Tensors and differential geometry

Doruk Efe Gokmen

March 8, 2019

Contents

I Tensor algebra 3

1 Vector spaces 31.1 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Tensors 62.0.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Operations on tensors . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Change of bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Vector component transformations . . . . . . . . . . . . . 82.2.2 Covector component transformations . . . . . . . . . . . . 92.2.3 Tensor component transformations . . . . . . . . . . . . . 9

2.3 Tensor equations and coordinate free forms . . . . . . . . . . . . 9

II Differential geometry 9

3 Topological spaces 9

4 Topological manifolds 114.1 Charts and atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1 Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . 13

5 Tangent and cotangent spaces 135.1 Tangent vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1.1 Chart induced basis . . . . . . . . . . . . . . . . . . . . . 145.2 Cotangent vector spaces . . . . . . . . . . . . . . . . . . . . . . . 15

5.2.1 Chart induced basis . . . . . . . . . . . . . . . . . . . . . 155.3 Basis transformations . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3.1 Vectors and covectors . . . . . . . . . . . . . . . . . . . . 155.3.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Pushforward and pullback 16

7 Embeddings and submanifolds 187.1 Manifolds with boundaries . . . . . . . . . . . . . . . . . . . . . . 18

1

8 Differentiable bundles 188.1 Tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Covector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Lie flows and Lie derivative 239.1 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.2 Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Affine connections and covariant derivative 2510.1 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 25

11 Torsion and curvature 2611.1 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.2.1 Ricci curvature . . . . . . . . . . . . . . . . . . . . . . . . 2611.2.2 Ricci scalar . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11.3 Bianchi identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.4 The geometric meaning of curvature . . . . . . . . . . . . . . . . 26

12 Pseudo-Riemannian manifolds and Levi-Civita connection 2612.1 Pseudo-metric manifolds . . . . . . . . . . . . . . . . . . . . . . . 2612.2 Musical isomorphisms and raising, lowering of indices . . . . . . 2712.3 Fundamental theorem of Riemannian geometry . . . . . . . . . . 2712.4 Signature of a metric . . . . . . . . . . . . . . . . . . . . . . . . . 27

13 Geodesics 2713.1 Length of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14 Differential forms 2814.1 Volume forms on pseudo-Riemannian manifolds . . . . . . . . . . 28

15 Integration on manifolds 2815.1 Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 2815.2 Interior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 2815.3 Stokes’ theorem on manifolds . . . . . . . . . . . . . . . . . . . . 28

III Symplectic geometry and classical mechanics 28

16 Symplectic forms 28

17 Lagrangian submanifolds 28

2

18 Hamiltonian vector fields 2818.1 Hamiltonian phase flows . . . . . . . . . . . . . . . . . . . . . . . 28

Part I

Tensor algebra

1 Vector spaces

We will focus on vector spaces over R.

Definition I.1. An d dimensional vector space is the triple (V,+, ·), whereV ⊂ Rd, + is the vector addition with properties, ∀{v, w, u} ∈ V

1. Commutativity v + w = w + v,

2. Associativity v + (w + u) = (v + w) + u,

3. ∃ an additive neutral element v + 0 = v,

4. ∃ an additive inverse v + (−v) = 0.

and · is scalar multiplication (s-multiplication) with properties, ∀{v, w} ∈ Vand {a, b} ∈ R

1. Associativity ∀v ∈ V , a · (b · v) = (ab)v,

2. Distributivity of vector in addition of scalars, (a+ b) · v = a · v + b · v,

3. Distributivity of scalar in addition of vectors, a · (v + w) = a · v + a · w,

4. ∃ a multiplicative neutral element 1 · v = v.

From thereon we will denote the vector space in short by V . The dimen-sionality d of V is determined by the number of elements in the (Hamel) basisset B = {e1, · · · , ed}. All v ∈ V can be written as v = viei, where we use theEinstein summation convention.

1.1 Linear maps

Definition I.2. A linear map φ between two vector spaces V and W is ahomomorphism1, the set of all such linear maps is denoted by Hom(V,W ) :={φ : V →W |φ : linear}.

Example I.1. The derivative δ ∈ Hom(C∞M, C∞M).

δ : C∞M→ C∞Mf 7→ f ′. (1)

1A homomorphism is a map between two vectorspaces that preserves the operations ofthese structures i.e. a map between two groups.

3

Lemma I.1. If φ, ψ ∈ Hom(V,W ) he composition φ ◦ ψ ∈ Hom(V,W ).

This can easily be verified by showing linearity.

Example I.2. The second derivative δ2 := δ ◦ δ ∈ Hom(C∞M, C∞M).

δ2 : C∞M→ C∞Mf 7→ f ′′. (2)

Note that Hom(V,W ) is also a vector space when equipped with elementwiseaddition and s-multiplication.

⊕ : Hom(V,W )×Hom(V,W )→ Hom(V,W )

(ψ, φ) 7→ ψ ⊕ φ(ψ ⊕ φ)(v) = ψ(v) + φ(v) (3)

� : R×Hom(V,W )→ Hom(V,W )

(c, ψ) 7→ c� ψ(c� ψ)(v) = cψ(v). (4)

1.2 Dual spaces

Definition I.3. The dual space V ∗ of a vector space V is the vector spaceof homomorphisms Hom(V,R) = {φ : V → R|φ : linear}. φ ∈ V ∗ are calledcovectors.

Below we give three examples.

Example I.3. Inner product with another vector w ∈ V

〈w, ·〉 : V → Rv 7→ 〈w, v〉 ∈ R. (5)

Example I.4. Definite integration

I : C∞M→ R

f 7→ I(f) =

∫ b

a

dxf(x) = f(a)− f(b) ∈ R. (6)

Example I.5. Gradient (informal)2 ∇f of a function f in R3 is an element ofV ∗, where V is the vector space of directions in R3

∇f : V → Rv 7→ (∇f)(v) ∈ R. (7)

2It will be seen that this argument with ∇f is formalised using the exterior derivativedf = df = fidxi ∈ V ∗. The directional derivative with direction v ∈ V is thus simply(df)(v) = vifi.

4

Proposition I.1. Given the basis B for the vector space V , the basis B∗ ={ε1, · · · εd} of V ∗ is completely defined by the condition

εiej = δij , (8)

where δij is the Kronecker symbol.

This can be verified by checking linear independence and spanning propertiesof set B∗. Note that this also shows that dimV = dimV ∗.

Theorem I.1. In finite dimensions, double dual vector space is isomorphic tothe original vector space: (V ∗)∗ := V ∗∗ ≡ V .

