σ∗pα .

• The k-th face of a standard p-simplex ∆p = (P0, P1, . . . , Pp) (a face oppositeto to the vertex Pk) is the convex set in Rp generated by an ordered p-tupleof points in Rp

∆(k)p−1 = (P0, . . . , Pk, . . . , Pp)

where the k-th point Pk is omitted.

• Since this points lie in Rp and Rp−1 a face is not a standard (p − 1)-simplex.It can be rather described as a singular simplex in Rp defined by the uniqueaffine map

fk : ∆p−1 → ∆p

whose image in Rp is exactly ∆(k)p .



• Such a map is uniquely defined as follows. Let ∆p−1 = (Q0, . . . ,Qp−1),where Qi ∈ R

p−1, be the standard (p−1)-simplex inRp−1 and ∆p = (P0, . . . , Pp),where Pi ∈ R

p, be the standard p-simplex in Rp. Then

fk(Qi) = Pi for i = 0, . . . , k − 1

andfk(Qi) = Pi+1 for i = k, . . . , p − 1 .

• If Q is a point in Rp−1 with coordinates (xi) = (x1, . . . , xp−1), then P = fk(Q)is the point in Rp with coordinates (yµ) = (y1, . . . , yp), where

yµ = Aµ(k) jx

j + bµ(k) .

The above requirements fix the matrix A(k) and the vector b(k) uniquely.

• Problem. Find the maps fk.

• Let M and V be manifolds and ϕ : V → M be a differentiable map. Letσp : ∆p → V be a singular p-simplex in V .

• Then the composition map

ϕ σp : ∆p → M

defines a singular p-simplex in M.

• Thus, the compositionσp fk : ∆p−1 → M

defines a singular (p−1)-simplex in M, which is the k-th face of the singularp-simplex σp in M.

• The boundary ∂∆p of the standard p-simplex ∆p is defined as follows. Forp > 0, we define

∂(P0, P1, . . . , Pp) =

p∑k=0

(−1)k(P0, . . . , Pk, . . . , Pp) ,

that is, the boundary is the formal sum

∂∆p =

p∑k=0

(−1)k∆(k)p−1 .

For p = 0, we define∂∆0 = 0 .



• Examples.

• The boundary of a standard simplex is not a simplex, but an integer (p−1)-chain.

• Let G be an Abelian group. A singular p-chain on M with coefficients inG is a finite formal sum

cp =

r∑k=1

gkσkp

of singular simplexes σkp : ∆p → M with coefficients gk which are elements

of the group G.

• Examples.

• Let S p(M) be the set of all singular p-simplexes in M. Then, a p-chain is afunction

cp : S p(M)→ G ,

such that its value is not equal to zero only for finitely many simplexes.These simplexes are exactly σk

p listed in the formal sum, and the values ofthe function cp are exactly the coefficients of the formal sum, that is

cp(σkp) = gk .

• Alternatively, a p-chain can be thought of as a finite subset of S p(M) × G,that is, a finite set of ordered pairs

cp = (σkp, gk)rk=1

• The notation of a p-chain as a sum, or as a function, is useful since we candefine the addition of p-chains simply as addition of corresponding func-tions. If two p-chains cp and c′p have the same p-simplex σp in both ofthem, then in the sum cp + c′p we add the corresponding group elements gand g′. That is, if

cp = gσp + . . . , and c′p = g′σp + . . . ,

thencp + c′p = (g + g′)σp + . . . .



• We denote the identity element of G by 0 and define the zero p-chain simplyby

0p = 0 .

• Then the set of all singular p-chains on M with coefficients in G forms anAbelian group, called the singular p-chain group of M with coefficients inG and denoted by Cp(M; G).

• A chain with integer coefficients, when G = Z, is called an integer chain.

• The standard p-simplex ∆p is an integer p-chain in Rp with only one term:cp = 1 · ∆p. That is, it is an element of Cp(Rp;Z).

• The boundary of the standard p-simplex in Rp,

∂∆p =

p∑k=0

(−1)k∆(k)p−1

is an integer (p − 1)-chain in Rp, that is, an element of Cp−1(Rp;Z).

• Let M and V be closed manifolds. Let F : M → V be a map of M intoV and σp : ∆p → M be a singular p-simplex in M. Then the compositionF σp : ∆p → V is a singular p-simplex in V . We denote it by

F∗σp = F σp .

• The induced chain homomorphism

F∗ : Cp(M; G)→ Cp(V; G)

is defined by: for any gk ∈ G and σkp ∈ S p(M),

F∗

r∑k=1

gkσkp

=

r∑k=1

gkF∗σkp

• Let F : M → V and E : V → W be two maps of manifolds and F∗ :Cp(M; G) → Cp(V; G) and E∗ : Cp(V; G) → Cp(W; G) be the correspond-ing induced chain homomorphisms. Then

(E F)∗ = E∗ F∗ .



• Let σp : ∆p → M be a singular p-simplex in M. Then its boundary ∂σp isthe integer (p − 1)-chain in M defined by

∂σp = σp∗(∂∆p) .

• In more detail

∂σp = σp∗(∂∆p)

=

p∑k=0

(−1)kσp∗(∆(k)p−1) =

p∑k=0

(−1)k(σp fk) .

Recall that σp∗(∆(k)p−1) = (σp fk) is the k-th face of the singular p-simplex

σp.

• That is, the boundary of the image of ∆p is the image of the boundary of ∆p.

• The boundary of any singular p-chain with coefficients in G is defined by,for any gk ∈ G, σk

p ∈ S p(M),

∂

r∑k=1

gkσkp

=

r∑k=1

gk∂σkp .

• This leads to the boundary homomorphism

∂ : Cp(M; G)→ Cp−1(M; G) .

• Let F : M → V be a map, σp be a singular p-simplex in M and F∗σp be theinduced singular p-simplex in V . Then

∂(F∗σp) = ∂(F σp)= (F σp)∗(∂∆p)= (F∗ σp∗)(∂∆p)= F∗[σp∗(∂∆p)]= F∗(∂σp)

• More generally, let

cp =

r∑k=1

gkσkp

be a p-chain on M and F∗cp be the induced p-chain on V .



• Then∂(F∗cp) = F∗(∂cp) .

• Therefore,∂ F∗ = F∗ ∂ ,

in other words, the boundary of an image is the image of the boundary.

• Thus, we obtain a commutative diagram

F∗Cp(M; G) → Cp(V; G)

∂ ↓ ∂ ↓Cp−1(M; G) → Cp−1(V; G)

F∗

which is a fancy way to say that for any p-chain cp ∈ Cp(M; G) on M wehave

F∗(∂cp) = ∂(F∗cp) .

•Theorem 7.2.1 The boundary of a boundary is zero, that is,

∂2 = 0 .Proof :

1. For a standrad p-simplex ∆p we have

∂∂∆p =

p∑k=0

(−1)k∂∆(k)p

=

p∑k=0

(−1)k∂(P0, . . . , Pk, . . . , Pp)

=

p∑k=0

(−1)kk−1∑j=0

∂(P0, . . . , P j, . . . , Pk, . . . , Pp)

+

p∑k=0

(−1)kp∑

j=k+1

∂(P0, . . . , Pk, . . . , P j, . . . , Pp)

= 0

because of the pairwise cancellation.



2. Then, for a singular p-simplex σp

∂∂σp = ∂[σp∗(∂∆p)] = σp∗∂(∂∆p) = σ∗(0) = 0

7.2.1 Examples• Cylinder.

• Mobius Band.



7.3 Singular Homology Groups

7.3.1 Cycles, Boundaries and Homology Groups

• We can define the singular p-chains with coefficients in a field K.

