geometry and topology€¦ · geometry and topology nima moshayedi notes, february 1, 2015 contents

GEOMETRY AND TOPOLOGY

NIMA MOSHAYEDI

Notes, February 1, 2015

Contents

Part 1. Topology 21. Topological spaces 22. Tools to construct topological spaces 42.1. Cell complexes 83. Connectedness 83.1. Separation Axioms 94. Compactness 105. Homotopy 116. The Fundamental group 136.1. Functionality of the Fundamental group 147. Coverings and the Fundamental group of the Circle 158. The Seifert- van Kampen theorem 178.1. Generators and Relations 189. Compact surfaces 199.1. The Fundamental group of RP n 209.2. Construction of new Surfaces from known ones: Connected Sums 209.3. Polygonal representation of surfaces 219.4. Strategy of the Proof of the Classification Theorem 2310. The Euler characteristic 26

Part 2. Geometry 2811. Curves 2811.1. Curves in Euclidean Space (3 Dimension) 3012. Smooth Surfaces in R3 3213. The First Fundamental Form 3613.1. Angles between Curves 3813.2. Areas 3914. isometric surfaces 4015. The Second Fundamental Form 4116. The Gaussian curvature 4417. The Gauss-Bonnet theorem 4518. Morse Functions 4719. Geodesics 48

Date: February 1, 2015.1

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Part 1. Topology

1. Topological spaces

Definition 1.1. Metric SpaceA metric on a set X is a function d : X ×X → [0,∞), (x, y) 7→ d(x, y) such that for

all x, y, z ∈ X:

(i) d(x, y) = 0⇐⇒ x = y(ii) d(x, y) = d(y, x)

(iii) d(x, z) ≤ d(x, y) + d(y, z)

(X, d) is called a metric space.

Definition 1.2. ContinuityLet f : X → Y be a function between metric spaces (X, dX) and (Y, dY ). f is continu-

ous at x ∈ X if for all ε > 0, there exists a δ > 0 such that

dX(x, x′) < δ =⇒ dY (f(x), f(x′)) < ε

for all x′ ∈ X. More formally, ∀ε > 0,∃δ > 0 s.t. f(BXδ (x)) ⊂ BY

ε (f(x)).

Definition 1.3. NeighborhoodU is called a neighborhood of x if there exists an ε > 0 such that Bε(x) ⊂ U .

Proposition 1. Let f : X → Y be a function between metric spaces and let x ∈ X. Thenf is continuous if and only if for any neighborhood V of f(x), there exists a neighborhoodU of x such that f(U) ⊂ V .

Definition 1.4. OpenA subset O of a metric space (X, d) is open if it is a neighborhood of each of its points.

Proposition 2. Let f : X → Y be a function between metric spaces. Then f is contin-uous if and only if for any open subset O′ ⊂ Y , f−1(O′) is open. In words we say, f iscontinuous if and only if the reciprocal image of any open set is open.

GEOMETRY AND TOPOLOGY 3

Definition 1.5. TopologyLet X be a set. A set O of subsets of X is a topology on X if the following holds.

(i) ∅, X ∈ O(ii) Any union of elements of O is in O. Oi ∈ O, i ∈ I =⇒

⋃i∈I Oi ∈ O.

(iii) Any finite intersection of elements of O belongs to O. O1, ..., On ∈ O =⇒⋂ni=1 Oi ∈

O.

Definition 1.6. Topological SpaceA topological space is a pair (X,O), where X is a set and O a topology on X. The

sets O ∈ O are called the open sets. A subset A ⊂ X is called closed set if and only if(X \ A) ∈ O. The closed sets are exactly the complements of the open sets.

Definition 1.7. Zariski TopologyThe Zariski topology on R, C is defined such that the closed subsets are the zero loci of

polynomials. Hence the open sets are ∅, (R,C), (R,C) \F , F ⊂ (R,C) is a finite subset.

Definition 1.8. BasisLet (X,O) be a topological space. B ⊂ O is a basis for the topology O if and only if

any open set O ∈ O is the union of elements of B.

Lemma 1. Let X be a set and B ⊂ P(X) a set of subsets of X s.t. X =⋃U∈B U . Then

the following are equivalent

(1) There is a topology on X with basis B.(2) If U, V ∈ B, U ∩ V 6= ∅, x ∈ U ∩ V , then ∃W ∈ B s.t. x ∈ W ⊂ U ∩ V .

Definition 1.9. Characterization of interior, closure and boundaryLet (X,O) be a topological space, x ∈ X and A ⊂ X a subset.

(1) V ⊂ X is a neighborhood of x if ∃U ∈ O such that x ∈ U ⊂ V .(2) x is an inner point of A if and only if A is a neighborhood of x.

(3) The set A of inner points is called the interior of A(4) x is a point of closure of A if and only if each neighborhood of x intersect A

non-trivially. The set A of all the points of closure is called the closure of A. Apoint in A \ A is a limit point of A.

(5) The points of closure of A that are not inner points are boundary points. The set∂A of boundary points of A is the boundary of A.

Lemma 2. Let (X,O) be a topological space and let A ⊂ X be a subset.

(1) A is open. A = A⇐⇒ A is open.(2) A is closed. A = A⇐⇒ A is closed.

(3) A = A t ∂A.

(4) ∂A = A ∩X \ A(5) X \ A = (X \ A)o, A = X \ (X \ A)o.

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Lemma 3. The following hold.

(1) A is the smallest closed set containing A. A is the largest open set contained inA.

(2) ∅ = ∅, X = X, ¯A = A, A ∪B = A ∪ B(3) ∅ = ∅, X = X,

˚A = A, (A ∩B)o = A ∩ B.

Definition 1.10. Homeomorphismf is called a homeomorphism if and only if it is bijective, continuous and f−1 is con-

tinuous.

Definition 1.11. Finer and Coarser TopologiesLet O1,O2 be two topologies. O1 is finer than O2 if O1 ⊃ O2, i.e. every set in O2 is

open in O1. Then O2 is coarser than O1. This gives a partial order on topologies.

Lemma 4. The following are equivalent:

(1) O1 is finer than O2.(2) id : (X,O1)→ (X,O2) is continuous.(3) id : (X,O2)→ (X,O1) is open.

2. Tools to construct topological spaces

Definition 2.1. Subspace TopologyLet (X,O) be a topological space and A ⊂ X a subset of X. OA = {O ∩ A | O ∈ O}

is called the subspace topology on A. (A,OA) is called a subspace of X.

Proposition 3. Let (X,O) be a topological space and A ⊂ X.

(1) The inclusion ιA : (A,OA)→ (X,O) is continuous.(2) OA is the coarsest topology on A such that ιA is continuous.

Proposition 4. Let f : X → Y and X =⋃ni=1Ai with Ai closed for all 1 ≤ i ≤ n. Then

f is continuous if and only if fi := f |Ai : Ai → Y are continuous for all 1 ≤ i ≤ n.

Definition 2.2. Product Topology ILet X, Y be two topological spaces and let X × Y be the product space. We declare

W ⊂ X×Y to be open if and only if for all (x, y) ∈ W , there exists U ⊂ X, V ⊂ Y opensuch that (x, y) ∈ U × V ⊂ W . The topology defined on this way is the product topology.

Lemma 5. The product topology is the coarsest such that the projections X × Y → Xand X × Y → Y are continuous.


Definition 2.3. Product Topology IILet {Xi}i∈I be a set of topological spaces and let

∏i∈I Xi be their product as sets. Let

Op be the set of subsets of∏

i∈I Xi that can be obtained by unions and finite intersection

of the sets p−1i (U), for U open in X (pj :

∏i∈I Xi → Xj projection). Then Op is the

product topology on∏

i∈I Xi.

Remark 2.1. Op is not the topology generated by products of open sets! The latter is thebox topology, which is finer. The product topology and the box topology coincide when Iis a finite index set.

Definition 2.4. Disjoint Union (Topological sense)Let X, Y be two sets and define the disjoint union X q Y = X × {0} ∪ Y × {1}. Let

(X,OX) and (Y,OY ) be topological spaces. The disjoint union (topological sense) is thetopological space (X q Y,O) where O = {U q V | U ∈ OX , V ∈ OY }.