Proof. We define a map

[·] : V → V ∗∗.

v 7→ [v], (9)

such that [v](v∗) := v∗(v). It suffices to show that [·] is bijective. Injectivityimplies that the image of [·] is a subset of V ∗∗ and surjectivity shows that it isin fact the whole set.

Let v ∈ V , φ ∈ V ∗ and λ ∈ R,

[λv + w]φ = φ(λv + w)

= λφ(v) + φ(w)

= (λ[v] + [w])φ, ∀φ=⇒ [λv + w] = λ[v] + [w], (10)

hence the map [·] is linear.Since [·] is linear it is injective iff ker[·] = {0}3. Suppose that ∃v 6= 0 such

that [v](φ) = 0 ∀w.

[v](φ) = φ(v) = εiφiejvj = φiv

jδij = φivi !

= 0

=⇒ vi = 0 =⇒ v = 0, (11)

which contradicts our assumption, hence ker[·] = {0}, which means that [·] isinjective.

Consider now the basis B∗∗ = {[e1], · · · , [ed]} of V ∗∗. By the above defini-tion, it acts on the dual basis B∗ as follows

[ei](εj) = εj(ei) = δji . (12)

as mentioned before, this is sufficient to define the basis B∗∗ with the samedimensionality as B and B∗. This means that the map [·] is surjective becausethe only subset of a vector space with the same dimension is the vector spaceitself. This proves that [·] is bijective, and thereby that V ∗∗ ≡ V .

In what follows, we will identify V and V ∗∗ via this isomorphism.

3The kernel ker is also known as the null space.

5

2 Tensors

Definition I.4. Let V be a vector space, and let r, s ∈ N04. An ( rs )-tensor T

is a multilinear map

T : (V ∗)×r × V ×s → R(εi1 , · · · , ejs) 7→ T (εi1 , · · · , ejs) := T i1···irj1···js , (13)

where T i1···irj1···js are (dimV )r+s real numbers and which are the componentsof tensor in the basis

B⊗r⊗ (B∗)⊗s = {ei1 ⊗ · · ·⊗ eir ⊗ εj1 ⊗ · · ·⊗ εjs : i1, · · · , js ∈ {1, · · · , d}}, (14)

with the understanding that the elements ei ∈ V can be regarded as elementsof [ei] ∈ V ∗∗.

Example I.6. A ( 00 )-tensor T : V ∗ → R is a real number.

Example I.7. A ( 10 )-tensor T : V ∗ → R is an element of the vector space

isomorphic to V , hence is a vector.

Definition I.5. The set

T rs := Hom((V ∗)×r × V ×s,R

)= V ⊗r ⊗ (V ∗)⊗s (15)

of all such homomorphisms, equipped with element-wise addition and s-multiplicationis a vector space, called ( rs )-tensor space.

This indicates that an ( rs )-tensor can be constructed by an outer product ofr vectors and s covectors. Thus, its components determine a tensor completely

T = T i1···irj1···js ei1 ⊗ · · · ⊗ eir ⊗⊗εi1 ⊗ · · · ⊗ εis . (16)

This can be seen by proving that B⊗r ⊗ (B∗)⊗s is a valid basis for ( rs )-tensorsby verifying linear independence and spanning properties. Suppose that

T i1···irj1···js ei1 ⊗ · · · ⊗ εis = 0. (17)

Acting on the tuple (εq1 , · · · , eps) we get

0 = (T i1···irj1···js ei1 ⊗ · · · ⊗ εis)(εq1 , · · · , eps)

= T q1···qrp1···ps , (18)

which implies that 17 only has the trivial solution, proving linear independenceof the proposed basis.

4N ∪ {0}.

6

Span can also be verified easily by acting by T on an arbitrary tuple (φ(1), · · · , v(s))

T(φ(1), · · · , v(s)

)= T

(ei1

(φ(1)

)εi1 , · · · , εjs

(v(s))ejs

)= T

(εi1 , · · · , ejs

)ei1

(φ(1)

)· · · εjs

(v(s))

=[T(εi1 , · · · , ejs

)ei1 ⊗ · · · ⊗ εjs

] (φ(1), · · · , v(s)

), (19)

where in the first step we have decomposed vectors and covectors into their com-ponents in the bases B and B∗, in the second step we have used multilinearity.What we have found is true for all tuples

(φ(1), · · · , v(s)

), thus

T = T(εi1 , · · · , ejs

)ei1 ⊗ · · · ⊗ εjs

= T i1···irj1···js ei1 ⊗ · · · ⊗ εjs , (20)

i.e. any tensor can be decomposed into the proposed basis as claimed. This alsomeans that one can denote a tensor by its components, which from now on wewill do from time to time.

Example I.8. Let T be a ( 11 )-tensor. If we fix v ∈ V , T (·, v) : V ∗ → R, thus

T ∈ V ∗∗, which is isomorphic to V .

2.0.1 Metric tensor

Definition I.6. The metric tensor g on V is a symmetric ( 02 )-tensor, i.e.

g(v, w) = g(w, v) and positive definite, i.e. g(v, v) ≥ 0 and g(v, v) = 0 ⇐⇒v = 0.

Because of these properties, g can also be thought of as an inner product,i.e.

g : V × V → R(v, w) 7→ g(v, w) := 〈v, w〉. (21)

In components g = gijεi ⊗ εj . Inserting only one vector v ∈ V , g induces a

natural map (isomorphism) from V → V ∗

Z : V → V ∗

v 7→ g(v, ·) := vZ

= gijεi(vkek)εj = vigijε

j , (22)

i.e. vZ

= φ = φjεj , with φj = vigij .

2.1 Operations on tensors

Tensor product

7

Definition I.7. Let T be an ( rs )-tensor and let T be an ( rs )-tensor. The tensorproduct of T and T is defined via the action on a (r + r)-tuple of covectorsφ(i) ∈ V ∗ and (s+ s)-tuple of vectors v(i) ∈ V :

(T ⊗ T )(φ(1), · · · , φ(r+r), v(1), · · · , v(s+s))

= T (φ(1), · · · , φ(r), v(1), · · · , v(s))T (φ(r+1), · · · , φ(r), v(s+1), · · · , v(s)).(23)

Tensor contraction

Definition I.8. Let T be an(r+1s+1

)-tensor and let m ≤ r and r < n ≤ s, the

tensor contraction is the map C(m,n)(T ) : T r+1s+1 → T rs

C(m,n)(Ti1··· ,m,··· ,ir+1

j1··· ,n,··· ,js+1)ei1 ⊗ · · · ⊗ em ⊗ · · · ⊗ εn ⊗ · · · εjs+1

= Ti1··· ,m,··· ,ir+1

j1··· ,n,··· ,js+1εn(em)ei1 ⊗ · · · ⊗ εjs+1

= Ti1··· ,m,··· ,ir+1

j1··· ,n,··· ,js+1δnmei1 ⊗ · · · ⊗ εjs+1

:= Ti1··· ,m,··· ,ir+1

j1··· ,n,··· ,js+1ei1 ⊗ · · · ⊗ εjs+1 , (24)

where i denotes the omitted indices.