• Furthermore, we can define the multiplication of p-chains by elements ofthe field K, called the scalars by, for any a, bi ∈ K,

a

r∑i=1

biσip

=

r∑i=1

abiσip .

• The chain groups Cp(M,K) with coefficients in a field K become infinite-dimensional vector spaces.

• In this case the boundary homomorphism becomes a linear transformation(operator) in a vector space.

• Let M be a manifold and G an Abelian group. A singular p-chain zp in Mwhose boundary is 0 is called a singular p-cycle.

• The set of all p-cycles in M

Zp(M; G) = zp ∈ Cp(M; G) | ∂zp = 0

is a subgroup of the chain group Cp(M; G) called the p-cycle group.

• Obviously the p-cycle group is the kernel of the boundary homomorphism

Zp(M; G) = Ker ∂p ,

where∂p : Cp(M; G)→ Cp−1(M; G) .

• In the case, when G = K is a field, then Zp(M; K) is a vector subspace ofthe vector space Cp(M; K).

• A singular p-chain bp in M that is the boundary of a singular (p + 1)-chainis called a p-boundary.


7.3. SINGULAR HOMOLOGY GROUPS 187

• The set of all p-boundaries in M

Bp(M; G) = bp ∈ Cp(M; G) | bp = ∂cp+1 for some cp+1 ∈ Cp+1(M; G)

is a subgroup of the chain group Cp(M; G) called the p-boundary group.

• Obviously the p-boundary group is the image of the boundary homomor-phism

Bp(M; G) = Im ∂p+1 .

• Since every p-boundary is a p-cycle (because of ∂2 = 0) the group Bp(M; G)is a subgroup of Zp(M; G).

• In the case, when G = K is a field, then Bp(M; K) is a vector subspace ofthe vector space Zp(M; K).

• Let M be a manifold and G be an Abelain group. We say that two p-cyclesare homologous if they differ by a boundary.

• The set of equivalence classes of p-cycles homologous to each other, thatis, the quotient group

Hp(M; G) = Zp(M; G)/Bp(M; G) ,

is called the p-th homology group.

• In the case when the coefficient group G is a field G = K, all the groups, Zp,Bp and Hp are vector spaces.

7.3.2 Simplicial Homology• The important fact is that if M is a compact manifold, then the vector space

Hp(M; K) is finite dimensional. (This can be proved but we will not do that).

• Let M be a compact n-dimensional manifold. Then there is a triangulationof M by finitely many n-simplexes diffeomorphic to the standard n-simplex∆n.

• Thus, M is a union of finitely many n-simplexes, which are either disjointof intersect along common r-simplexes with r = 0, 1, . . . , n − 1.



• The set of all these simplexes form a finite simplicial complex with somecoefficient group G.

• All the simplicial chain groups Cp(M; G), Zp(M; G) and Bp(M; G) are finitelygenerated Abelian groups.

• Therefore, the homology group Hp(M; G) is a finitely generated group.

• Since any simplicial cycle can be described as a singular cycle, there arehomomorphisms

Zp → Zp, Bp → Bp

and the induced homomorphism

Hp → Hp .

•

Theorem 7.3.1 Let M be a compact manifold and G be an Abeliangroup. Then the singular homology groups are isomorphic to the sim-plicial homology groups

Hp(M; G) = Hp(M; G)

and are finitely generated Abelian groups.

Proof : Nontrivial.

•Corollary 7.3.1 Let M be a compact manifold and K be field. Thenthe homology groups Hp(M; K) are finite-dimensional vector spaces.

Proof : Follows from above theorem.

7.3.3 Betti Numbers and Topological Invariants• Let M be a compact manifold. The dimensions of the real homology groups

Hp(M;R) are called the Betti numbers, that is,

Bp(M) = dim Hp(M;R) .

• The Betti number Bp is the maximal number of linearly independent p-cycles modulo a boundary.



• Let M and V be manifolds, G be a n Abelain group, F : M → V be a mapand F∗ : Cp(M; G) → Cp(V; G) be the induced homomorphism of chaingroups.

• Since the induced homomorphism F∗ commutes with the boundary homo-morphism ∂, the groups Zp(M; G) and Bp(M; G) are closed under F∗.

• Therefore, the homomorphism F∗ naturally acts on the homology groups

F∗ : Hp(M; G)→ Hp(V; G) .

• If F : M → V is a homeomorphism, then there is the inverse homeomor-phism F−1 : V → M and the inverse induced homomorphism

F−1∗ : Hp(V; G)→ Hp(M; G) .

• In this case, the induced homomorphism F∗ is an isomorphism.

•

Theorem 7.3.2 Let M and V be compact homeomorphic manifoldsand G be an Abelian group. Then their homology groups are isomor-phic, that is, for any p

Hp(M; K) Hp(V; G) .


• Thus, homology groups are topological invariants.

•

Corollary 7.3.2 Let M and V be compact manifolds. If there is anAbelian group G and an integer p such that their homology groups arenot isomorphic, that is,

Hp(M; K) Hp(V; G) ,

then the manifolds M and V are not homeomorphic to each other.

• Remark. The converse is not true.



7.3.4 Some Theorems from Algebraic Topology• A manifold M is path-connected (or just connected) if every two points in

M can be connected by a piecewise-smooth curve.

• Let M be a manifold and G be an Abelian group.

• A point p in a manifold M is a 0-chain. Since ∂p = 0, then each point is a0-cycle (by definition).

• A smooth map C : [0, 1]→ M defines a singular 1-simplex and

∂C = C(1) −C(0)

is a 0-chain.

• More generally, for any g ∈ G, then gC is a singular 1-simplex and

∂(gC) = gC(1) − gC(0) .

• A piecewise-smooth map C : [0, 1]→ M defines a 1-chain (as a formal sum

c1 =

r∑k=1

gCk

of smooth pieces with the same coefficient) and

∂c1 = gC(1) − gC(0)

is a 0-chain.

•

Theorem 7.3.3 Let M be a compact connected manifold and G be anAbelian group. Then

H0(M; G) = Gp = gp | g ∈ G, p ∈ M .

Proof :

1. In a connected manifold any two 0-simplexes with the same coefficientare homologous.

2. Moreover, for any point p ∈ M, g , 0, and any element g ∈ G, thereis no 1-chain c1 such that ∂c1 = gp.



3. Therefore, a multiple gp of a single point is not a boundary for anyg , 0.

4. Thus, any point p ∈ M is a 0-cycle that is not a boundary.

5. Moreover, for any g ∈ G and any p ∈ M the 0-chain gp is a 0-cyclethat is not a boundary.

• In particular,H0(M;Z) = 0,±p,±2p, . . .

andH0(M;R) = R

is a one-dimensional vector space.

•

Corollary 7.3.3 Let M be a compact connected manifold. Then thezero Betti number is equal to

B0(M) = 1 .


•

Theorem 7.3.4 Let M be a compact manifold consisting of k con-nected pieces M1, . . .Mk. Then

H0(M;R) = Rp1 + · · · + Rpk ,

where pi ∈ Mi, i = 1, . . . , k, meaning

Rp1 + · · · + Rpk =

k∑i=1

ai pi

∣∣∣ ai ∈ R, pi ∈ Mi, i = 1, . . . , k

.

Proof :

• In this caseH0(M;R) = Rk

is a k-dimensional vector space.



•

Corollary 7.3.4 Let M be a compact manifold consisting of k con-nected pieces. Then the zero Betti number is equal to

B0(M) = k .


• Let V be a p-dimensional oriented closed (compact without boundary) man-ifold.