Lemma 6. The disjoint union topological is the finest on such that the inclusions

X ↪→ X q Yand

Y ↪→ X q Yare continuous.

Definition 2.5. Quotient TopologyLet (X,O) be a topological space, ∼ an equivalence relation on X and π : X → X/ ∼,

x 7→ [x] the projection onto the equivalence class. The quotient topology is defined by:

O ⊂ X/ ∼ is open⇐⇒ π−1(O) is open in X

Lemma 7. The following hold.

(1) π is continuous.

(2) Let f : X/ ∼→ Y . Then f is continuous iff f := f ◦ π is continuous.

Xf

- Y

X/ ∼

f

-

π

-

Lemma 8. The quotient topology is the finest such that π is continuous.

Example 2.1. We have the following examples:

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(1) Collapsing of a subspace to a point

Let X be a topological space and A ⊂ X. Let X/A = X/ ∼, where x ∼ y ⇐⇒x = y or x, y ∈ A. Now let X = [0, 1] and A = {0, 1} ⊂ X. Then X/A = S1. ForX = Bn and A = ∂Bn ∼ Sn−2 we get X/A = Sn.

(2) Cone of a space and suspension

Let X be topology space. CX = (X × [0, 1])/(X × {0}) is called the cone of X.

ΣX = (X × [0, 1])/(X × {0} ∪X × {1}) is called the suspension of X.

The operation Σ might seem random, but it is of fundamental importance in alge-braic topology.

(3) Gluing at a point

Let X, Y be topological spaces, x0 ∈ X and y0 ∈ Y . The spaces (X, x0), (Y, y0) arepointed topological spaces. The glued topological space is given by

(X, x0) ∨ (Y, y0) = ((X q Y )/({x0} q {y0}), x0 ∼ y0)


(4) Quotient by a group action

Let G be a group acting continuously on a topological space X, i.e.

G×X → X, (g, x) 7→ g · xwith e·x = x, (g1 ·g2)·x = g1 ·(g2 ·x) ∀g1 ·g2 ∈ G, x ∈ X and g· : X → X is continuousfor all g ∈ G. The orbit of x is Gx = {g · x | g ∈ G}. Define x ∼ y ⇐⇒ ∃g ∈ G s.t.g · x = y, i.e. y ∈ Gx. Then X/G := X/ ∼ is the orbit space. More over

π : X → X/G, x 7→ [x] π−1([x]) = Gx.

We have the following examples:(a) X = Sn, G = Z2 = {±1}, 1 · x = x, (−1) · x = −x. In Sn/Z2, one identifies the

antipode of Sn. Hence Sn/Z2 = RP n, the real projective space.(b) X = R, G = Zn. g ·x = x+g action by integer translation. X/G = Rn/Zn = T n,

the n- dimensional torus.

(5) Gluing of topological spaces

Let X, Y be topological spaces, A ⊂ X and ϕ : A→ Y continuous. Define

X ∨ϕ Y = X q Y/ ∼ϕ,where a ∼ϕ b⇐⇒ a = b or a = ϕ(b).

Example 2.2. We have the following examples:(a) X = [0, 1], Y = R. A = {0, 1}, ϕ(0) = 0 and ϕ(1) = 1 (gluing a 1-cell).

(b) Let D2 := {x ∈ R2 | ‖x‖ ≤ 1} be the two dimensional unit disc and S1 =∂D2 = {x ∈ R2 | ‖x‖ = 1} be the unit circle. Let Y be a topological space and

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ϕ : S1 → Y be continuous. The quotient is the gluing of a 2-cell. We can alsogo on with higher dimensional cells. This is a very general method to constructtopological spaces.

2.1. Cell complexes.

• Let S0 be a set of points (discrete space)• Glue a 1-ball B1 to S0 via ϕ1

1 : ∂B1 → S0

• Repeat as many times as needed, with maps {ϕ1n : ∂B1 → S0}

• We get the topological space S1, the 1.scelleton• Glue a 2-ball B2 to S1 via ϕ2

1 : ∂B2 → S1

• Repeat as many times as needed, with maps {ϕ2n : ∂B1 → S0}

• We get the topological space S2, the 2-scelleton• Continue by attaching n-balls Bn to Sn−1 via ϕnk : ∂Bn → Sn−1.

3. Connectedness

Definition 3.1. Connected SpaceA topological space (X,O) is called connected if X cannot be decomposed into the union

of two disjoint non-empty open subspaces, i.e. X = O1 ∪ O2, O1, O2 ∈ O, O1 6= ∅ 6=O2 =⇒ O1 ∩O2 6= ∅ for X connected.

Lemma 9. (X,O) is connected iff ∅ and X are the only subspaces of X that are bothopen and closed.


(1) (a, b) ∪ (c, d), a < b < c < d is disconnected.(2) (a, b) ⊂ R is connected for a < b. Indeed, assume ∃U, V open with U ∪ V = (a, b)

and U ∩ V = ∅ with U, V non-empty. Let u ∈ U , v ∈ V and assume u ⊂ v. LetS = {s ∈ (a, b) | (u, 1) ⊂ U} and so = supS ∈ (a, b), s0 ≤ v. Then s0 6∈ U becauseU is open and s0 6∈ V because V is open. This is a contradiction, which proves theclaim.

Proposition 5. Let X be a topological space and A ⊂ X connected (in the inducedtopology). Then A ⊂ B ⊂ A =⇒ B is connected.


Proposition 6. Let f : X → Y be continuous. Then if X is connected =⇒ f(X) isconnected.

Proposition 7. Intermediate Value TheoremLet X be a connected space, f : X → R continuous and t, s ∈ f(X). Then f takes

every values between t and s.

Definition 3.2. Path-connected SpaceX is called path-connected if for each x, y ∈ X, there exists a continuous function

ϕ : [0, 1]→ X with ϕ(0) = x and ϕ(1) = y. We say that ϕ is a path from x to y.

Proposition 8. If X is path-connected then X is connected.

Remark 3.1. There are spaces that are connected but not path-connected.For example let f : (0,∞)→ R such that f(x) = sin

(1x

). Let X be a space given by

X = {(x, f(x)) | x ∈ (0,∞)} ∪ {{0} × [0, 1]} ⊂ R

with the topology induced from the metric topology on R2. Then X is connected but notpath-connected.

3.1. Separation Axioms.

Definition 3.3. T1-space

(1) X is called a T1-space if for all x, y ∈ X with x 6= y there exists neighborhoodsUx of x and Uy of y such that x 6∈ Uy and y 6∈ Uy.

(2) X is called a T2-space, or a Hausdorff space if for all x, y ∈ X with x 6= y,there exists neighborhoods Ux of x and Uy of y such that Ux ∩ Uy = ∅.

Any T2-space is also a T1-space.


(1) R with the metric topology is T2.(2) R with the strange topology in which the only open sets are of the form (−∞, a) is

neither T2 nor T1.(3) R with the Zariski topology is T1 but not T2.

Lemma 10. A metric space with the induced topology is Hausdorff.

Definition 3.4. Let X be a topological space and let {xn}n∈N be a sequence of points ofX. Then x is a limit of {xn}n∈N iff for each neighborhood U of x, there exists nU ∈ Nsuch that xn ∈ U for each each n ≥ nU .

Lemma 11. In a Hausdorff space any sequence has at most one limit point.

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Definition 3.5. Metrizable SpaceA topological space (X,O) is metrizable, when there exists a metric on X that inducesO. If (X,O) is metrizable, then it is T2. The Zariski topology on R is not metrizable.

4. Compactness

Definition 4.1. Compact SpaceA topological space (X,O) is called

(1) quasi-compact if every cover of X by open sets admits a finite subcover. In general

X =⋃i∈I

Ui, Ui ∈ O =⇒ ∃i1, ..., in ∈ I s.t. X = Ui1 ∪ · · · ∪ Uin

(2) compact if X is quasi compact and Hausdorff.


(1) R is quasi-compact in the Zariski topology.(2) The closed cube in Rn with the metric topologyy is compact. Indeed, let Q1 =

[a1, b1]×· · ·×[an, bn] ⊂ Rn, where a = diam(Q1) = (∑n

i=1(bi − ai)2)1/2

= supx,y∈Q1d(x, y).