2.2 Change of bases

2.2.1 Vector component transformations

Consider bases B and B for V (dimV=d), the so called covariant forwardtransformation between these bases is

ei = Λjiej . (25)

We can define the inverse transformation such that(Λ−1

)ik

Λji := Λ ik Λji = δjk

=⇒(Λ−1

)ikei =

(Λ−1

)ik

Λjiej = ek

or more cleanly

ej =(Λ−1

)ijei, (26)

which is a contravariant transformation.This immediately yields the transformation of vector components because

vectors are basis independent objects v = viei = viei

viei = viΛjiej

=⇒ vi = Λjivi, (27)

vi =(Λ−1

)jivi. (28)

We see that the forward transformation of components is contravariant and theinverse transformation is covariant.

8

2.2.2 Covector component transformations

Consider bases B∗ and B∗ for V ∗. Consider

εi(ej) = εi((

Λ−1)kjek

)=(Λ−1

)ij

=(Λ−1

)imδmj =

(Λ−1

)imεm(ej),

which holds for all εj , thus we conclude that on contrary to vector transforma-tions, the forward covector transformation

εi =(Λ−1

)ijεj (29)

is a contravariant transformation. Likewise the inverse transformation is covari-ant

εi = Λij εj . (30)

In analogy to the vector case, by φ = φiεi = φiε

i, we find the componenttransformations

φi = Λjiφj (31)

φi =(Λ−1

)jiφj . (32)

2.2.3 Tensor component transformations

Since any tensor can be decomposed into the basis B⊗r ⊗ (B∗)⊗s, based on thefact that tensors are basis independenct, we can easily infer that the transfor-mation rule for the tensor components is the following

T i1···irj1···js = T q1···qrp1···ps(Λ−1

)i1q1· · ·Λpsjs . (33)

2.3 Tensor equations and coordinate free forms

If a tensor T equals zero, i.e. T = 0, then all of its components are equal to zero,and all linear transformations of 0 vector map it to 0. This means that T = 0 isa coordinate independent statement. This also means that any tensor equationT = R holds coordinate independently, and implies that all coordinates of R isequal to all coordinates of T with respect to any given basis.

Part II

Differential geometry

3 Topological spaces

Topology is the study of geometry, in particular, of inherent connectivity ofobjects without introducing the notion of distance. A topology O on a set Mis given by declaring which subsets of M are open. Thus, the axioms that a

9

topology satisfies are the are the abstraction of the properties that open setshave.

Definition II.1. A topology O on a set M, is the collection of all subsetso ⊆M which we declare to be open sets that satisfy the following three axioms.If Oα ∈ O, then

• ∅,M∈ O,

•⋂αOα ∈ O,

•⋃αOα ∈ O.

We can go on to define the notion of a topological space.

Definition II.2. A topological space is a tuple (M,O), i.e. it is a set M forwhich the topology O has been specified.

Here are two extremal examples of topologies, which can easily be verifiedto satisfy these axioms.

Example II.1. The trivial topology is the one for which the only open sets are∅ and M.

Example II.2. The discrete topology O is the power set P(M), i.e. the onefor which exactly all of the subsets of M are open sets.

A topology with many open sets is called strong, and vice versa. Thus, thetrivial topology is the weakest, whilst the discrete topology is the strongest.

Many simple examples can be given on finite set of symbols.

Example II.3. Sierpinski topology : LetM = {ξ, χ}, and let O = {∅,M, {ξ}}.

Example II.4. Let M = {π, a, 28,#}, and let O = {∅,M, {π}, {28}, {π, 28}}.

Of particular importance is the standard topology Os on Rd, in fact, whenwe talk about Rd, we usually talk about the topological space (Rd,Os), wherethe ordered set Rd is equipped with this topology.

Example II.5. Standard topology on Rd is the set Os of all O ⊆ Rd such that∀(x1, · · · , xd) ∈ O, ∃ an open ball defined by

Br(x1, · · · , xd) := {(x1, · · · , xd)|d∑i=1

‖xi − xi‖ ≤ r}, (34)

such that Br(x1, · · · , xd) ⊆ O.

To proceed, we will now introduce the notion of continuity and isomorphismto characterise the maps f : (M,O) → (N ,Q) between topological spaceson sets M,N with topologies O,Q, respectively. The following definition ofcontinuity aims to capture the intuition that there are not jumps and separationsin the map. In fact, before introducing it, we recall the classical definition ofcontinuity.

10

Definition II.3. Let X and Y be metric spaces. A function f : X → Yis continuous at x0 ∈ X, if ∀ε > 0, ∀x ∈ X, ∃δ > 0 : ‖x − x0‖ < δ =⇒‖f(x)− f(x0)‖ < ε.

Definition II.4. Topological continuity : A map f : (M,O) → (N ,Q) is con-tinuous at a ∈ M if, ∀ open neighbourhoods Q ∈ Q of f(a), ∃ an open neigh-bourhood O ∈ O such that for any b ∈ O =⇒ f(b) ∈ q.

In other words, f is continuous if the preimage of any open subset of thetarget of f is open, i.e. f−1(X) ∈ O ∀X ∈ Q.

For example, if O is discrete topology, then f : (M,O)→ (N ,Q) is contin-uous for any topology Q on N . This can be seen by noting that regardless ofthe topology on the target space, any subset of M is open, obviously includingthe preimage of f .

Note also that composition of two continuous functions is also continuous.

Definition II.5. A map ∃f is a homeomorphism (topological isomorphism)5 ifit is a bijection, continuous, and f−1 is also continuous.

Our aim in this part is to systematically equip a set of d-tuple numbers Mwith structure. We have already made progress by first introducing the notionof topology to study connectivity in the tuple (M,O). Now we will requirethat (M,O) locally homeomorphic to (Rd,Os), and finally we will introducethe notion of differentiability and impose the conditon thereof.

4 Topological manifolds

Definition II.6. A d-dimensional topological manifold is a topological space(M,O) such that every point in M is contained in an open subset U ∈ Othat is locally homeomorphic to an open subset x(U) of (Rd,Os). In otherwords, ∀P ∈ M, ∃ an open subset such that P ∈ U , and a homeomorphismx : (M,O)→ (Rd,Os).

The homeomorphism x among with U have a specific name.