• Then a triangulation of V defines an integer p-cycle, denoted by [V] so that

∂[V] = 0 .

• This does not work for non-orientable closed manifolds. A triangulation ofa non-orientable closed manifold V gives an integer p-chain, which is not ap-cycle since

∂[V] , 0 .

• Example. Klein bottle.

• By changing the coefficient group G sometimes one can get a p-cycle evenfor non-orientable manifolds.

•

Theorem 7.3.5 Let M be an n-dimensional compact manifold and Vbe a closed oriented p-dimensional submanifold of M. Let G be anAbelian group and g be an element of G. Consider a triangulation of Vin p-triangles and assign to each p-triangle in this triangulation of Vthe same coefficient g ∈ G. Then g[V] defines a p-cycle zp in Hp(M; G).

Proof : Without proof.

• Remark. A p-cycle is a generalization of the concept of a closed orientedsubmanifold.

• In the case of real homology groups, when G = R, the following theorem istrue.



•

Theorem 7.3.6 Let M be an n-dimensional compact manifold. Everyreal p-cycle zp in Hp(M;R) is homologous to a finite formal sum

zp ∼

r∑k=1

akVk

of closed oriented p-dimensional submanifolds Vk of M with real coef-ficients ak.

Proof : Nontrivial.

•

Theorem 7.3.7 Let M be an n-dimensional manifold and G be anAbelian group. Let zp and z′p be two cycles in Hp(M; G) that can bedeformed into each other. Then they are homologous to each other

zp ∼ z′p .

Proof :

1. Since the deformation defines a deformation chain cp+1 such that

∂cp+1 = z′p − zp .

•

Proposition 7.3.1 Let M be an n-dimensional closed manifold and Gbe an Abelian group. Then for p > n the singular homology groupsHp(M; G) are trivial

Hp(M; G) = 0 .

Proof :

1. Singular homology groups are isomorphic to the simplicial homologygroups.

2. Since there are no simplicial complexes of dimension p > n then allsimplicial homology groups are trivial for p > n.



7.3.5 Examples.• Sphere S n.

• From the facts that S n is connected, orientable and closed it follows that

H0(S n; G) = Hn(S n; G) = G ,

Hp(S n; G) = 0 , for p , 0, n ,B0(S n) = Bn(S n) = 1 ,Bp(S n) = 0 , for p , 0, n .

• Torus T 2.

H0(T 2; G) = H2(T 2; G) = G ,

H1(T 2; G) = GA + GB ,B0(T 2) = 1, B1(T 2) = 2, B2(T 2) = 1,

where A and B are the basic 1-cycles.

• Klein Bottle K2.

• Since K2 is connected closed non-orientable it follows that

H0(K2;Z) = Z,

H2(K2,Z) = 0 ,H1(K2;Z) = ZA + Z2B ,

where A and B are the basic 1-cycles, and

H0(K2;R) = R,

H2(K2,R) = 0 ,H1(K2;R) = RA ,B0(K2) = 1, B1(K2) = 1, B2(K2) = 0 .

• Real Projective Plane RP2.

• RP2 is connected closed non-orientable.

H0(RP2;Z) = Z,

H2(RP2,Z) = 0 ,H1(RP2;Z) = Z2A ,



where A is the basic 1-cycle, and

H0(RP2;R) = R,

H2(RP2,R) = 0 ,H1(RP2;R) = 0 ,B0(RP2) = 1, B1(RP2) = 0, B2(RP2) = 0 .

• Torus T 3.

• T 3 is a connected closed orientable manifold.

H0(T 3;Z) = H3(T 3,Z) = Z ,

H1(T 3;Z) = ZA + ZB + ZC ,

H2(T 3;Z) = ZD + ZE + ZF ,

where A, B and C are basic 1-cycles, and D, E and F are basic 2-cyles,

B0(T 3) = 1, B1(T 3) = B2(T 3) = 3, B2(T 3) = 1 .

• Real Projective Space RP3.

• RP3 is connected closed orientable.

H0(RP3;R) = H3(RP3,R) = R ,

H1(RP3;R) = H2(RP3;R) = 0 ,B0(RP3) = B3(RP3) = 1, B1(RP3) = B2(RP3) = 0 .



7.4 de Rham Cohomology Groups• Let M be a manifold and G = R be the coefficient group.

• Then Cp(M;R), Zp(M;R), Bp(M;R) and Hp(M;R) are vector spaces.

• For simplicity we will denote them in this section simply by Cp(M), Zp(M),Bp(M) and Hp(M).

• Let Cp(M) = C∞(Λp(M)) be the space of smooth p-forms on M.

• We will call the closed p-form p-cocyles and the space

Zp(M) = αp ∈ Cp(M) | dαp = 0

of all closed p-forms on M, the cocycle group.

• The exact p-forms on M are called the p-coboundaries and the space

Bp(M) = αp ∈ Zp(M) | αp = dβp+1 for some βp+1 ∈ Cp+1(M)

of all exact p-forms on M is called the coboundary group.

• Both Zp(M) and Bp(M) are vector spaces with real coefficients.

• Recall that the exterior derivative is a map

dp : Cp(M)→ Cp+1(M)

such thatKer dp = Zp(M)

andIm dp−1 = Bp(M) .

• Two closed forms are said to be equivalent (or cohomologous) if they differby an exact form.

• The collection of all equivalence classes of closed forms is the quotientvector space

Hp(M) = Z p(M)/Bp(M)

called the p-th de Rham cohomology group.


7.4. DE RHAM COHOMOLOGY GROUPS 197

• de Rham cohomology groups are vector spaces.

• Let

cp =

r∑k=1

akσkp

be a real p-chain in M, and α be a p-form on M.

• We define the integral of α over cp by

⟨α, cp

⟩=

∫cp

α =

r∑k=1

ak

∫σk

p

α .

• Thus every p-form on M defines a linear functional on Cp(M)

α : Cp(M)→ R ,

bycp

α7−→

⟨α, cp

⟩.

• The space of all p-forms can be naturally identified with the dual spaceC∗p(M)

Cp(M) C∗p(M) .

• Furthermore, by Stokes theorem we have for a (p − 1)-form⟨dα, cp

⟩=

⟨α, ∂cp

⟩.

• Thus, for every p-cycle zp, that is, if ∂zp = 0,⟨dα, zp

⟩= 0 ,

and for every closed form α, that is, if dα = 0,⟨α, ∂cp

⟩= 0 .

• More generally, let αp ∈ Z p(M) be a closed p-form, βp+1 ∈ Cp+1(M) be a(p+1)-form, zp ∈ Zp(M) be a p-cycle and cp+1 ∈ Cp+1(M) be a (p+1)-chain.Then ⟨

α + dβ, zp + ∂cp

⟩=

⟨α, zp

⟩.



• Therefore, for every equivalence class [αp] ∈ Hp(M) of closed forms wecan define a linear functional Hp(M)→ R on the space of homology groupsby, for any [zp] ∈ Hp(M), ⟨

[αp], [zp]⟩

=⟨αp, zp

⟩.

This is well defined since the right hand side does not depend on the choiceof representatives in the equivalence classes.

• This naturally identifies the space of cycles with the space of cocycles

Z p(M) Z∗p(M) .

• For a closed p-form αp and a p-cycle zp the value of the functional⟨α, zp

⟩=

∫zp

αp

is called the period of the form αp on the cycle zp.

• We conjecture thatHp(M) H∗p(M) .

•

Proposition 7.4.1 . Let M be a closed manifold. Then for any linearfunctional ϕ : Hp(M)→ R on homology groups there is a closed p-formαp such that

ϕ(zp) =⟨αp, zp

⟩.