By contradiction: Assume that Q1 ⊂⋃i∈I Ui with no finite subcover. Split Q1 into

2n smaller cubes Q11, ..., Q

2n

1 by splitting each intervals into 2 equal pieces. Then

there is at least one of the small cubes Q2 ∈ {Q11, ..., Q

2nm1 } that is not covered by any

subcover of {Ui}i∈I . Let diam(Q2) = a2. Now iterate the procedure. Then we get

Q1 ⊃ Q2 ⊃ Q3 ⊃ · · · with diam(Qi) = a2i

for i ∈ N. None of the Qk are covered

by a finite subcover of {Ui}i∈I . As diam(Qk)k→∞−−−→ 0, there exists a unique x0 ∈ Rn

such that x0 ∈⋂∞k=1 Qk. Then x0 ∈ Ui0 for at least one io ∈ I. As Ui0 is open,

x0 ∈ Bε(x0) ⊂ Ui0 for ε > 0 small enough. Hence there is a k such that Qk ⊂ Ui0 .But this is a contradiction and therefore Q1 is compact.

Proposition 9. Let X be quasi-compact and f : X → Y continuous. Then f(X) isquasi-compact. If Y is Hausdorff, then f(X) is compact.

Proposition 10. The following hold.

(1) Let X be quasi-compact and A ⊂ X a closed subset. Then A is quasi compact.(2) Let X be Hausdorff and A ⊂ X a compact subset. Then A is closed.

Proposition 11. Let (X, d) be a metric space and let A ⊂ X be a compact subset. ThenA is bounded in the sense that ∃r > 0 and x ∈ X such that A ⊂ Br(x).

Proposition 12. Let (X, d) be a metric space. Then we have the following implicationfor a subset A ⊂ X:

A is compact =⇒ A is bounded and closed.

But not the reverse!


Theorem 12. Heine-BorelIn Rn, the compact subsets are the closed and bounded subsets.

Proposition 13. Let X be quasi-compact, Y Hausdorff and f : X → Y continuous andbijective. Then f is a homeomorphism, i.e. f−1 is continuous.

Theorem 13. Bolzano-WeierstrassLet (X, d) be a metric space and let X be compact. Then every sequence in X has a

convergent subsequence.

Theorem 14. TychonoffLet X be the product space X =

∏λ∈ΛXλ with the product topology. Then X is compact

iff each Xλ is compact.

5. Homotopy

Definition 5.1. Homotopy and Homotopic FunctionsLet X, Y be two topological spaces and f, g : X → Y be continuous functions. Then f

is homotopic to g (written f ' g) if there exists a continuous function H : X× [0, 1]→ Ysuch that:

(1) H(x, 0) = f(x) for all x ∈ X(2) H(x, 1) = g(x) for all x ∈ X

We say H is a homotopy from f to g.

Remark 5.1. H defines a continuous family ht for t ∈ [0, 1], of functions from X to Y ,ht(x) = H(x, t) sucht that h0 = f and h1 = g. We say f can be continuously deformed intog.


(1) Let f, g : X → Rn be continuous functions. Then f ' g, because H : X× [0, 1]→ Rn

with H(x, t) = (1− t)f(x) + tg(x) is continuous and H(x, 0) = f(x), H(x, 1) = g(x).(2) Let f, g : S1 → R2 \ {0}. We will show that f is not homotopic to g.

Lemma 15. Let X, Y be two topological spaces. Then ' is an equivalence relation onthe set of continuous function from X to Y .

Definition 5.2. Homotopy ClassThe equivalence class [f ], with respect to ', of a continuous function f : X → Y is

called the homotopy class of f .

Lemma 16. Let f, f ′ : X → Y and g, g′ : Y → Z be continuous functions sucht thatf ' f ′ and g ' g′. Then g ◦ f ' g′ ◦ f ′.

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Definition 5.3. Homotopy Equivalence

(1) f : X → Y is called a homotopy equivalence if there exists a continuous functiong : Y → X such that g ◦ f ' idX and f ◦ g ' idY .

(2) If f is a homotopy equivalence, then X and Y are called homotopic equivalent,X ' Y .

(3) X is said to be contractible if X ' ∗, where ∗ is a point.


(1) Let X = S1 and Y = R2 \ {0}. Then S ' R2 \ {0}. Indeed, see S1 ⊂ R2 \ {0}. Takethe inclusion f : S1 → R2 \ {0}, x 7→ x and the radial projection g : R2 \ {0} → S1,y 7→ y

|y| . Then g ◦ f = idS1 . We need to show that f ◦ g is homotopic to idR2\{0}.

Take H : R2 \ {0} × [0, 1] → R2 \ {0}, (y, t) 7→ (1 − t) y|y| + ty. Then H(y, 0) = y

|y| ,

H(y, 1) = y. Therefore f ◦ g ' idR2\{0} and hence S1 ' R2 \ {0}.(2) Rn is contractible. Indeed, take f : Rn → {0}, x 7→ 0 and g : {0} → Rn, 0 7→ 0.

Then g◦f : Rn → Rn, x 7→ 0 and f ◦g = id{0}. Now H : Rn× [0, 1]→ Rn, (x, t) 7→ txis a homotopy between g ◦ f and idRn .

Definition 5.4. Relative homotopicLet X, Y be two topological spaces and A ⊂ X a subspace. Let f, g : X → Y be

continuous functions such that f |A= g |A. f is called homotopic to g relative to A(written f ' g rel A) if there exists a homotopy H from f to g that is constant on A,i.e. H : X × [0, 1] → Y , H(x, 0) = f(x), H(x, 1) = g(x), H(a, t) = f(a) = g(a) for alla ∈ A.

Remark 5.2. When A = ∅, we recover the usual notion of homotopy.

Definition 5.5. Deformation retractLet X be a topological space and A ⊂ X. A is called a deformation retract of X if there

exists a continuous function f : X → X such that f(X) ⊂ A, f |A= id and f ' idX relA.

Remark 5.3. Notice that following hold.

• f is a homotopy equivalence between X and A relative to A.• If A is a deformation retract of X, one can shrink X continuously onto A, keeping

the points of A fixed.


6. The Fundamental group

Definition 6.1. Homotopy of PathsLet X be a topological space. A path in X is a continuous function f : [0, 1] → X. A

homotopy of paths (with fixed endpoints) between paths p1, p2 : [0, 1]→ X is a homotopybetween p1 and p2 relative to {0, 1}, i.e. a map F : [0, 1]× [0, 1] → X such that ft(x) =F (x, t), ft(0) = x0, ft(1) = x1, f0(x) = p1 and f1(x) = p2. The homotopy of pathsdefines an equivalence relation on the space of paths from x0 to x1.

Definition 6.2. Loop with Base pointA loop based at x0 ∈ X is a path starting and ending at x0, i.e. a continuous map

f : [0, 1]→ X such that f(0) = f(1) = x0. We write

π1(X, x0) = {[f ] | f : [0, 1]→ X, f(0) = f(1) = x0}for the set of homotopy classes of loops based at x0. We call x0 the base point.

Definition 6.3. ConcatinationLet f, g : [0, 1] → X be two paths in X such that f(1) = g(0). We can define the

concatenation f · g of f and g as the path

f · g(t) =

{f(2t), t ∈ [0, 1

2]

g(2t− 1) t ∈ [12, 1]

Corollary 1. The concatenation of homotopy classes of paths is well defined, i.e. iff0 ' f1 and g0 ' g1 =⇒ f0 · g0 ' f1 · g1. Therefore [f ] · [g] := [f · g] is well defined.

Lemma 17. π1(X, x0) is a group with respect to the concatenation.

Definition 6.4. π1(X, x0) is called the fundamental group of X.

Lemma 18. Auxiliary LemmaThe reparameterization of a loop does not change its homotopy class.

Lemma 19. There is a group isomorphism βh : π1(X, x1)→ π1(X, x2) between π1(X, x1)and π1(X, x2) defined by

βh([f ]) = [h · (f · h)].