4.1 Charts and atlas

Definition II.7. Let U ⊆ (M,O), and V ⊆ (Rd,Os) be open subsets. A chartis a tuple (U, x) such that the chart map x : U → x(U) = V is a homeomor-phism. Any x defines a d-tuple of coordinate functions (x1, · · · , xd), xi : U → R.

Using this definition, one can define a topological manifold as a topologicalspace (M,O) such that ∀P ∈M, ∃ a chart.

Definition II.8. An atlas A is the collection of all charts (Uα, xα) such that⋃α Uα =M. Conversely,

⋃{U : (U, x) ∈ A} =M.

5Geometrically, a homeomorphism is a relabelling that preserves the structure of the topo-logical space induced by O.

11

Remark II.1. A curve γ : R→M is continuous ⇐⇒ its chart representationx ◦ γ : R → Rd is continuous ∀ charts (U, x). This can be seen by noting thatcomposition of any continuous functions is again continuous.

Definition II.9. Let (U, x) and (V, y) be a charts for the topological space(M,O) such that U ∩V 6= ∅. A chart transition map is y◦x−1 : (U, x)→ (V, y).

Remark II.2. Note that since by construction, both Im{y◦x−1}, Im{x◦y−1} ∈Os, chart transition maps are homeomorphisms.

Before we continue with differentiable topological manifolds, let us give ex-amples of low-dimensional topological manifolds.

Example II.6. 2-dimensional plane. A chart (R2, x) for the 2-d plane couldbe given by the Cartesian coordinates x = (x1, x2), with respect to a givenorigin. Another one is the polar coordinates(

y−1 (R>0 × (0, 2π)) , y =(y1, y2

)), (35)

with y1 being the distance from the origin and y2 the angle with respect to theCartesian x1-axis. The corresponding chart transition map from Cartesian topolar coordinates is

y ◦ x−1 =(√

(x1)2 + (x2)2, arctan(x2/x1

)). (36)

Example II.7. Circle. The (1-d sphere) circle S1 has an atlasA = {(U, u), (V, v)}such that U = (−π, π) and V = (0, 2π) and both u and v are angles with respectto the Cartesian x1-axis. The chart transition map is

v ◦ u−1 =

{u, u ≥ 0,

u+ 2π, u < 0.(37)

Example II.8. Torus. Torus can be defined analogously as S1×S1. One possi-ble chart is

(u−1 ((0, 2π)× (0, 2π)) , u

)and another is

(v−1 ((0, 2π)× (−π, π)) , v

),

where the two components of v and u are polar and azimuthal angles. The charttransition map is

v ◦ u−1 =

{(u1, u2), u2 ≥ 0,

(u1, u2 + 2π), u2 < 0.(38)

Example II.9. Sphere. Likewise, a chart for sphere S2 is(u−1 ((0, π)× (0, 2π)) , u

),

where the domain consists of the sphere with a meridian cut out.

4.2 Differentiable manifolds

Definition II.10. Let W ≡ U ∩ V 6= ∅ where U, V ∈ O. A differentiablemanifold (M,O,A) is a topological manifold (M,O) together with an atlas Asuch that all of its chart transition maps are differentiable functions y ◦ x−1 :x(W )→ y(W ). Here, by differentiability, we mean f ∈ C∞-functions.

12

In this case, one also talks about smooth manifolds.

Remark II.3. We say that a curve γ : R → M differentiable if its chartrepresentation γ ◦ x−1 : R → Rd is differentiable. It is sufficient to verify thiscondition on a single set of charts x such that

⋃{U : (U, x) ∈ A} ⊇ Im{γ} ⊂ M,

because a chart transition map y◦x−1 to any other set of charts y is differentiableby definition on differentiable manifolds.

Note that this also justifies why we have chosen to impose the condition ofdifferentiability on chart transition maps.

Remark II.4. Similarly, we say that a function f :M→ R is differentiable, ifits chart representation f ◦ x−1 : Rd → R for any chart (U, x) is differentiable.We call the set of all such functions C∞M.

4.2.1 Diffeomorphisms

Definition II.11. A diffeomorphism is an isomorphism of smooth manifolds.A map φ : (M,OM,AM) → (N ,ON ,AN ) is called differentiable if its chartrepresentation y ◦φ◦x−1 : x(U ∩φ−1(V ))→ y(V ◦φ(U)) for any charts (U, x) ∈AM and (V, y) ∈ N . We call the set of all such functions C∞(M,N ).

5 Tangent and cotangent spaces

5.1 Tangent vector spaces

Definition II.12. Let M,O,A be a differentiable manifold. A tangent vectoris the linear map vγ,p : C∞M→ R induced by a C1-curve γ : R→M at pointp = γ(λ0) ∈M, such that

vγ,p : f 7→ vγ,p(f) := (f ◦ γ)′(λ0) = f ′(γ(λ0)). (39)

One can also think of tangent vectors vγ,p as an equivalence class of C1-curves with the same slope at point p.

Definition II.13. The tangent space TpM of the differential manifold (M,O,A)at point p ∈M is the collection of all tangent vectors vγ,p.

Proposition II.1. TpM is a vector space.

Proof. Essentially, what we need to show is that TpM is equipped with welldefined addition and s-multiplication. We will do this by proving that vγ,p ≡

13

cvα,p + vβ,p ∈ TpM for vα,p, vβ,p ∈ TpM, and c ∈ R. Let f ∈ C∞M

(cvα,p + vβ,p)(f) = [c(f ◦ α)′ + (f ◦ β)′]∣∣p

=[c(f ◦ x−1 ◦ x ◦ α)′ + (f ◦ x−1 ◦ x ◦ β)′

] ∣∣p

=[c∂i(f ◦ x−1) ◦ (xi ◦ α)′ + ∂i(f ◦ x−1) ◦ (xi ◦ β)′

] ∣∣p

=[c(xi ◦ α)′ + (xi ◦ β)′

] ∣∣p∂i(f ◦ x−1)

∣∣x(p)

=[c(xi ◦ α) + (xi ◦ β)

]′ ∣∣p∂i(f ◦ x−1)

∣∣x(p)

≡ (xi ◦ γ)′∣∣p∂i(f ◦ x−1)

∣∣x(p)

=(f ◦ x−1 ◦ x ◦ γ

)′ ∣∣p

= (f ◦ γ)′∣∣p

= vγ,p(f), ∀f=⇒ cvα,p + vβ,p = vγ,p ∈ TpM, (40)

where we have set the curve γ(λ) = x−1(c(xi ◦ α) + (xi ◦ β)

)(λ).

Derivation property The elements X ∈ TpM have the derivation propertyon functions f, g ∈ C∞M. Let X = vγ,p for p ∈M and let γ(0) = p

Proof.