Proof : Difficult.

•

Corollary 7.4.1 . Let M be a closed manifold. Let k = Bp(M) be the p-th Betti number. Let z(1)

p , . . . , z(k)p , be a basis of p-cycles in the homology

groups Hp(M) and π1, . . . , πk be arbitrary real numbers. Then there isa closed p-form αp such that⟨

αp, z(i)p

⟩= πi, i = 1, 2, . . . , k .

Proof :


7.4. DE RHAM COHOMOLOGY GROUPS 199

•

Proposition 7.4.2 . Let M be a closed manifold. Let αp ∈ Z p(M) be aclosed p-form on M such that for any p-cycle zp ∈ Zp(M)⟨

αp, zp

⟩= 0 .

Then the p-form αp is exact.

Proof : Difficult.

•

Theorem 7.4.1 de Rham Theorem. Let M be a closed manifold. Thenthe map

Hp(M)→ H∗p(M) ,

that associates to each equivalence class [αp] of closed p-forms a linearfunctional Hp(M)→ R on the homology group Hp(M) defined by

[zp][αp]7−→

⟨αp, zp

⟩,

is an isomorphism.

Proof :

7.4.1 Examples• Torus T 2.

• Closed Surfaces in R2.



7.5 Harmonic Forms• Let (M, g) be a closed oriented n-dimensional Riemannian manifold with a

Riemannian metric g and the Riemannian volume n-form vol .

• Let Λp(M) be the bundle of p-forms.

• Recall that there is a natural fiber inner product on Λp defined by

〈α, β〉 =1p!

gi1 j1 · · · gip jpαi1...ipβ j1... jp ,

and the corresponding fiber norm

||α|| =√〈α, α〉 .

• Also, there is a duality between p-forms and (n − p)-forms defined by theHodge star operator

∗ : Λp → Λn−p ,

that maps any p-form α to a (n − p)-form ∗α dual to α defined as follows.

• For each p-form α the form ∗α is the unique (n − p)-form such that for anyp-form β

β ∧ ∗α = 〈β, α〉 vol .

• In components, this means that

(∗α)ip+1... in =1p!εi1... ipip+1...in

√|g|gi1 j1 · · · gip jpα j1... jp

=1p!

1√|g|

gip+1 jp+1 · · · gin jnεj1... jp jp+1... jnα j1... jp .

• Recall that the Hodge star maps forms to pseudo-forms and vice-versa.

• Recall also that for any p-form α,

∗2α = (−1)p(n−p)α ,

meaning that∗−1 = (−1)p(n−p) ∗ .


7.5. HARMONIC FORMS 201

• The coderivative is a linear map

δ : Λp → Λp−1

defined byδ = ∗−1d∗ = (−1)(n−p+1)(p−1) ∗ d ∗ .

• The exterior derivative and the coderivative satisfy the important conditions

d2 = δ2 = 0 .

• Problem. Show that in local coordinates the coderivative of a p-form α isthe (p − 1)-form δα with components

(δα)i1...ip−1 = gi1 j2 · · · gip−1 jp

1√|g|∂ j

( √|g|g jk1g j2k2 · · · g jpkpαk1k2...kp

)

• Now we define the L2-inner product of p-forms by

(α, β)L2 =

∫Mα ∧ ∗β =

∫M〈α, β〉 vol ,

and the L2-norm||α||L2 =

√(α, α)L2 .

• This makes the space C∞(Λp(M)) of smooth p-forms an inner-product vec-tor space.

• The completion of C∞(Λp(M)) in the L2-norm gives the Hilbert spaceL2(Λp(M)) of square-integrable p-forms.

• Let A : H → H be an operator on a Hilbert space H. The adjoint of theoperator A with respect to the inner product of the space H is the operatorA∗ : H → H defined by, for any ϕ, ψ ∈ H,

(Aϕ, ψ) = (ϕ, A∗ψ) .



•

Theorem 7.5.1 Let M be a closed orientable Riemannian manifold.Then the adjoint of the exterior derivative is the negative coderivative

d∗ = −δ .

That is, for any p-form α and any (p + 1)-form β,

(dα, β) = −(α, δβ) .

Proof :

1.

•

Theorem 7.5.2 Let M be a compact orientable Riemannian manifoldwith boundary. Then for any p-form α and any (p + 1)-form β,

(dα, β) + (α, δβ) =

∫∂Mα ∧ ∗β .

Proof :


• Notice that in case of a manifold with boundary the coderivative is the neg-ative adjoint of the exterior derivative only on the forms that satisfy one ofthe two types of boundary conditions:

α|∂M = 0 or ∗ α|∂M = 0 .

• Let g be the Riemannian metric and ∇ be the corresponding Levi-Civitaconnection. It defines a natural connection on the bundle of p-forms. TheLaplacian on p-forms is the operator

∆ : C∞(Λp(M))→ C∞(Λp(M))

defined by∆ = gi j∇i∇ j .



• The Hodge Laplacian on p-forms is the operator

L : C∞(Λp(M))→ C∞(Λp(M))

defined byL = dδ + δd = (d + δ)2 .

• The operators d and δ commute with the Hodge Laplacian, i.e.

dL = Ld , δL = Lδ .

•

Theorem 7.5.3 For any p there holds

L = ∆ + W ,

where W : Λp → Λp is an endomorphism on the bundle of p-formscalled the Weitzenbock endomorphism.

Proof :

1.

• Weitzenbock endomorphism is a linear combination of Riemann curvaturetensor, that is, when acting on p-forms W has the form

W i1...ipj1... jp = Fmni1...ip

kl j1... jpRkl

mn ,

where Fmni1...ipkl j1... jp is constructed only from the Kronecker symbol δi

j andthe metric gi j and gi j.

• Problem. Obtain the expression for the Weitzenbock endomorphism forp-forms. Hint: replace partial derivatives by covariant derivatives and usethe definition of the curvature.

• A p-form α is called harmonic if

Lα = 0 .

•



•

Theorem 7.5.4 Let M be a closed Riemannian manifold. Then a p-form α is harmonic if and only if it is closed and coclosed, that is,

dα = δα = 0 .Proof :

1. Use0 = (Lα, α) = ||dα||2 + ||δα||2 .

• On a closed Riemannian manifold the Laplacian and the Hodge Laplacianare self-adjoint elliptic operators.

•

Theorem 7.5.5 Hodge Theorem. Let M be a closed Riemannianmanifold. Then:

1. The vector spaceH p of harmonic p-forms is finite-dimensional.

2. The equationLα = ρ

has a solution if and only if ρ is orthogonal toH p.

Proof :

1.

•

Theorem 7.5.6 Let M be a closed Riemannian manifold. Then any p-form β is a sum of an exact form dα, a coexact form δγ and a harmonicform h, that is,

β = dα + δγ + h .

In other words, there is an orthogonal decomposition (called the Hodgedecomposition)

Λp = Im dp−1 + Im δp+1 +H p .

Proof :

1.



•

Corollary 7.5.1 Let M be a closed Riemannian manifold. Then anyclosed p-form β is a sum of an exact form dα and a harmonic form h,that is,

β = dα + h .

Proof :

1.

•

Corollary 7.5.2 Let M be a closed Riemannian manifold. Then eachde Rham class of cohomologous closed p-forms has a harmonic repre-sentative.

Let k = Bp(M) be the p-th Betti number. Let z(1)p , . . . , z

(k)p , be a basis of

real p-cycles in the real homology groups Hp(M) and π1, . . . , πk be ar-bitrary real numbers. Then there is a unique harmonic p-form hp suchthat ⟨

hp, z(i)p

⟩= πi, i = 1, 2, . . . , k .