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Remark 6.1. Notice that the following hold.

• If X is not path-connected there is no path from x0 to x1. Then we can haveπ1(X, x0) 6' π1(X, x1).• If X is path-connected, we can speak of the fundamental group π1(X) without refer-

ence to a base point.

Definition 6.5. Simply Connected SpaceX is said to be a simply connected topological space if X is path-connected and the

fundamental group is trivial. There are two important facts:

(i) If X is simply connected, any loop in X can be contracted continuously to a point.(ii) A contractible space is simply connected.

Lemma 20. Let F : [0, 1] × [0, 1] → X be a continuous function. Let α(t) = F (0, t),β(t) = (1, t), γ(s) = F (s, 0) and δ(1) = F (s, 1). Then δ ' αγβ.

6.1. Functionality of the Fundamental group. Let f : X → Y be a continuous functionbetween two topological spaces X and Y . Let x0 ∈ X and y0 = f(x0). Given a pathp : [0, 1] → X, we can push it forward using f as follows. f∗p : [0, 1] → Y is a path in Ydefined by f∗p(t) = f(p(t)). If P : [0, 1]× [0, 1]→ X is a path homotopy between p0 and p1,then f∗P is a path homotopy between f∗p0 and f∗p1. Therefore f∗ passes to a well definedfunction on homotopy classes of paths in X. We have a well defined map

f∗ : π1(X, x0)→ π1(Y, y0 = f(x0)), [σ] 7→ f∗[σ] = [f ◦ σ]

Proposition 14. Homeomorphic spaces have identical fundamental groups.

If f is a homeomorphism, then f induces an isomorphism on the space of loops based at x0

and y0. It also maps bijectively the homotopies between loops. Hence f∗ is an isomorphismbetween π1(X, x0) and π1(Y, y0).

The basic problem in topology is to classify topological spaces up to homeomorphisms. Thefundamental group is useful for this purpose because spaces admitting different fundamentalgroups are distinct up to homeomorphisms. Moreover π1 is an example of a topologicalinvariant.

Proposition 15. If φ : X → Y is a homotopy equivalence between two topological spacesX and Y , then φ∗ : π1(X, x0) → π1(Y, φ(x0)) is an isomorphism. Therefore homotopyequivalent spaces have isometric fundamental groups.


7. Coverings and the Fundamental group of the Circle

Definition 7.1. Covering MapLet E and X be two topological spaces. A continuous function p : E → X is called a

covering map iff for all x ∈ X, there exists a neighborhood U of x such that p−1(U) =⋃β Vβ with p |Vβ : Vβ → U being homeomorphisms and the Vβ are disjoint for all β.

Such a neighborhood will be called admissible. p−1(x) is a disjoint set, called the fiberof the covering over x. If the fiber is a finite set of k points, the covering is called ak-sheeted covering.


(1) Let p : R → S1 be a map such that t 7→ (cos 2πt, sin 2πt). Then p−1(x) = Z forx ∈ S1.

(2) Let pn : S1 → S1 be a map such that (cos 2πt, sin 2πt) 7→ (cos 2πnt, sin 2πnt). Thenp−1(x) = {1, ..., n} for x ∈ S1.

(3)

16 N. MOSHAYEDI

Definition 7.2. Equivalent CoveringsTwo coverings p1 : E1 → X and p2 : E2 → X are called equivalent if there is a

homeomorphism f : E1 → E2 satisfying p2 ◦ f = p1.

Proposition 16. Lifting LemmaLet X,E and Y be topological spaces. Let p : E → X be a covering map. Let σ :

Y × [0, 1] → X be a continuous family of paths in X parameterized by Y , such thatσ(y, ·) = f(y) for a continuous function f : Y → X. Assume that f is lifted to a

function f : Y → E such that p◦ f = f . Then there exists a unique lift σ : Y × [0, 1]→ E

such that p ◦ σ = σ (σ(y, 0) = f(y)).

Remark 7.1. The proposition sais that if we fix x0 ∈ p−1(x0), then there is a unique path σin E lifting σ and starting at x0. σ is called the lift of σ starting at x0.

Proposition 17. Path Homotopy LiftingLet X and E be two topological spaces. Let p : E → X be a covering map. Let σ0 and

σ1 be paths in X with x0 = σ0(0) = σ1(0), σ0(1) = σ1(1) and σ0 ' σ1 rel {0, 1}. Letx0 ∈ p−1(x0) and σ0, σ1 be the lifts of σ0, σ1 respectively, starting at x0. Then

σ0 ' σ1 rel {0, 1}.

Corollary 2. Let σ0, σ1 be two paths in X such that x0 = σ0(0) = σ(0), σ0(1) = σ1(1)and σ0, σ1 be their lifts at x0 ∈ p−1(x0). If σ1(1) 6= σ0(1), then σ0 and σ1 are not pathhomotopic. Hence the endpoint of the lifts can give information about the homotopy classof a path.

Proposition 18.π1(S1, 1) ' Z

Proposition 19. Brower’s Fixed point Theorem for the DiscLet D2 = {x ∈ R2 | ‖x‖ ≤ 1} and f : D2 → D2 be continuous. Then there is a fixed

point of f , i.e. a point x ∈ D2 such that f(x) = x.


Proposition 20. Borsuk-UlamLet f : S2 → R2 be continuous. Then there exists x ∈ S2 such that f(x) = f(−x).

This also holds for f : Sn → Rn.

Corollary 3. Let S2 ⊂ A1∪A2∪A3 and Ai closed for i ∈ {1, 2, 3}. Then there is at leastone Ai containing a pair of antipodal points, i.e. there exists x ∈ S2 with x,−x ∈ Ai.

8. The Seifert- van Kampen theorem

We want to find a tool to compute the fundamental group of a topological space.

Definition 8.1. Free productThe free product ∗αGα of the groups Gα is defined as follows. As a set, ∗αGα is the set

of finite words g1 · · · gm for m ≥ 0, where gi ∈ Gαi and such that αi 6= αi+1 (such wordsare called reduced.) Equivalently one can allow arbitrary words, but make identificationg1 · · · gigi+1 · · · gm = g1 · · · g · · · gn whenever gi, gi+1 ∈ Gα, gigi+1 = g. The product in∗αGα is given by the concatination of words (possibly followed by reduction):

(g1 · · · gm)(h1 · · ·hn) = g1 · · · gmh1 · · ·hn.The right hand side can be reduced if gm and h1 belong to the same group and the

process goes further if gm = h−11 . The neutral element is the empty word, written 1. The

inverse of g1 · · · gm is given by g−1m · · · g−1

1 .

Example 8.1.

Z︸︷︷︸{an|n∈Z}

∗ Z︸︷︷︸{bn|n∈Z}

(i) a2b−2ab2 ∈ Z ∗ Z(ii) (a2b−1ab2)(b2a) = a2b−2ab4a

(iii) a2b−2ab2(b−2a−1) = a2b−2

Theorem 21. Universal property of the Free productLet ϕα : Gα → H be a group homomorphism. Then there exists a unique ϕ : ∗αGα → H

such thatϕ(g1 · · · gn) = ϕα1(g1) · · ·ϕαn(gn), gi ∈ Gαi .

For α ∈ {1, 2}, we get the following commutative diagram:

G1⊂ - G1 ∗G2

� ⊃ G2

H

ϕ

?

................. �

ϕ 2ϕ1

-

18 N. MOSHAYEDI

8.1. Generators and Relations. A useful way to describe many groups is as follows.Z ∗ · · · ∗ Z =: Fn is the free group on n generators. Its elements are all reduced words inthe generators. Pick words w1, ..., wk. Elements of the form gwig

−1, for g ∈ Fn, generate anormal subgroup N of Fn. We write

Fn/N = 〈g1, ..., gn | w1 = 1, ..., wk = 1〉

with g1, ..., gn the generators of Fn. The elements of Fn/N are all the words in the gener-ators, subject to the relation wi = 1. Fn/N is said to admit a presentation with generatorsand relations.

Example 8.2.