X(fg) = vγ,p(fg) = (fg ◦ γ)′(0) = (fg ◦ x−1 ◦ x ◦ γ)′(0)

=[(f ◦ x−1) ◦ (g ◦ x−1) ◦ (x ◦ γ)

]′(0)

=[∂i(f ◦ x−1)(g ◦ x−1) + (f ◦ x−1)∂i(g ◦ x−1)

](x(p))(xi ◦ γ)′(0)

= g(p)[∂i(f ◦ x−1) ◦ (xi ◦ γ)′

](0) + f(p)

[∂i(g ◦ x−1) ◦ (xi ◦ γ)′

](0)

= g(p)vγ,p(f) + f(p)vγ,p(g) = X(f)g(p) + f(p)X(g). (41)

Proposition II.2. Any linear map X : C∞M→ R that satisfies the derivationproperty at some p ∈M is an element of TpM.

Proof. content...

5.1.1 Chart induced basis

Since TpM is a vector space, we can use a basis to decompose X ∈ TpM.Instead of using an arbitrary basis, we can use the basis that is induced by thechart.

Definition II.14. The chart induced basis for TpM is given by the set of vec-

tors{(

∂∂xi

)p

}. These unit vectors are defined through their action on functions

f ∈ C∞M. Let x(0) = p, then(∂

∂xi

)p

(f) = ∂i(f ◦ x−1)(0). (42)

14

Proposition II.3. Chart induced basis of TpM is a valid basis.

Proof. text

5.2 Cotangent vector spaces

Definition II.15. The cotangent space T ∗pM is the dual space of TpM.

Example II.10. A typical example of an element of T ∗pM is the differentialdf of a function f ∈ C∞M, defined as its action on a vector X ∈ TpM via

df : TpM→ RX : 7→ X(f). (43)

5.2.1 Chart induced basis

Proposition II.4. The set {(dxi)p}, forms a valid basis for T ∗pM, .

Proof. To see this, observe that

(dxi)p

(∂

∂xj

)p

=

(∂

∂xj

)p

xi = δij . (44)

Hence we have shown that {(dxi)p} is the dual basis of{(

∂∂xi

)p

}.

Because it is the dual basis of{(

∂∂xi

)p

}, {(dxi)p} is called the chart induced

basis of T ∗pM.

5.3 Basis transformations

5.3.1 Vectors and covectors

We can easily obtain the transformation of basis elements for vectors in TpM(∂

∂xi

)p

(f) = ∂i(f ◦ x−1)(x(p))

= ∂i(f ◦ y−1 ◦ y ◦ x−1)(x(p))

= ∂j(f ◦ y−1)(y(p))∂i(yj ◦ x−1)(x(p))

=

(∂yj

∂xi

)p

(∂

∂yj

)p

(f), ∀f

=⇒(

∂

∂xi

)p

=

(∂yj

∂xi

)p

(∂

∂yj

)p

, (45)

which we call the inverse transformation. By symmetry, we readily have theforward transformation (

∂

∂yi

)p

=

(∂xj

∂yi

)p

(∂

∂xj

)p

. (46)

15

Do covector transformations - deduce by tensor transformation rules (covariantto contravariant and vice versa), infer the component transformations

5.3.2 Tensors

Hence write the transformation for tensor components from one chart x toanother y.

6 Pushforward and pullback

Definition II.16. Let φ : (M,OM,AM) → (N ,ON ,AN ) be a map betweendifferentiable manifolds. The pushforward φ∗ on vectors of the map φ is thecomposition map (or the transfer map) defined ∀p ∈M as

φ∗ : TpM→ Tφ(p)Nvγ,p 7→ vφ◦γ,φ(p). (47)

Acting on a function f ∈ C∞M, we get

(φ∗vγ,p)(f) = vφ◦γ,φ(p)(f) = (f ◦ φ ◦ γ)(0) = vγ,p(f ◦ φ).

In other words, for df ∈ T ∗φ(p)N

df(φ∗X) = (φ∗X)(f) = X(f ◦ φ). (48)

or even6

(φ∗X)(f) = X(φ∗f). (49)

Thus we see that the pushforward φ∗ on a vector precomposes the function thatthe vector acts on with φ.

Proposition II.5. The pushforward is linear, i.e. for X,Y ∈ TpM and c ∈ R

(φ∗(cX + Y ))(f) = cφ∗(X(f)) + φ∗(Y )(f). (50)

Proof. One can easily see this by using 49.

Definition II.17. Let ω ∈ T ∗pM and X ∈ TpM. Similarly, pullback φ∗ oncovectors is the precomposition map

φ∗ : T ∗φ(p)M→ T ∗pMω 7→ φ∗(ω), (51)

such thatφ∗(ω)(X) := ω (φ∗(X)) . (52)

6We will see shortly, that φ∗ is the precomposition map, called the pullback.

16

For a function f ∈ C∞M, which can be regarded as a ( 00 )-tensor, we have

the precomposition(φ∗f)(p) = (f ◦ φ)(p), (53)

defined by the pullback.

Example II.11. For instance if ω = df ∈ T ∗pM, then

φ∗(df)(X) = df(φ∗X) = X(f ◦ φ) = (d(f ◦ φ))(X), ∀X=⇒ φ∗(df) = d(f ◦ φ) = d(φ∗f). (54)

which is consistent with 48 and 52. We see that the pullback commutes with theexterior derivative7 d for 0-forms. One can show that this holds for differentialforms of any degree.

Proposition II.6. Another property of the pullback is the chain rule:

φ∗(fω) = (φ∗f)(φ∗ω). (55)

Proof. Let us use a test vector field X ∈ ΓM (see differentiable bundles). Then

[φ∗(fω)(X)]p = (fω)φ(p)(φ∗X)p = (f ◦ φ)pωφ(p)(φ∗X)p

= (φ∗f)p [(φ∗ω)(X)]p

= [φ∗f · (φ∗ω)(X)]p ∀p ∈M ,∀X ∈ ΓM=⇒ φ∗(fω) = (φ∗f)(φ∗ω). (56)

Remark II.5. If φ is a diffeomorphism, then one can also define the pullbackof vectors and pushforward of covectors. Hence in this case, pushforward andpullback can be defined for tensors.

Let φ : (M,OM,AM) → (N ,ON ,AN ) be a diffeomorphism. The pullbackof a vector may be defined as follows

Tφ(p)M→ TpMvφ◦γ,φ(p) = vα,q 7→ φ∗(vα,p) = vφ−1◦α,φ−1(q) = (φ−1)∗(vα,p),

=⇒ φ∗(vα,p) = (φ−1)∗(vα,p) (57)

where we set γ = φ−1 ◦ α and p = φ−1(q). Thus it also follows that

ω(φ∗(X)) = ω((φ−1)∗(X)

)= (φ−1)∗(ω)(X), (58)

where in the second step, we used 52.Similarly, we can define the pushforward on covectors by the formula

φ∗(ω) = (φ−1)∗(ω). (59)

7See later.