Proof :

1.

• The metric g is said to have positive Ricci curvature if its Ricci tensor ispositive-definite.

•

Corollary 7.5.3 Bochner Theorem Let M be a closed Riemannianmanifold with positive Ricci curvature. Then the first Betti number van-ishes, i.e.

B1(M) = 0 .

That is there are no harmonic 1-forms on M.

Proof :

1. Let h be a harmonic 1-form. Then

0 =12

∫M

∆ 〈h, h〉 vol =

∫M

Ri jhih jvol + ||∇ jhi||2 ≥ 0 .



2. Thus h = 0.

• Remark. The elements of the first homology group H1(M,G) are equiva-lence classes of 1-cycles.

• The 1-cycles are closed oriented curves (loops) on M.

• If a closed curve can be deformed to a point, then it is a boundary of asurface (a 2-simplex).

• That is, a closed curve that can be contracted to a point is a trivial 1-cycle.

•

•

Corollary 7.5.4 For any simply connected manifold M and any Abeliangroup G,

H1(M,G) = 0 .


7.6. RELATIVE HOMOLOGY AND MORSE THEORY 207

7.6 Relative Homology and Morse Theory

7.6.1 Relative Homology• Let M be a compact Riemannian n-dimensional manifold with smooth bound-

ary ∂M andi : ∂M → M

be the inclusion map.

• Let xi, i = 1, . . . , (n − 1), be the local coordinates on the boundary ∂M andxµ, µ = 1, . . . , n, be the local coordinates on M in a patch U close to theboundary. Then the inclusion map is defined locally by

xµ = xµ(x) .

• Close to the boundary there exists a system of local coordinates (xµ) = (xi, r)so that the boundary is described by the equation

r = 0 .

The coordinate r can be chosen to be the normal geodesic distance from theboundary so that the vector ∂r is normal to the boundary.

• The inclusion map in this case is given by

xi = xi, r = 0 .

Therefore,∂xk

∂x j = δkj ,

∂r∂x j = 0 .

• A p-form α on M is called normal to ∂M if

i∗α = 0 ,

that is,∂xµ1

∂x j1· · ·

∂xµp

∂x jpαµ1...µp(x(x)) = 0 .

• This just means that a normal p-form α vanishes on tangent vectors to theboundary.



• In other words, a p-form α is normal to the boundary if

α(v1, . . . , vp)∣∣∣∣∂M

= 0

for any tangent vectors v1, . . . , vp.

• In the special coordinate system (xi, r) this means that a p-form is normal ifall purely tangential components vanish on the boundary

α j1... jp |∂M = 0 .

• Obviously,i∗dr = 0 .

Therefore, a p-form α is normal if there is a (p − 1)-form β such that

α = β ∧ dr .

• A p-form α is called tangent to the boundary if the (n − p)-form ∗α isnormal, that is,

i∗ ∗ α = 0 .

• In other words, a p-form α is tangent to the boundary if

α(v1, . . . , vp−1,N)∣∣∣∣∂M

= 0

for any tangent vectors v1, . . . , vp−1 and a normal vector N, meaning that theinterior product with a normal vector vanishes

iNα∣∣∣∣∂M

= 0 .

• In local coordinates, this condition becomes

ελ1...λn−p µ1...µp

∂xλ1

∂x j1· · ·

∂xλn−p

∂x jn−pgµ1ν1 · · · gµpνpαν1...νp

∣∣∣∣∂M

= 0 .

• In the special coordinate system (xi, r) this just means that the componentswith at least one index along the normal direction vanish

α j1... jp−1r

∣∣∣∂M

= 0 .



•

Proposition 7.6.1 Let M be a compact Riemannian manifold withsmooth boundary. Let α and β be two p-forms on M, which are eitherboth normal to the boundary or both tangent to the boundary. Then

(dα, β) = −(α, δβ) .

That is, on normal or tangent p-forms

d∗ = −δ .

Proof : Obvious.

•

Theorem 7.6.1 Let M be a compact Riemannian manifold withsmooth boundary. Let k = Bp(M) be the p-th Betti number. Letz(1)

p , . . . , z(k)p , be a basis of real p-cycles in the real homology groups

Hp(M) and π1, . . . , πk be arbitrary real numbers. Then there is a uniquetangent harmonic p-form hp such that

dh = δh = 0

and ⟨hp, z(i)

p

⟩= πi, i = 1, 2, . . . , k .

Proof :

1.

• More generally,



•


p , . . . , z(k)p , be a basis of real p-cycles in the real homology groups

Hp(M) and π1, . . . , πk be arbitrary real numbers. Let γ be a closed(n − p)-form on ∂M such that for every (n − p)-cycle yn−p on ∂M that isa boundary of a (n − p + 1)-chain cn−p+1 in M, that is,

i∗γ = ∂cn−p+1 ,

the value of the integral of γ over yp vanishes,⟨γ, yp

⟩∂M

=

∫∂Mγ = 0 .

Then there is a unique harmonic p-form hp such that

dh = δh = 0 ,

i∗ ∗ h = ∗h|∂M = γ ,⟨hp, z(i)

p

⟩= πi, i = 1, 2, . . . , k .

Proof :

1.

• Let M be a compact manifold with boundary. A relative p-cycle (mod ∂M)is a p-chain on M whose boundary lies on ∂M.

• A p-cycle with no boundary is called an absolute p-cycle.

• From the point of view of relative homology any p-chain that lies on ∂M isneglected.

• Example.

• Two relative p-cycles are homologous (mod ∂M) if they differ by a trueboundary and a p-chain that lies on ∂M,

c′p − cp = ∂up+1 + vp , v ⊂ ∂M .



• A relative boundary (mod ∂M) is a sum of an absolute boundary and achain that lies on ∂M.

• Example.

• The relative homology group is the quotient group of relative cycles mod-ulo the relative boundaries

Hp(M, ∂M; G) = Zp(M, ∂M; G)/Bp(M, ∂M; G) .

•


p , . . . , z(k)p , be a basis of real relative p-cycles of M (mod ∂M) in the

real homology groups Hp(M, ∂M;R), that is,

Hp(M, ∂M;R) =

k∑i=1

Rci .

Let π1, . . . , πk be arbitrary real numbers. Then there is a unique normalharmonic p-form hp on M such that

dh = δh = 0 ,

and ⟨hp, z(i)

p

⟩= πi, i = 1, 2, . . . , k .

Proof :

1.

7.6.2 Morse Theory• Let M be a closed manifold and f : M → R be a smooth function on M.

• Example.

• A point x0 ∈ M is called a critical point of the function f if

d f∣∣∣x0

= 0 ,∂ f∂xi

∣∣∣∣∣∣x0

= 0 .



• A real number a ∈ R is called a critical value of f if the inverse imagef −1(a) ⊂ M contains at least one critical point.

• A critical point is called inessential if it can be removed by a small defor-mation of the function f .

• A critical point x0 is called non-degenerate if the Hessian

Hi j =∂2 f∂xi∂x j

is non-degenerate, that is,

det Hi j

∣∣∣x0, 0 .

• The Hessian defines a linear operator on the tangent space

H : TxM → TxM

with the matrixHi

j = gikHk j .

• For a non-degenerate critical point the Hessian operator has non-zero realeigenvalues.

• The number of the negative eigenvalues (counted with multiplicities), thatis the dimension of the subspace of the tangent space on which the Hessianis negative-definite, is called the Morse index of the critical point.