Z2 = 〈g | g2 = 1〉

Z× Z = 〈g1, g2 | g1g2 = g2g1〉

Definition 8.2. Amalgamated productLet A,G1, G2 be groups and ψ1 : A→ G1, ψ2 : A→ G2 be group homomorphisms. Let

N be the normal subgroup of G1∗G2 generated by all elements of the form ψ1(a)(ψ2(a))−1.Then the free product of G1 and G2 with amalgamated subgroup A is given by

Ga ∗A G2 =G1 ∗G2

N.

G1 ∗AG2 is composed of the same words as G1 ∗G2, in which the extra relation ψ1(a) =ψ2(a), a ∈ A has been imposed.

Theorem 22. Universal property for the Amalgamated product

A

G1

i1-�

ψ 1

G1 ∗A G2�i2

G2

ψ2

-

H

ϕ

?

................. �

γ 2γ1

-

Whenever there is H, γ1, γ2 making the diagram commute, ∃!ϕ : G1 ∗A G2 → H com-pleting it.

Example 8.3. Consider the case where a topological space X is the union of subspaces. LetX = U1 ∪U2, where U1, U2 and U1 ∩U2 are all path-connected. We pick x0 ∈ U1 ∩U2. Writeγi : Ui ↪→ X and ki : U1∩U2 ↪→ Ui. Define also ιi : π1(Ui, x0)→ π1(X, x0) where ιi = γi∗ and


ψi : π1(U1 ∩ U2, x0)→ π1(Ui, x0) where ψi = ki∗. Therefore we get the following diagram:

π1(U1 ∩ U2)ψ1- π1(U1)

π2(U2)

ψ2

?

ι2- π1(X)

ι1

?

Note that X is the universal object for the diagram. The Seifert-van Kampen theorem saisthat π1(X) is the universal object for the diagram above.

Theorem 23. Seifert-van KampenIf we have everything as in the example above, then

π1(X) = π1(U1) ∗π1(U1∩U2) π1(U2),

π1(U1 ∩ U2)

π1(U2)ι1 -

�

ψ 2

π1(X) �ι2

π1(U1)

ψ1

-

H

ϕ

?

................. �

γ 2γ1

-

i.e. for any group H and γi : π1(Ui)→ H such that γ1 ◦ψ2 = γ2 ◦ψ1, ∃!ϕ : π1(X)→ Hthat makes the diagram commute.

Remark 8.1. The Seifert-van Kampen theorem allows us to compute π1(X) from the knowl-edge of π1(U1), π1(U2), π1(U1 ∩ U2), ψ1, and ψ2.

Example 8.4. Let X = S1 ∨ S1. Then π1(X) = π1(S1) ∗0 π1(S1) = Z ∗ Z.

9. Compact surfaces

Definition 9.1. ManifoldA topological space M is called a manifold if

(1) M is Hausdorff(2) M has a countable basis of its topology(3) M is locally homeomorphic to Rn for some n. I.e. for each point x ∈ M , there

exists an open neighborhood of x that is homeomorphic to an open subset of Rn.

Remark 9.1. n is called the dimension of the manifold M .


20 N. MOSHAYEDI

• Rn, Sn, T n = S1 × · · · × S1︸︷︷︸n

are manifolds of dimension n.

• The wedge of 2 circles is not a manifold because the crossing point does not satisfycondition (3).

Definition 9.2. SurfaceA surface is a 2-dimensional manifold.

Example 9.2. R2, S2, T 2,RP 2 := S2/± id, Klein bottle.

Remark 9.2. The fundamental group of Sn is π1(Sn, x0) = 0 (trivial) for all n ≥ 2.

Proposition 21. Invariance of the DomainLet U ⊂ Rn, V ⊂ Rm be open subsets. Then there is no homeomorphism between U

and V if n 6= m.

Remark 9.3. This implies that a manifold has a well defined dimension (on each connectedcomponent).

9.1. The Fundamental group of RP n. RP n is the quotient of S2 by the antipodal map.Use the parameterization S1 × [−π

2, π

2]→ S2, (~x, t) 7→ cos t(~x, 0) + sin t(~0, 1). Consider also

the antipodal map given by (~x, t) 7→ (−~x,−t). We have a covering p : S2 → RP 2. Take U+ =p(open upper hemisphere) ⊂ RP 2 and U2 = p(S1 ' (−π

4, π

4)). Therefore we get U1 ∩ U2 '

Moebius-central circle = cylinder. We have π1(U1) = 1, π1(U2) = Z and π1(U1 ∩ U2) = Z.Moreover for a ∈ π1(U1 ∩ U2), ψ2(a) = 2a ∈ π1(U2). Therefore π1(RP 2, x0) = Z2.

Z - 1

Z

·2

?- π1(RP 2)

?

9.2. Construction of new Surfaces from known ones: Connected Sums.

Definition 9.3. Connected SumLet F1 and F2 be two connected surfaces. The connected sum F1#F2 is constructed as

follows.

(1) Delete an open disc in F1 and one in F2.(2) One obtains two surfaces with boundary, i.e. two topological spaces locally home-

omorphicc to R2 or {(x, y) ∈ R2 | x > 0} (bounderies).(3) The boundaries are a circle in each surface. Identify them with a homeomorphism.

After the identification, one gets a closed surface, F1#F2.

Proposition 22. F1#F2 is independent of

(1) The choice of the open discs in F1 and F2,(2) The homeomorphism used to identify the boundary circles.

Moreover # is associative: (F1#F2)#F3 ' F1#(F2#F3).


Example 9.3.

T 2

T 2#T 2

T 2#T 2#T 2

Theorem 24. Classification of Compact Closed SurfacesAny compact closed surface is homeomorphic to one of the following:

(1) S2

(2) A connected sum of k tori T 2, k ≥ 1.(3) A connected sum of k projective spaces RP 2, k ≥ 1.

Moreover, these surfaces are all distinct up to homeomorphism.

9.3. Polygonal representation of surfaces. The following way of continuing surfaces isuseful.

• Let P be a polygon with an even number of sides, say 2n.• Orient and label edges with n labels, so that each label is used twice.• Identify the edge carrying the same label by mean of homeomorphisms preserving

the orientation of the edges. I.e. take the quotient topological space.

This construction always yields surfaces.

22 N. MOSHAYEDI

We call this construction a polygonal presentation of a surface.

Let us check in one example that the quotient is locally homeomorphic to R2.

We see why we need to identify the edges 2 by 2.

• An unidentificated edge would look like =⇒ Not locally homeomorphic to R2.• 2 identified edges would look like

The surfaces we are familiar with can all be presented by quotients of polygons:

But the presentation is not unique! For instance

The operation of taking the connected sum can easily be presented in terms of polygons:

Notation: We will denote a polygonal presentation of a surface by the sequences of labelsof its edges, going clockwise, with a ()−1 representing the edges oriented counterclockwise.

Defined only up to cyclic permutation. We call pairs of edges appearing as · · · a · · · a · · · or· · · a−1 · · · a−1 · · · straight and pairs appearing as · · · a · · · a−1 · · · or · · · a−1 · · · a · · · opposed.


Lemma 25. The connected sum of k tori can be presented by

(a1b1a−11 b−1

1 )(a2b2a−12 b−1

2 ) · · · (akbka−1k b−1

k )

Lemma 26. The connected sums of k projective spaces can be presented by

(a1a1)(a2a2) · · · (akak)

9.4. Strategy of the Proof of the Classification Theorem. Do following steps:

(1) Show that any compact, closed surface admits a polygonal presentation.(2) Show that any polygonal presentation is homeomorphic to one of the canonical pre-

sentation above.(3) Show that the surfaces corresponding to the canonical presentations are distinct up

to homeomorphism (for this, use the fundamental group).

We will not prove (1), but here is the idea: Let F be a compact and closed surface. Asit’s compact, it admits a finite covering by open discs. With some work, one can use thecovering to show that F admits the structure of a CW-complex.

Definition 9.4. CW-complexA CW-complex is a topological space computed inductively as follows:

• Start from a discrete set of points S0, the 0-sceletton.• Take a union of 1-balls (line segments)

⋃αB

1α and a continuous map f1 :

∂(⋃αB

1α) → S0, and glue along f1 (i.e. quotient by x ∼ f1(x), ∀x ∈ ∂(

⋃αB

1α)).