17

This implies by 58 a similar relation as 52

φ∗(ω)(X) = ω(φ∗(X)). (60)

Thus we see that in the case of a diffeomorphism φ, we see that the distinctionbetween pushforward and pullback is lost to some extent. In particular, for afunction, we have

φ∗φ∗f = (φ−1)∗φ∗f = f ◦ φ ◦ φ−1 = f. (61)

Using the extended definition in diffeomorphisms we can see that, for an( rs )-tensor, the pullback is defined as follows

φ∗(T )(ω(1), · · · , X(s)

)= T

((φ−1)∗

(ω(1)

), · · · , φ∗X(s)

). (62)

Furthermore, for any tensor such as T, S we have the properties

φ∗(T ⊗ S) = φ∗T ⊗ φ∗S, (63)

andφ∗Tr(T ) = Tr(φ∗(T )). (64)

This extension of pullbacks will be useful in defining the Lie flow and deriva-tive of tensor fields.

7 Embeddings and submanifolds

Definition II.18. Let N andM be two manifolds. A smooth map f :M→Nis called an embedding, if its derivative, (i.e. the pushforward f∗) is injective,and if f is a diffeomorphism fromM to its image f(M), which is a submanifoldof N .

Example II.12. A curve is a 1-dimensional submanifold in a manifold.

7.1 Manifolds with boundaries

8 Differentiable bundles

To define a tensor field, we need to lift the notion of differentiability (which wasdefined on points in M) to the collection of tangent spaces TpM.

Definition II.19. A differentiable bundle is a triple (E, π,B), where

• E is a 2d-dimensional differentiable manifold, called the total space,

• B is a d-dimensional differentiable manifold, called the base space, and

18

• π : E → B (usually assumed to be surjective8) is a differentiable map,called projection.

Informally, E contains d-dimensional objects and their d-dimensional labels,among with other 2d-dimensional objects. B can be thought of as the set oflabels, and the map π gives out the label of a labelled object in E.

Definition II.20. A fibre over p ∈ B is the preimage of the projection π−1(p) ⊆E.

A fibre at p can be thought of as the set of objects, which have the samelabel p defined by the projection π.

Definition II.21. A (cross) section σ : B → E is a differentiable map suchthat σ(p) ∈ π−1(p) ∀p ∈ B, i.e. the image of σ is in the fibre.

A section σ maps from the labels in the base space B to the correspondingobjects in the total space E of the differentiable bundle.

8.1 Tangent bundle

Definition II.22. The tangent bundle of a d-dimensional differentiable mani-fold (M,O,A), which we will denote in short by M, is a bundle (TM, π,M),where TM is a 2d-dimensional manifold consisting of the disjoint union9

TM =⊔p∈M

TpM =⋃p∈M

(p, TpM), (65)

and π is the natural projection from TM to M, that maps any element fromTM that belongs to TpM to p.

Here we identify TM with the total space E, the differentiable manifolditself M with the base space B. It is also important to note that the fibre atp ∈ M is the tuple (p, TpM). It follows that the section σ(p) ∈ TpM, i.e. thesection maps point p to a tangent vector.

The topology OTM and the atlas ATM are defined in terms of the charts.

The atlas ATM. ATM is defined as set of all charts (TU, x) obtained from achart (U, x) from the atlas AM = {(Uα, xα) :

⋃α Uα =M} of M by

TU =⊔p∈U

TpM, (66)

and

x : TU → R2d

X 7→(x1(π(X)), · · · , xd(π(X)), (dx1)π(X)(X), · · · , (dxd)π(X)(X)

). (67)

8So that all points in B is reached.9Union all sets together, but remember their labels, p in this case.

19

Proposition II.7. (TU, x) is a valid chart.

Proof. By definition, a chart map should be a homeomorphisms, and for differ-entiable manifolds, chart transition maps are differentiable.

First we will show that x is invertible by explicitly constructing the inverse

x−1 : Rd+d → TU(ξ1, · · · , ξd, ξd+1, · · · , ξ2d

)7→ ξd+i

(∂∂xi

)x−1(ξ1,··· ,ξd)

, i ∈ {1, · · · d} (68)

Now we will show that chart transition maps are differentiable. One canconstruct another chart (TV, y) as above. Let p = x−1(ξ1, · · · , ξd), then thechart transition map is

y ◦ x−1 : x(TU ∩ TV )→ y(TU ∩ TV )(x(p), ξd+1, · · · , ξ2d

)7→(y(p), (dy1)p

[ξd+i

(∂∂xi

)], · · · , (dyd)p

[ξd+i

(∂∂xi

)p

]).

(69)

The first d-tuple entries on the right hand side are the elements of the charttransition map y ◦ x−1 : (U, x) → (V, y), which are differentiable by definition,since M is a differentiable manifold. The remaining d entries are of the form

(dyj)p

[ξd+i

(∂∂xi

)p

]= ξd+i(dyj)p

[(∂∂xi

)p

]= ξd+i

(∂∂xi

)pyj

= ξd+i∂i(yj ◦ x−1)(ξ1, · · · , ξd),

(70)

which depends linearly on ξd+1, · · · , ξ2d and smoothly on ξi, · · · , ξd, becauseyj ◦ x−1 are smooth functions10, again by definition.

The topology OTM. We can now use ATM to construct OTM on TM.

Definition II.23. The topology OTM on TM is defined by the requirementthat a subset O ⊆ TA is open ⇐⇒ x(O ∩ TU) is an open set with respect tothe standard topology Os on R2d, ∀(TU, x) ∈ ATM.

This means that the preimage of x is always open, and hence it is continuous.Since x is invertible (as shown), this also ensures that x is a homeomorphism.

One can define the cotangent bundle T ∗M analogously, as a disjoint unionof contangent spaces T ∗pM.

8.2 Vector fields

Definition II.24. A vector field X on the differential manifold M is a differ-entiable map from X :M→ TM that is a section, i.e. Xp := X(p) ∈ TpM =π−1(p). The set of vector fields is denoted by Γ(TM).

10Recall that we required that the chart transition maps are infinitely many times differen-tiable in differentiable manifolds.

20

Γ(TM) is naturally equipped with addition and s-multiplication, which canbe defined point-wise. Let X,Y ∈ Γ(TM) and s ∈ R, then

(X + sY )p := Xp + sYp. (71)

Therefore, Γ(TM) forms a vector space.Furthermore, for we define a multiplication · with a function f ∈ C∞M by

(f ·X)p := f(p)Xp. (72)

Note that this condition is called C∞-linearity. With the previous condition,this implies that Γ(TM) is a C∞M-vector space. Note that C∞-linearity is amuch stricter condition than R-linearity.