•

Lemma 7.6.1 Morse Lemma. Let M be a closed manifold and f :M → R be a smooth real-valued function on M. Let x0 be a nondegen-erate critical point of f with Morse index p. Then in a neighborhood ofx0 there exist local coordinates (x1, . . . , xp, y1, . . . , yn−p) such that

f (x, y) = f (x0) − (x1)2 − · · · − (xp)2 + (y1)2 + · · · + (yn−p)2 .

Proof :

1.



• The number of critical points of index p is called the p-th Morse type num-ber and denoted by Mp.

• Let t be a formal variable. The polynomial

M(t) =

n∑p=0

Mptp

is called the Morse polynomial.

• For each real number a ∈ R we define

Ma = x ∈ M | f (x) ≤ a

and

M−a = x ∈ M | f (x) < a .

• A real number a is called a homotopically critical value of the function fif some relative homology group Hp(Ma,M−

a ; G) is non-zero.

• It turns out that for non-degenerate critical points a value is homotopicallycritical if and only if it is critical.

• That is, for non-degenerate critical points the critical values of f are pre-cisely the values at which new relative cycles appear.

• Let Bp(M) = dim Hp(M) be the Betti numbers. The polynomial

P(t) =

n∑p=0

Bptp

is called the Poincare polynomial.



•

Theorem 7.6.4 Morse Theorem. Let M be a closed manifold andf ; M → R be a smooth function. Suppose that the function f has onlynon-degenerate critical points. Let Mp be the Morse type numbers, Bp

be the Betti numbers, M(t) be the Morse polynomial and P(t) be thePoincare polynomial. Then there is a polynomial

Q(t) =

n−1∑p=0

qptp

with non-negative coefficients, qp ≥ 0, such that

M(t) − P(t) = (1 + t)Q(t) ,

that is,Mp − Bp = qp + qp−1 .

Proof :

1.



•

Corollary 7.6.1 There hold:

1. Weak Morse inequalities

Mp ≥ Bp ,

2. In particular, the total number of critical points is bounded belowby the sum of all Betti numbers

n∑p=0

Mp ≥

n∑p=0

Bp .

3. Strong Morse inequalities

M0 ≥ B0

M1 − M0 ≥ B1 − B0

. . .

Mn − Mn−1 + · · · + (−1)nM0 = Bn − Bn−1 + · · · + (−1)nB0 .

4. In particular,n∑

p=0

(−1)pMp =

n∑p=0

(−1)pBp

Proof :

1.


Chapter 8

Appendix A: Linear Algebra

8.1 Mathematical Background

8.1.1 Sets and Maps• Sets, elements.

• Empty set

• Subset

• Proper subset

• Union, intersection

• Difference of sets

• Complement

• De Morgan’s laws

• Maps, domain, codomain

• Identity map

• Image, inverse image, range

• Injections, surjections, bijections

• Cardinality

217

218 CHAPTER 8. APPENDIX A: LINEAR ALGEBRA

• Finite sets, countably infinite sets, countable sets, uncountable sets

• Composition

• Inverse map

• Binary relations

• Associative binary relations

• Identity element

• Inverse elements

• Equivalence relation

• Equivalence classes

• Partition

• Ordered sets

• Cartesian product

8.1.2 Groups• Groups

• Subgroups

• Cyclic group

• Permutation group

• Transpositions (elementary permutations)

• Even and odd permutations

• Matrix groups

• Abelian groups

• A map from a group G to a group H is a homomorphism.


8.1. MATHEMATICAL BACKGROUND 219

• A bijective homomorphism is an isomorphism.

• A homomorphism from a group to itself is an endomorphism.

• A bijective endomorphism is an automorphism.

• An action of a group G on a set X is a map

G × X → X

given by(g, x) 7→ gx

and satisfying the conditions: for all x ∈ X and all g, h, ∈ G,

ex = x

(gh)x = g(hx)

• A point x ∈ X is called a fixed point of the action of the element g ∈ G if

gx = x .

• The action of the group G on a set X is free if it does not have any fixedpoints except when g = e.

8.1.3 Rings and Fields• A nonempty set R is an associative ring with unit if it has two associative

binary operations with identities, addition, +, and multiplication, ·, such thataddition is commutative and every element has an additive inverse and forall a, b, c ∈ R the distributive law holds:

a(b + c) = ab + ac, (b + c)a = ba + ca

• Additive identity is called zero and denoted by 0.

• Multiplicative identity is called unit and denoted by 1.

• An associative ring with unit is called a division ring if every non-zeroelement has a multiplicative inverse.

• A division ring F is called a field if the multiplication is commutative.



8.2 Vector Spaces• A nonempty set V is an vector space over a field F if it has an associative

commutative binary operation with identity, called addition, +, and an op-eration F × V → V called scalar multiplication, ·, such that every elementhas an additive inverse and for all scalars a, b ∈ F and all vectors u, v ∈ Vthere hold:

a(u + v) = au + av

(a + b)v = av + bv

a(bv) = (ab)v

and for the multiplicative identity in F

1v = v.

• Subspace

• Direct sum

• Linear independence

• Spanning set

• Basis

• Finite-dimensional vector spaces

• Dimension

• Components of a vector

8.3 Linear Maps• Linear maps (homomorphisms)

• Kernel, Nullity

• Image, rank

• Isomorphisms


8.4. MATRIX REPRESENTATIONS 221

• Endomorphisms

• Automorphisms

• Idempotent maps

• Projections

• Rank-Nullity Theorem. Let T : V → W be a linear map. Then

dim Ker T + dim Im T = dim V

• Quotient spaces

8.4 Matrix Representations• Let T : V → V be an endomorphism and ei be a basis of V . Then

Tei =

n∑j=1

T jie j

The matrix (T ij) is a matrix representation of the endomorphism T in the

basis ei.

• The representation of the composition of endomorphisms is given by theproduct of their matrices.

• The representation of the inverse endomorphism is given by the inverse ma-trix.

• The rank of a square matrix is the number of linearly independent rows (orcolumns).

• The rank of an endomorphism is equal to the rank of its representation ma-trix.

• The maximal rank of an endomorphism is equal to the dimension of thespace.



8.5 Dual Space

• A linear functional on a vector space V is a linear map

α : V → F.

• The set V∗ of all linear functionals on a vector space V is called the dualspace of V , also denoted by Hom (V,F).

• The dual space V∗ is a vector space over the same field.

• The elements of the dual space V∗ are called covectors and the action of acovector α ∈ V∗ on a vector v ∈ V is denoted by 〈α, v〉 and called a pairing.

• The dual space V∗ has a dual basis θi defined by

〈θi, e j〉 = δij,

where

δij =

1, if i = j,0, if i , j

is called the Kronecker symbol.

• The dimension of the dual space V∗ is equal to the dimension of the spaceV ,

dim V∗ = dim V.

• Any vector v ∈ V and any covector f ∈ V∗ can be expanded as

v =

n∑i=1

viei, α =

n∑i=1

αiθi.

The scalars vi and αi are called the components of the vector v and covectorα,

vi = 〈θi, v〉, αi = 〈α, ei〉


8.6. CHANGE OF BASIS 223

8.6 Change of Basis• Let ei be a basis of a vector space V .

• Let e′j be another basis of V; then

e′j =

n∑i=1

Aijei,

where Aij is the change of basis matrix.

• The transpose B = (A−1)T = (AT )−1 of the inverse matrix A−1 is called thecontragredient matrix of A.