This yields the 1-sceletton S1.• Take a union of n-balls

⋃αB

1α and a continuous map fn : ∂(

⋃αB

1α)→ Sn−1 and

glue along fn. The balls Bnα are called the cells of dimension n (n-cells) of the

complex.

Example 9.4. Any polygonal presentation of a surface is a CW-complex.

Theorem 27. Any surface admits a CW-complex structure.

• Given a CW-complex, triangulate the cells.

• Cut the triangulation so that it lays fat, keeping track of the identification

24 N. MOSHAYEDI

• Delete the inner edges =⇒ polygonal presentation of the surface

We therefore assume that a surface F admits a polygonal presentation.

2) We now want to show that it is homeomorphic to one of the canonical presentation.We do this in several steps.

A)

Proposition 23. A polygonal presentation can always be replaced by an equivalent onethat contains only one vertex. Equivalent means here that the associated surfaces arehomeomorphic.

B) Make the straight pairs of edges contiguous

C) Pairs of labels a and b appearing as · · · a · · · b · · · a−1 · · · b−1 · · · up to cyclic permutationare called crossed quadruplets.

Proposition 24. The crossed quadruplets can be made contiguous · · · aba−1b−1 · · ·

D) No uncrossed pairs. All the straight pairs and crossed quadruplets are contiguous.Can they be repeated by lonely opposed pairs? No! If there were such pairs left, withoutanother pair turning it into a crossed quadruplet, then P can not be identified with Q in thequotient. No uncrossed pairs are left.

E) If there are straight pairs, replace the crossed quadruplets by straight pairs.


2 new straight pairs and 1 less crossed quadruplet. Use B) to make the straight pairscontiguous. One can eliminate all crossed quadruplets if there is at least one straight pair.We showed that any polygonal presentation is equivalent to a canonical one.

3) We now show that the surfaces obtained from the canonical presentations are all distinctup to homeomorphisms.

Idea: Compute their fundamental groups and show that they are all distinct. As thefundamental group is preserved by homeomorphisms, this will show 3). To compute thefundamental group of a surface given by a polygonal presentation, we proceed for the exampleof the torus above. We assume that there is only one vertex.

π1(U1) = Z ∗ Z ∗ Z ∗ Z

π1(U2) = 1

Now with Seifert-van Kampen we get

π1(F ) = 〈a, b, c, d | a2bd−1c−1d−1b−1c = 1〉.

This example above generalizes to any polygonal presentation with one vertex and we getthe following theorem.

Theorem 28. The fundamental group of a surface obtained from a polygonal presentationwith labels a1, ..., an and whose circumference forms the word w in the labels is

π1(F ) = 〈a1, ..., an | w = 1〉

Therefore the fundamental group of any surface is one of the following

(1) π1(S2)(2) π1(T 2# · · ·#T 2︸︷︷︸

n

) = 〈a1, b1, ..., an, bn | a1b1a−11 b−1

1 · · · anbna−1n b−1

n 〉

(3) π1(RP 2# · · ·#RP 2︸︷︷︸n

) = 〈a1, ..., an | a21 · · · a2

n = 1〉

Are any of the these groups isomorphic?

26 N. MOSHAYEDI

Given two group presentations, it is very hard to decide whether the describe isomorphicgroups. In fact the problem is undecidable: no algorithm can solve correctly every instanceof the problem. The trick is to abelianize them.

Definition 9.5. AbelianizationGiven a group G admitting the presentation 〈a1, ..., an | w1 = 1, ..., wk = 1〉, the

abelianization Gab of G is the group given by the presentation

〈a1, ..., an | w1 = 1, ..., wk = 1, aiaj = ajai, i, j = 1, ..., n〉i.e. it is the group admitting the same relation, but whose generators commute (hence

Gab is abelian).

Of course, if two groups have different abelianizations, they are different. We have

πab1 (S2) = 1

πab1 (T 2# · · ·#T 2︸︷︷︸n

) = 〈a1, b1, ..., an, bn | aiaj = ajai, aibj = bjai, bibj = bjbi〉 ' Z× · · · × Z︸︷︷︸2n

πab1 (RP 2# · · ·#RP 2︸︷︷︸n

) = 〈a1, ..., an | aiaj = ajai, (a1 · · · an)2 = 1〉 ' Z× · · · × Z︸︷︷︸n−1

×Z2

These groups are all non-isomorphic. The fundamental groups are non-isomorphic. Thecanonical presentations yield distinct surfaces up to homeomorphism.

10. The Euler characteristic

The fundamental group is a topological invariant, but it’s not always easy to deal with .We define here a simpler topological invariant of surface.

Observation: We notice the following:

• Tetrahedron: 4 vertices, 6 edges, 4 faces =⇒ 4− 6 + 4 = 2.

• Cube: 8 vertices, 12 edges, 6 faces =⇒ 8− 12 + 6 = 2.

• Octahedron: dual to the cube =⇒ 6− 12 + 8 = 2.

• Dodecahedron: 20 vertices, 30 edges, 12 faces =⇒ 20− 30 + 12 = 2.

• Icosahedron: dual to the dodecahedron =⇒ 12− 30 + 20 = 2


The platonic solids can be seen as CW-complex representation of the sphere. For any suchCW-complex representation, we have

χ(S2) = #vertex−#edges + #faces = 2

Definition 10.1. Euler CharacteristicGiven a CW-complex with kn n-cells, the Euler characteristic χ is defined by

χ = k0 − k1 + k2 − k3 ± ... =∞∑n=0

(−1)nkn,

where we assume that the CW-complex is finite dimensional, i.e. that kn = 0 forn > N .

Poperties of the Euler Characteristic:We have following properties:

• χ is a homotopy invariant, i.e. homotopy equivalent CW-complexes have the sameEuler characteristic.• χ is additive under disjoint union, i.e. χ(X q Y ) = χ(X) + χ(Y ).• χ is multiplicative under products, i.e. χ(X × Y ) = χ(X)χ(Y )• If M is a k-sheeted covering of M , then χ(M) = k · χ(M)

Using the canonical presentations, we can easily compute the Euler characteristic of anysurface.

(1) χ(S2) = 2, as we already saw.(2) χ((T 2)#n) = 1− 2n+ 1 = −2n+ 2. In particular χ(T 2) = 0(3) χ((RP 2)#n) = 1− n+ 1 = −n+ 2.

Remark 10.1. Note following

• χ((T 2)#n) = χ((RP 2)#2n) despite the fact that (T 2)#n and (RP 2)#2n are not homo-topy equivalent. The Euler characteristic is less refined than the fundamental groupas a homotopy invariant.• For oriented surfaces (i.e. (1) and (2)), one sometimes prefers the genus g, defined

by

g =2− χ

2The genus counts the number of holes in an oriented surface.

28 N. MOSHAYEDI

Part 2. Geometry

11. Curves

In the following I denotes any non-empty interval in R, or R itself.

Definition 11.1. Parameterized CurveA parameterized curve is a C∞-differentiable function c : I → Rn. We write a dot for

the tangent map

c(t) =

c1(t)...

cn(t)

, c(t) =

c′1(t)...

c′n(t)

Recall that C∞-differnetiable (or C∞ for short) means that the derivative of any order

exist.

Definition 11.2. Regular CurveA curve c is called regular if c(t) 6= 0 for all t ∈ I.

Definition 11.3. Change of Parameterization and ReparameterizationLet c : I1 → Rn be a parameterized curve. A change of parameterization is a bijective

function ϕ : I2 → I1, for I2 some interval on R, such that ϕ and ϕ−1 are C∞. The curvec = c ◦ ϕ is also a parameterized curve and is called the reparameterization of c by ϕ.

Lemma 29. If c is regular, then c is regular.

Definition 11.4. Orientation-preserving and Orientation-reversingThe change of parameterization ϕ is called orientation-preserving if ϕ′(t) > 0 ∀t ∈ I2,

and orientation-reversing if ϕ′(t) < 0 ∀t ∈ I2.