Action of X ∈ Γ(TM) on functions. Vector fields have a natural multi-plicative action on functions:

X : C∞M→ C∞Mf 7→ Xf, (73)

such that point-wiseXf(p) := Xpf ∈ R. (74)

This is to say that X is local, i.e. (Xf)(p) depends only on Xp. However,explicitly imposing this condition is not necessary, as it will be seen that C∞-linearity directly implies locality. Note that this also implies the derivationproperty

X(fg)(p) = Xp(fg) = (Xpf)g(p) + f(p)(Xpg) = ((Xf) · g + f · (Xg))(p), ∀p=⇒ X(fg) = (Xf) · g + f · (Xg). (75)

Using 74 and 72, we can calculate the following

((X + g · Y )f) (p) = (X + g · Y )pf

= (Xp + g(p) · Yp)f= Xf(p) + g(p)(Y f)(p)

= (Xf + g · (Y f))(p), ∀p,=⇒ (X + g · Y )f = Xf + g · (Y f), (76)

which indicates that, instead of point-wise calculation, we can also define ad-dition and C∞-multiplication of vector fields on their action on C∞M. Inparticular, the C∞-associativity (g · Y )f = g · (Y f) is another manifestation oflocality.

Proposition II.8. C∞-linearity implies locality, i.e. (f ·X)p = f(p)Xp =⇒(Xf)(p) depends only on Xp.

21

Proof. Let us now prove the locality claim without referring to 74. Let X and Ybe vector fields such that Xp = Yp. Hence the condition for locality implies forthe present case, that (Xf)(p) = (Y f)(p). Let g ∈ C∞M be such that g(p) = 0and the vector field Z be defined as

Z(q) =

{0, q = p,

g(q)−1(Xq − Yq), otherwise.(77)

((X − Y )f)(p) = (X − Y )p(f) = (g · Z)p(f) = g(p)Zp(f) = 0. (78)

8.3 Covector fields

Analogously to vector fields, one can define covector fields, where the set ofall covector fields Γ(TM) is equipped with addition, s-multiplication and C∞-linearity. The action of a covector field on vector fields is defined as

φ : Γ(TM)→ C∞MX 7→ φ(X), (79)

which point-wise reduces to the usual relation

(φ(X))(p) = φp(Xp) ∈ R. (80)

Let us calculate the following

(φ(fX))(p) = φp((fX)p) = φp(f(p)Xp) = f(p)φp(Xp) = (f · φ(X)) (p)

=⇒ φ(fX) = f · φ(X). (81)

Remark II.6. If φ(fX) = f · φ(X) ∀f ∈ C∞M, we say that ω is C∞-linear.

As in the vector case, we again have the locality condition implied by C∞-linearity.

Proposition II.9. Given a C∞M-linear map ω : Γ(TM)→ C∞M, ω is local,i.e. (ω(X))(p) only depends on Xp.

Proof. Again, it suffices to show that if Xp = Yp, then (ω(X))(p) = (ω(Y )). Weuse the same construction, the vector field Z ∈ Γ(TM) and g ∈ C∞M as inthe vector case, so that X − Y = g · Z. Then

ω(X)(p)− ω(Y )(p) = (ω(X − Y )) (p) = (ω(g · Z)) (p)

= (g · φ(Z)) (p) = g(p)φ(Z)(p) = 0.(82)

We are now ready to generalise the formalism that we have established forvector fields and dual vector fields to tensor fields.

22

8.4 Tensor fields

Definition II.25. An ( rs )-tensor field is a C∞-linear map

T : Γ(T ∗M)× · · · × Γ(TM)→ C∞M(φ(1), · · · , X(s)

)7→ T

(φ(1), · · · , X(s)

). (83)

Analogously, C∞-linearity implies that T(φ(1), · · · , Xs

)only depends on

φi(p) and X(s) at point p.

9 Lie flows and Lie derivative

9.1 Flows

Definition II.26. A (Lie) flow on M is a one-parameter group of diffeomor-phisms φt :M→M, parametrised by t ∈ R, such that

φt ◦ φs = φt+s, (84)

and with φ0 = id and φ−t = φ−1t .

Note that the flow φt can be thought of as a ( 11 )-tensor, whose components

are (φt)ij = φt

(dxi, ∂

∂xj

)in chart (U, x).

Remark II.7. At any point p ∈M the flow defines a curve γ passing throughγ(t) = φt(p), and hence also a vector

Xp = vγ,γ(t). (85)

In this way, a flow defines an entire section (vector field) X : p 7→ Xp.Conversely, under suitable conditions, a vector field X defines a unique flow

φXt with the above group properties. This follows from the theory of differentialequations.

Example II.13. Let (U, x) be a chart. The vector field(∂∂xi

)induces a flow

αit along the coordinate direction xi, which is explicitly given as

αit :M→Mp 7→ x−1(x1(p), · · · , xi(p) + t, · · · , xd(p)). (86)

The given curve is then γ(t) = αit(p) We can verify that this indeed is theinduced flow by

(∂∂xi

)as follows. Let f ∈ C∞M,

vγ,γ(t)f = (f ◦ γ)′(t) = (f ◦ αit)′(t)= (f ◦ x−1 ◦ x ◦ αit)′(t)= ∂j(f ◦ x−1)(xj ◦ αit)′(t)= δji ∂j(f ◦ x

−1)(γ(t)) = ∂i(f ◦ x−1)(γ(t)) =(∂∂xi

)γ(t)

f, ∀f

=⇒ vγ,γ(t) =(∂∂xi

)γ(t)

, (87)

23

because the second factor is constant for each j except when it is equal to j, inwhich case it is 1.

Lemma II.1. Such flows commute: αit ◦ αjs = αjs ◦ αit.

This will be proven when we will show that the Lie bracket of two vectorfields vanishes ⇐⇒ the flows generated by them commute.

9.2 Lie derivative

Definition II.27. The Lie derivative of an ( rs )-tensor field along a vector fieldX is the map

LX : Γ(TM)× · · · × Γ(T ∗M)→ Γ(TM)× · · · × Γ(T ∗M)

T 7→ LXT := limt→0

(φXt )∗T − Tt

= ddt (φ

Xt )∗T

∣∣∣t=0

.

(88)

It follows from this definition that point-wise, the Lie derivative is given by

(LXT )p = limt→0

(φXt )∗TφXt (p) − Tpt

=d

dt(φXt )∗TφX

t (p)

∣∣∣t=0

. (89)

One can easily see that the Lie derivative of a tensor is also a tensor, becausethe difference of any two tensors is again a tensor.