• Then

v =

n∑i=1

viei =

n∑i=1

v′ie′i

• Therefore, the components of the vectors transform covariantly

v′ j =

n∑i=1

(A−1) jivi

• The dual basis transforms according to

θ′ j =

n∑i=1

(A−1) jiθ

i

and the components of covectors transform covariantly

α′j =

n∑i=1

Aijαi

8.7 Inner Product Spaces• Let V be a vector space over C. A sesquilinear form on V is a map

g : V × V → C,

satisfying the properties: for all u, v,w ∈ V and a, b ∈ C,

g(u, av + bw) = ag(u, v) + bg(u,w)

g(v, u) = g(u, v).



• The form g is linear in the second entry and anti-linear in the first entry

g(av + bw, u) = ag(v, u) + bg(w, u)

• The sequilinear form g is non-degenerate if

g(u, v) = 0

for all v implies u = 0.

• A non-degenerate sequilinear form is called an inner product on V .

• A vector space with an inner product is an inner product space.

• An inner product g is positive definite if for any non-zero vector u,

g(u, u) > 0.

• Two vectors u and v are orthogonal if

g(u, v) = 0.

• A vector u is a unit vector if

g(u, u) = 1.

• A set of vectors vi is orthonormal if

g(vi, v j) = δi j

• Theorem. Every inner product space has an orthonormal basis.

• There is a natural isomorphism between a real vector space V and its dualV∗.

• Riesz Lemma. Let V be a real vector space and V∗ is its dual space. Thenthe map

ψ : V → V∗,

defined for any u ∈ V by

〈ψ(v), u〉 = g(v, u),

is an isomorphism.


8.7. INNER PRODUCT SPACES 225

• Let ei and θi be the bases in real inner product vector spaces V and V∗.

• The matrixgi j = g(ei, e j)

is called the metric.

• Let (gi j) denote the matrix inverse to (gi j).

• Let

v =

n∑i=1

viei

• Then

ψ(e j) =

n∑i=1

g jiθi

and

ψ(v) =

n∑j=1

v jθj

where

v j =

n∑i=1

g jivi

are called the covariant components of the vector v.

• The inverse isomorphismψ−1 : V∗ → V,

is defined by〈α, u〉 = g(ψ−1(α), u),

• That is,

ψ−1(θ j) =

n∑i=1

g jiei

and for any

α =

n∑i=1

αiθi

ψ−1(α) =

n∑j=1

α je j,



where

α j =

n∑i=1

g jiαi

are called the contravariant components of the covector f .

8.8 Pullback and Adjoint• Every linear map

A : V → W

defines an linear map (called the dual map or the pullback)

A∗ : W∗ → V∗

as follows: for any α ∈ W∗, A∗α is a covector on V such that for any v ∈ V

〈A∗α, v〉 = 〈α, Av〉

• Let eini=1 be a basis in V and fµmµ=1 be a basis in W; let θi and σµ be the

dual bases in V∗ and W∗.

• Let Aµi be the corresponding matrix representation of the map A such that

Aei =

m∑µ=1

Aµi fµ

and for any v ∈ V

Av =

m∑µ=1

(Av)µ fµ, (Av)µ =

n∑i=1

Aµivi

• Then

A∗σµ =

n∑i=1

Aµiθ

i.

and for any ω ∈ W

A∗ω =

n∑i=1

(A∗ω)iθi, (A∗ω)i =

m∑µ=1

Aµiωµ


8.8. PULLBACK AND ADJOINT 227

• The matrix of the dual map A∗ is just the transpose of the matrix of the mapA.

• Every linear mapA : V → V

on a inner product vector space V defines an linear map (called the adjointmap)

A† : V → V

as follows: for any w ∈ V , A†w is a vector such that for any v ∈ V

g(A†w, v) = g(w, Av).

• The matrix of the adjoint map A† is just the Hermitian conjugate, (A)T , ofthe matrix of the map A.


Chapter 9

Appendix B: Multivariable Calculus

9.1 Basic Theorems of Multivariable Calculus

• The Euclidean space Rn is the set of n-tuples x = (x1, . . . , xn) = (xi), i =

1, . . . , n.

• The standard basis in Rn is the set of vectors

ei = (0, . . . , 1, . . . , 0)

• Let f : U → R be a real valued function.

• The partial derivative of f at a point x with respect to xi is defined by

∂ f∂xi = lim

t→0

1t[ f (x + tei) − f (x)]

• It will be denoted by

∂i f =∂ f∂xi

• A function f is smooth if it has continuous partial derivatives of any order.

• The set of smooth functions is denoted by C∞(Rn).

• The composition of smooth functions is smooth.

229

230 CHAPTER 9. APPENDIX B: MULTIVARIABLE CALCULUS

• Let F : Rn → Rm be a map (vector-valued function),

x 7→ y = F(x) = (F1(x), . . . , Fm(x)) = (Fµ(x)),

where µ = (1, . . . ,m).

• The map F is smooth if every component Fµ is smooth.

• The derivative DF of F at x is the m× n matrix of partial derivatives calledthe Jacobian matrix

DF =

(∂Fµ

∂xi

).

• Chain Rule. Let F : Rn → Rm and G : Rm → Rk be two smooth maps. TheJacobian matrix of the composition H = G F is equal to the product of theJacobian matrices

D(G F) = DGDF.

• Let us denoted the coordinates of Rn,Rm,Rk by xi, yµ, za respectively withi = 1, . . . , n; µ = 1, . . . ,m; a = 1, . . . , k, so that

y = F(x), z = H(y) = G(F(x))

Then∂za

∂xi =

m∑µ=1

∂za

∂yµ∂yµ

∂xi

• If n = m, the determinant of the Jacobian matrix

det DF = det(∂F j

∂xi

)is called the Jacobian.

• Let U,V ⊂ Rn be two open sets and F : U → V be a homeomorphism.

• The map F is called a diffeomorphism if both F and F−1 are smooth.

• Inverse Function Theorem. Let F : Rn → Rn be a smooth map. Leta ∈ Rn be a point such that the Jacobian matrix DF(a) is non-degenerate ata. Then:


9.1. BASIC THEOREMS OF MULTIVARIABLE CALCULUS 231

1. there exists a neighborhood U of the point a such that the set V = F(U)is open and the map F : U → V has a smooth inverse F−1 : V → U(that is, F is a diffeomorphism),

2. for any point x ∈ U

(DF−1)(y) = (DF(x))−1,

where x = F−1(y).

• Implicit Function Theorem. Let F : Rr ×Rm → Rm be a smooth map. LetDyF and DxF be matrices defined by

DyF =

(∂Fµ(x, y)∂yν

), DxF =

(∂Fµ(x, y)∂xi

),

where µ, ν = 1, . . .m; i = 1, . . . , r. Let c ∈ Rm be a point in Rm and W be theset of points in Rr × Rm defined by

W = F−1(c) = (x, y) ∈ Rr × Rm | F(x, y) = c

Suppose that the set W is non-empty. Let (a, b) be a point in W, that is,

F(a, b) = c,

such that the matrix DyF(a, b) is not degenerate at (a, b). Then:

1. there is a neighborhood U ⊂ Rr of the point a and a neighborhoodV ⊂ Rm of the point b and a unique smoth map

G : U → V,

such that V = G(U) and for any x ∈ U, (x,G(x)) ∈ W.2. For any point x ∈ U,

DxG = −(DyF)−1DxF.

• Proof. By the inverse function theorem one can invert (locally) the map(with fixed x)

z = F(x, y)

to express the coordinates yµ in terms of the coordinates zµ,

y = G(x, z).

• By fixing z = c we get the map

x 7→ y = G(x) = G(x, c) .