Definition 11.5. Length of a Regular CurveLet c : I → Rn be a regular parameterized curve. Then

L(c) =

∫I

‖c(t)‖dt ∈ [0,∞]

is the length of c. A curve with finite length is called rectifiable.

Lemma 30. If ϕ : I2 → I1 is a change of parameterization, then L(c ◦ ϕ) = L(c).

Definition 11.6. Parameterization by arc-lengthA curve c is parameterized by arc-length iff ‖c(t)‖ = 1 for all t ∈ I.

Lemma 31. Let c : I → Rn be a regular curve. Then one can reparameterize c byarc-length.


Definition 11.7. Plane CurveA plane curve is a curve whose target is R2.

Definition 11.8. CurvatureFor a curve c, not necessarily parameterized by arc-length, the curvature is defined as

Kc(t) =1

‖c(t)‖3〈Jc(t), c(t)〉,

where J =

(0 −11 0

).

Lemma 32. Let F : R2 → R2 be an orientation-preserving isometry and c : I → R2 acurve. Then

KF◦c(t) = Kc(t).

Lemma 33. Let c : [a, b]→ R2 be a curve parameterized by arc-length. Then there existsa C∞-function

θ : [a, b]→ Rsuch that

c(t) =

(cos θ(t)sin θ(t)

),

where θ is defined up to addition of 2πk for k ∈ Z.

Definition 11.9. Periodic CurveA periodic curve is a curve c : R → R2 such that c(t + L) = c(t) for all t ∈ R and

L > 0 is the smallest number such that this is true.

Definition 11.10. Rotation IndexLet c be a periodic curve and θ as in the Lemma 33. Then the expression

nc :=1

2π(θ(L)− θ(0)) ∈ Z

is called the rotation index of c.

30 N. MOSHAYEDI

Remark 11.1. As c(t+ L) = c(t), we get

(cos θ(t+ L)sin θ(t+ L)

)=

(cos θ(t)sin θ(t)

)and

nc =1

2π(θ(t+ L)− θ(t)) ∈ Z

for all t ∈ R.

Theorem 34. Let c be a periodic curve with period L and let Kc : R→ R be its curvature.Then

nc =1

2π

∫ L

0

Kc(t)dt.

Remark 11.2. Notice that the following hold:

• nc is invariant under (orientation-preserving) homeomorphisms of R2, which meansthat nc is a topological invariant.• Kc is only invariant under the isometries of R2, which means that Kc is only a

geometrical invariant.• Theorem 34. states that nevertheless,∫ L

0

Kc(t)dt

is invariant under homeomorphism of R2.

Definition 11.11. Simple CurveA periodic curve is simple iff it does not cross itself, i.e. iff c(t1) = c(t2) =⇒ t2−t1 ∈ kL

for k ∈ Z and L the period.

Theorem 35. Rotation index theorem (Hopf)The rotation index of a simple curve is ±1.

11.1. Curves in Euclidean Space (3 Dimension). Let c : I → R3 be a regular curve.We want to define at each point c(t) an adaptive frame (orthonormal basis of vectors). Weassume that c(t) and c(t) are linearly independent.


Definition 11.12. Tangent unit vector, Normal unit vector, Binormal unitvector, Oscillating plane

The tangent unit vector is given by

e1(t) =c(t)

‖c(t)‖.

The normal unit vector is given by

e2(t) =c(t)− 〈c(t), e1(t)〉e1(t)

‖c(t)− 〈c(t), e1(t)〉e1(t)‖The binormal unit vector is given by

e3(t) = e1(t)× e2(t)

The oscillating plane is given by

span(e1(t), e2(t)) = span(c(t), c(t)).

Lemma 36. The following hold

(1)〈ei(t), ej(t)〉+ 〈ei(t), ej(t)〉 = 0

(2)〈ei(t), ei(t)〉 = 0 =⇒ ei(t) ⊥ ei(t)

(3)〈e1(t), e3(t)〉 = 0.

Definition 11.13. Curvature and TorsionLet c be a curve in R3 with c(t) and c(t) linearly independent.Then the curvature of c is given by

K(t) := 〈e1(t), e2(t)〉and the torsion of c is given by

τ(t) := 〈e2(t), e3(t)〉.We write Kc and τc to specify the curve c.

Proposition 25. Frenet-Serret Formulas

(1)e1(t) = K(t)e2(t)

(2)e2(t) = −K(t)e1(t) + τ(t)e3(t)

(3)e3(t) = −τ(t)e2(t)

32 N. MOSHAYEDI

Proposition 26. Kc and τc are independent of the parameterization of the oriented curvec. They can be expressed as follows:

Kc(t) =‖c(t)× c(t)‖‖c(t)‖3

(1)

τc(t) =det(c(t), c(t),

...c (t))

‖c(t)× c(t)‖2(2)

Definition 11.14. Fundamental theorem of CurvesLet c be a regular curve c : I → R3 such that c(t) and c(t) are linearly independent for

all t ∈ I. Let T be an isometry of R3. Then

Kc(t) = KT◦c(t)

τc(t) = τT◦c(t).

Conversely, let K and τ be smooth functions from I to R, with K strictly positive.Then there is a curve c parameterized by arc-length such that

K(t) = Kc(t)

τ(t) = τc(t).

Any two such curves are selected by an isometry of R3.

Theorem 37. Total Curvature of space CurvesLet c : R → R3 be a periodic space curve (regular parameterized by arc-length), with

period ω. Then ∫ ω

0

Kc(t)dt ≥ 2π.

Equality can hold only if c ⊂ P ⊂ R3 for some plane P ⊂ R3.

Lemma 38. Let v ∈ S⊥ ⊂ R3. Then there exists tv ∈ [0, ω] such that

〈c(tv), v〉 = 0.

12. Smooth Surfaces in R3

We study here surfaces pictured as subspaces of R3.


Definition 12.1. Smooth Surface in R3

A smooth surface in R3 is a subset X ⊂ R3 such that for each point of x, there is anopen neighborhood U ⊂ X, an open set V ⊂ R2 and a map ~r : V → R3 such that

• ~r : V → U is a homeomorphism. ~r is a local parameterization of X

• ~r is C∞. If ~r(u, v) = (x(u, v), y(u, v), z(u, v)), u, v ∈ R, x, y, z : R2 → R, thenx, y, z have derivatives of all order in u, v.• at each point of X, ~ru := ∂~r

∂uand ~rv := ∂~r

∂vare linearly independent.

Remark 12.1. Recall the definition of a surface we had in the topology part:

(1) X is Hausdorff(2) X has a countable basis of its topology(3) X is locally homeomorphic to R2, i.e. for each point of X, there is an open neigh-

borhood U homeomorphic to an open subset V of R2.

That means a smooth surface in R3 is a (topological) closed surface in the sense of theabove.

We can also make sense of the notion of a smooth surface without referring to R3. WriteϕU : U → V for the homeomorphism in the definition of a topological surface. We callϕU a coordinate system on U . Then for ϕU : U → V and ϕU ′ : U ′ → V ′, we have thehomeomorphisms

ϕV V ′ := ϕU ′ ◦ ϕ−1U : V → V ′, V = ϕU(U ∩ U ′) ⊂ R2, V ′ = ϕU ′(U ∩ U ′) ⊂ R2.

Definition 12.2. DiffeomorphismA homeomorphism between subsets of Rn that is C∞ and such that its inverse is C∞

is called a diffeomorphism.

Definition 12.3. Smooth Surface (general)A smooth surface is a closed topological surface such that the homeomorphisms ϕV V ′

are all diffeomorphisms.

34 N. MOSHAYEDI

Definition 12.4. Smooth MapA smooth map between smooth surfaces X and Y is a continuous map f : X → Y such

that for each coordinate system ϕU : U → V with U ⊂ X, x ∈ U and ϕU ′ : U ′ → V ′ withU ′ ⊂ X, f(x) ∈ U ′, the composition

ϕU ′ ◦ f ◦ ϕ−1U

is a smooth map.

Example 12.1. We have the following examples of smooth surfaces in R3. Let ~e1, ~e2 and ~e3

be an orthonormal basis of R3.