Let us now investigate the case where LXY = 0 for a vector field Y ∈Γ(TM). This implies that (φXt )∗YφX

t (p) = Yp. Applying the pushforward onboth sides we have

(φXt )∗Y = (φXt )∗(φXt )∗Y

= (φX−t)∗(φXt )∗Y

= (φX−t ◦ φXt )∗Y = Y, or

LXY = 0 =⇒ (φXt )∗Yp = YφXt (p) ∀p ∈M. (90)

This means that the flow φXt is an isometry of Y , as Y is invariant under aflow generated by X. Therefore, if the Lie derivative of a vector field Y alonganother vector field X vanishes at point p ∈M, we say that Y is Lie transferred(or transported) along the flow generated by X.

Properties of Lie derivative

1. LX is an R-linear map

2. LXf = Xf for f ∈ C∞M

3. LX(T ⊗ S) = (LXT )⊗ S + T ⊗ (LXS), for two tensors T, S.

4. LX(TrT ) = Tr(LXT ) for some tensor T .

24

Lemma II.2. LXY = [X,Y ], for X,Y ∈ Γ(TM).

Proof. Let f ∈ X∞M. Then

(LX(Y ))pf = ddt

((φXt )∗YφX

t (p)

) ∣∣∣t=0

f

= ddt

((φX−t)∗YφX

t (p)f) ∣∣∣

t=0

= ddt

(YφX

t (p)(f ◦ φX−t)) ∣∣∣

t=0

= ddt

(YφX

t (p)(f ◦ φX−t)) ∣∣∣

t=0

= ddt

(YφX

t (p)f) ∣∣∣

t=0+ Yp

(ddt

(f ◦ φX−t

)) ∣∣∣t=0

= ddt

((Y f) ◦ φXt (p)

) ∣∣∣t=0− Yp(Xf)

= Xp(Y f)− Yp(Xf) = (X(Y f)− Y (Xf))p = [X,Y ]pf, ∀p. (91)

Lemma II.3. [X,Y ] = 0 ⇐⇒ φXt ◦ φYs = φYs ◦ φXt .

Proof.

10 Affine connections and covariant derivative

10.1 Parallel transport

Has the same properties as pushforward, pullback. As a consequence has theproperty (τ(t, u)ω)(X) = ω(τ−1(t, u)) = ω(τ(u, t)), cf. 52 and (τ(t, u)X)(f) =X(τ(u, t)f) cf. 49.

10.2 Covariant derivative

∇XY = 0 implies that Y is parallel transported along X. This is as in the caseof Lie transport, means that parallel transport along X is an isometry for Y .

10.3 Normal coordinates

One can make Christoffel symbols 0 at most at a single point in so called normalcoordinates.

Straight lines (which are geodesics in Levi-Civita connection) are con-strained by the equation of autoparallel transport.

25

11 Torsion and curvature

11.1 Torsion

Definition II.28. In the coordinate free form, the torsion is a ( 12 )-tensor field

defined asT (φ,X, Y ) := φ(∇XY −∇YX − [X,Y ]). (92)

In coordinates, one has

T ijk = T

(dxi,

∂

∂xj,∂

∂xk

)= dxi

(∇∂j

∂

∂xk−∇∂k

∂

∂xj

)= Γikj − Γijk ≡ Γi[kj].

(93)

We see that if the manifold is torsion-free, i.e. T = 0, then the connectioncoefficents are symmetric in the lower indices. The converse is also true.

11.2 Curvature

Definition II.29. In the coordinate free form, the Riemann curvature is a( 1

3 )-tensor field defined as

R(φ,Z,X, Y ) := φ(∇X∇Y Z −∇X∇Y Z −∇[X,Y ]Z). (94)

In coordinates

Rijmn = Γijn,m − Γijm,n + ΓirmΓrjn − ΓirnΓrjm. (95)

11.2.1 Ricci curvature

Definition II.30. Ricci curvature is a ( 02 )-tensor field defined as the contraction

of Riemann curvature tensor in first and third indices

Rij = Rρiρj . (96)

11.2.2 Ricci scalar

Definition II.31. Ricci scalar is given by

R = gijRij . (97)

11.3 Bianchi identities

11.4 The geometric meaning of curvature

12 Pseudo-Riemannian manifolds and Levi-Civitaconnection

12.1 Pseudo-metric manifolds

Definition II.32. A pseudo-Riemannian manifold is a differentiable manifold(M,O,A) together with a differentiable ( 0

2 )-tensor field g (the metric) that

26

at any point p ∈ M satisfies the following symmetry and non-degeneracyconditions.

• gp(X,Y ) = gp(Y,X) for X,Y ∈ TpM i.e. g is symmetric in its entries inany basis.

• If, for a given X ∈ TpM, gp(X,Y ) = 0 ∀Y ∈ TpM, then X = 0, i.e. g isnon-degenerate.

For non-pseudo-Riemannian manifolds, the second property is replaced withpositive definiteness.

12.2 Musical isomorphisms and raising, lowering of in-dices

Remark II.8. Note that the fact that the metric g is non-degenerate impliesthat the natural map [ : TpM→ T ∗pM induced by g (as introduced earlier) isan isomorphism.

12.3 Fundamental theorem of Riemannian geometry

The fundamental theorem of Riemannian geometry states that there exists aunique connection, called the Levi-Civita connection, such that

• the connection is torsion free: T (X,Y ) = 0 =⇒ ∇XY +∇YX = [X,Y ],and

• parallel transport is an isometry: ∇Zg = 0 ∀Z. In other words, the metricg is conserved under parallel transport, i.e. g is parallel transported.

Definition II.33. The Levi-Civita connection is specifically given in coordi-nates as

Γijk =1

2gim(gjm,k + gkm,j − gjk,m). (98)

12.4 Signature of a metric

Sylvester’s law of inertia

13 Geodesics

13.1 Length of curves

are given in terms of the metric

Definition of a geodesic : A geodesic between two points is a curve with ex-tremal length. Hence we use the Euler-Lagrange equations to find the conditionon geodesics.

27

In Levi-Civita connection, the geodesics (curves with extremal length)are strainght lines.

Proof. content...

14 Differential forms

14.1 Volume forms on pseudo-Riemannian manifolds

15 Integration on manifolds

15.1 Exterior derivative

15.2 Interior derivative

15.3 Stokes’ theorem on manifolds

Part III

Symplectic geometry andclassical mechanics

16 Symplectic forms

17 Lagrangian submanifolds

18 Hamiltonian vector fields

18.1 Hamiltonian phase flows

References

28