9.2 Coordinates• The Euclidean space Rn is the set of points x represented by n-tuples (xi) =

(x1, . . . , xn), with the numbers xi called coordinates of the point x.

• We may introduce the coordinate functions

ϕi(x) = xi

that assign the coordinates to the points. They provide a coordinate systemon Rn.

• Example. Unit sphere S 2.

S 2 =

x ∈ R3∣∣∣∣ 3∑

i=1

(xi)2 = 1

• Problem: how to describe points on S 2?

• There is no single coordinate system that adequately describes S 2.

• One needs at least two coordinate systems.

• Coordinates on S 2 are described by the coordinate map

ϕ : R3 → R2

• Let the sphere S 2 be centered at the origin with the north pole (0, 0, 1).

• Spherical coordinates are described by the coordinate map

ϕ1(x) = θ, ϕ2(x) = ϕ.

These coordinates are bad at both poles. Need to rotate them.

• Stereographic coordinates (with respect to the North (South) pole) are de-scribed by the coordinate map

u1 =x1

1 − r, u2 =

x2

1 − r,

or

v1 =x1

1 + r, v2 =

x2

1 + r,

wherer =

√1 − (x1)2 − (x2)2.


9.3. SMOOTH MAPS 233

• Coordinate systems are not unique. The change of coordinates is admissi-ble if the corresponding map is a diffeomorphism, that is, has a non-zeroJacobian.

Exercises: Compute the Jacobian for the change of coordinates

1. The change from the spherical to the stereographic coordinates is givenby the map

u1 =cosϕ sin θ1 − cos θ

= cotθ

2cosϕ, u2 =

sinϕ sin θ1 − cos θ

= cotθ

2sinϕ, .

2. The change from the North pole to the South pole stereographic coor-dinates is given by

v1 =u1

R2 , v2 =u2

R2 ,

whereR =

√(u1)2 + (u2)2

9.3 Smooth Maps• Let f : M → R be a function.

• Let p be a point and (U, ϕ) be a chart containing p.

• This defines a real valued function of several variables

f = f ϕ−1 : ϕ(U)→ R

called the local representation of the function f in the cordinate chart(U, ϕ). Usually we just use the same symbol to denote it.

• A function f is smooth if its local representation is smooth.

• LetF : M → N

be smap between two manifolds. Let n = dim M and m = dim N.

• Let p ∈ M be a point in M and (U, ϕ) be a chart in M containing p.



• Let (V, ψ) be the chart on N containing the point F(p) as well as the setF(U).

• The local representation of the map F is given by the m-vector valued func-tion of n variables

F = ψ F ϕ−1 : ϕ(U)→ ψ(V)

• Let xi, i = 1, . . . , n, be local coordinates in U and yµ, µ = 1, . . . ,m, be localcoordinates in V . Then

yµ = Fµ(x).

Again, to simplify notation we just denote this local representative by F.

• The map F is smooth if all its local representatives are smooth.

• The map F is called an immersion at a point p ∈ M if

n = dim M ≤ m = dim N

and the Jacobian matrix DF(p) at p has the maximal rank,

rank DF(p) = n.

• The manifold M is called an immersed submanifold of the manifold N ifthe map F is an immersion at every point p ∈ M.

• In this case the image F(M) can be equipped by local coordinates by theinverse function theorem.

• The image F(M) of an immersed submanifold can have self-intersections.

• The map F is called an submersion at a point p ∈ M if

n = dim M ≥ m = dim N

and the Jacobian matrix DF(p) at p has the maximal rank,

rank DF(p) = m.


9.3. SMOOTH MAPS 235

• The map F is called an embeding if it is an injective immersion and themap

F : M → F(M)

is a homeomorphism.

• The manifold M is called an embedded submanifold if the map F is anembedding.

• The image F(M) of an embedded submanifold does not have any self-intersections.

• Whitney Embedding Theorem.

1. Any n-dimensional topological manifold can be embedded in R2n+1.

2. Any n-dimensional smooth manifold can be embedded in R2n.

• Example. Klein Bottle K2 cannot be embedded in R3.

• Let M and N be two manifolds with

n = dim M ≥ m = dim N.

• Let F : M → N be a smooth map.

• A point p ∈ M is called a regular point of the map F if the Jacobian DF atp has the maximal rank (that is, F is a submersion at p )

rank DF(p) = m

and a critical point ifrank DF(p) < m.

• A point q ∈ N is called a regular value of the map F if the inverse imageF−1(q) is either empty or every point p in the inverse image F−1(q) is regular.

• Sard Theorem. The set of all regular values of a smooth map F : M → Nis dense in N.

• That is, the set of all critical values has a measure zero. We also says, thatalmost all values are regular.



• Regular Value Theorem. Let M be an n-dimensional manifold and N be am-dimensional manifold. Suppose n ≥ m and let r = n − m. Let

F : M → N

be a smooth map. Let q ∈ N be a regular value and W ⊂ M be the setdefined by

W = F−1(q) = p ∈ M | F(p) = q.

Then W is an embedded r-dimensional submanifold of M.

• Proof. This follows from the implicit function theorem.

• Let V be a neighborhood of q with local coordinates zµ, µ = 1, . . . ,m suchthat zµ(q) = cµ with some cµ.

• Let U = F−1(V) and (x1, . . . , xr, y1, . . . , ym) = (xi, yµ), µ = 1, . . . ,m, i =

1, . . . , r, be local coordinates in U.

• Since q is a regular value, every point p ∈ W is regular. So, the rank of theJacobian DF(p) is maximal

rank DF(p) = m

• Therefore, we can reorder the coordinaes (x, y) so that the m × m matrixDyF(x, y) is non-degenerate.

• By the inverse function theorem this means that locally one can invert themap (with fixed x)

z = F(x, y)

to express the coordinates yµ in terms of the coordinates zµ,

y = G(x, z).

• Now, by fixing z = c we get the map

y = G(x, z) = G(x).

• Notice that for any xF(x,G(x)) = c,

which means that the point (x,G(x)) ∈ W and, therefore, xi provide localcoordinates for W.


Bibliography

[1] W. M. Boothby, Introduction to Differentiable Manifolds and RiemannianGeometry, Academic Press, 2003

[2] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry—Methods and Applications, Parts I and II, Springer, 1984

[3] M. Fecko, Differential Geometry and Lie Groups for Physicists, CambridgeUniversity Press, 2006

[4] H. Flanders, Differential Forms, Academic Press, 1963

[5] T. Frankel, The Geometry of Physics, An Introduction, Cambridge Univer-sity Press, 1997

[6] V. G. Ivancevic and T. T. Ivancevic, Applied Differential Geometry, WorldScientific, 2007

[7] C. Isham, Modern Differential Geometry for Physicists, 2nd Edition, WorldScientific, 1999

[8] M. Nakahara, Geometry, Topology and Physics, Institute of Physics, 2003

[9] S. P. Novikov and I. A. Taimanov, Modern Geoimetric Structures andFields, AMS, 2006

[10] P. Renteln, Manifolds, Tensors and Forms, Cambridge University Press(2014)

[11] B. Schutz, Geometrical Methods of Mathematical Physics, Cambridge Uni-versity Press, 1980

[12] M. Spivak, A Comprehensive Introduction to Differential Geometry, Pub-lish or Perish, 1979

237

238 Bibliography


Notation

Add all common notationf : X → Y mapping (function) from X to Yf (X) range of fχA characteristic function of the set A∅ empty setN set of natural numbers (positive integers)Z set of integer numbersQ set of rational numbersR set of real numbersR+ set of positive real numbersC set of complex numbers

239

240 Notation


lecture notes introduction to di erential geometry...

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