(1) A sphere of radius a:

X = {~p ∈ R3 | ‖~p‖ = a}.We have the following parameterization: Let

V1 = (−π, π)× (0, π),

U1 = X \ {~p ∈ R3 | c1~e2 + c2~e3, c1 ≤ 0}and

~r1(u, v) = a sinu sin v~e2 + a cosu sin v~e2 + a cos v~e3.

Let

V2 = (−π, π)× (0, π),

U2 = X \ {~p ∈ R3 | c1~e1 + c2~e2, c2 ≥ 0}and

~r2(u, v) = a cos v~e1 − a cosu sin v~e2 + a sinu sin v~e3.

It is not possible to cover the whole sphere with a simple parameterization ~r : V →U .

(2) A torus with outer radius a+ b and inner radius a− b.

X = {~p ∈ R3 | ~p = (a+ b cosu)(cos v~e1 + sin v~e2) + b sin v~e3, (u, v) ∈ R2}.We have the following parameterization: Let

V1 = (−π, π)× (−π, π),


U1 = X \ {~p ∈ R3 | ~p = (a− b)(cos v~e1 + sin v~e2) or ~p = (a+ b cos v)~e1 + b sin v~e3, v ∈ R}

and

~r1(u, v) = (a+ b cosu)(cos v~e1 + sin v~e2) + b sinu~e3.

(3) A plane (con-compact). We have the following parameterization:

~r(u, v) = ~a+ u~b+ v~c,

for all ~a,~b,~c ∈ R3 with ~b and ~c linearly independent.Note that the plane is diffeomorphic to R2, hence admits a global parameterization

in terms of a single map ~r.

Definition 12.5. Change of ParameterizationLet ~r : V → R3 be a local parameterization of a smooth surface X ⊂ R3. A change

of parameterization is a diffeomorphism f : V ′ → V for some V ′ ⊂ R2. The newparameterization is ~r = ~r ◦ f : V ′ → R3. If f(x, y) = (u(x, y), v(x, y)), then

(~r ◦ f)x = ~ruux + ~rvvx(~r ◦ f)y = ~ruuy + ~rvvy

i.e. ((~r ◦ f)x(~r ◦ f)y

)=

(ux vxuy vy

)(~ru~rv

)= J

(~ru~rv

).

As f−1 is differentiable, the Jacobian matrix J is invertible and {(~r ◦ f)x, (~r ◦ f)y} arelinearly independent iff {~ru, ~rv} are linearly independent.

Example 12.2. The (x, y)-plane in R3 admits the parameterization

~r(x, y) = x~e1 + y~e2.

Define

f : R× (−π, π)→ R2 \ R− × {0}, f(r, θ) = (r cos θ, r sin θ).

The change of parameterization gives us another (local) parameterization of the (x, y)-plane:

~r ◦ f(r, θ) = r cos θ~e1 + r sin θ~e2.

36 N. MOSHAYEDI

Definition 12.6. Tangent SpaceThe tangent space (or tangent plane) TpX of a surface X at a point p ∈ X is the

vectorspace generated by

{~ru(~r−1(p)), ~rv(~r−1(p))}.

Proposition 27. The tangent space is independent of the local parameterization ~r.

Definition 12.7. Unit Normals to the SurfaceThe vectors ±~n = ± ~ru∧~rv

|~ru∧~rv | are two unit normals to the surface.

• The two unit normals are orthogonal to the tangent plane.• They are invariant under changes of parameterizations f that preserve the orien-

tation of R2.

13. The First Fundamental Form

Let c : I → R3 be a regular curve in R3 Let X be a surface, ~r : V → U be a localparameterization. Assume that c(I) ⊂ U . Then γ = (~r)−1 ◦ c is a curve in V . We havec = ~rvu+ ~rvv where γ(t) = (u(t), v(t)). As {~ru, ~rv} are linearly independent, the fact that cis regular implies that (u, v) = γ 6= 0 and hence γ is regular.

Let us try to compute the length of c in the parameter space V , i.e. using γ instead of c.Let I = (a, b). Then

∫ b

a

‖c(t)‖dt =

∫ b

a

(√c · c)dt(3)

=

∫ b

a

√(~ruu+ ~rvv) · (~ruu+ ~rvv)dt(4)

=

∫ b

a

√Eu2 + 2Fuv +Gv2dt,(5)

where E = ~ru · ~rv, F = ~ru · ~rv and G = ~rv · ~rv.

Definition 13.1. First Fundamental FormThe first fundamental form of a surface X in R3, relative to a local parameterization

~r : V → U ⊂ X, is the expression

I~r = E(du)2 + 2Fdudv +G(dv)2,

where E = ~ru · ~rv, F = ~ru · ~rv and G = ~rv · ~rv.

The first fundamental form is simply the quadratic form given by the standard scalarproduct on R3, (Q~v = ~v ·~v) restrictted to the tangent space of X and expressed in the basis{~ru, ~rv}. If ~v ∈ TpX with ~v = v1~ru + v2~rv, we should think of du and dv as linear forms onTpX selecting the component of ~v along ~ru and ~rv


du : TpX → R, ~v 7→ v1 dv : TpX → R, ~v 7→ v2.

Proposition 28. Under a change of parameterization f : V ′ → V , f(x, y) =(u(x, y), v(x, y)), we have

Edu2 + 2Fdudv +Gdv2 = Edx2 + 2F dxdy + Gdy2,

with (E F

F G

)=

(ux vxuy vy

)(E FF G

)(ux uyvx vy

).


(1) Consider the plane parameterized by

~r = (x, y) = x~e1 + y~e2 ~rx = ~e1, ~ry = ~e2 I~r = dx2 + dy2.

Change the parameterization to polar coordinates:

x = r cos θ, y = r sin θ.

then

dx = dr cos θ − r sin θdθ

dy = dr sin θ + r cos θdθ

Therefore

I~r = dx2 + dy2 = (dr cos θ − r sin θdθ)2 + (dr sin θ + r cos θdθ)2 = dr + r2dθ2.

In the following above we have

uvxy

→xyrθ

(E FF G

)→(

1 00 1

),

(E F

F G

)→(

1 00 r2

).

ux →dx

dr= cos θ, uy →

dx

dθ= −r sin θ, vx →

dy

dr= sin θ, vy →

dy

dθ= r cos θ.

(2) Consider the sphere with with radius a and the parameterization

~r(u, v) = a sinu sin v~e1 + a cosu sin v~e2 + a cos v~e3.

Then

~ru = a cosu sin v~e1 − a sinu sin v~e2

~rv = a sinu cos v~e1 + a cosu cos v~e2 − a sin v~e3

We have

38 N. MOSHAYEDI

E = ~ru · ~ru = a2 sin2 v, F = ~ru · ~rv = 0, G = ~rv · ~rv = a2.

Hence the first fundamental form is

I~r = a2dv2 + a2 sin2 vdu.

Remark 13.1. On a smooth surface X (not necessarily embedded in R3), the analog of thefirst fundamental form is a Riemannian metric and is defined as follows.

Given a coordinate system ϕU on an open set U ⊂ X, with coordinates (u, v), we specifythree functions E,F and G such that E(u, v) > 0, G(u, v) > 0 and E,G, F 2 > 0. Givenanother open set U ⊂ X such that U ∩ V 6= ∅, with coordinates (x, y) and function E,F and G satisfying the same properties. A Riemannian metric is not necessarily the firstfundamental form of some embedding of X in R3.

We can do more than measure the length of curves with the first fundamental form.

13.1. Angles between Curves. Assume c1, c2 are curves on X that intersect at p ∈ X.Let t1 = c−1

1 (p), t2 = c−12 (p). In the following all the quantities related to c1 are evaluated

at t1 and similarly for c2. For instance the angle θ between c1 and c2 at p is given by

cos θ =c1 · c2

|c1||c2|,

where the arguments t1 and t2 have been omitted. Suppose ~r : V → U is a local parame-terization and p ∈ U . We get curves γi = (~r)−1 ◦ ci in V and we write

γi(t) = (ui(t), vi(t)).

Let us decompose ci on the basis {~ru, ~rv} of TpX:

ci = ~ruui + ~rvvi.

Therefore

c1 · c2 = (~ruu1 + ~rvv1) · (~ruv2 + ~rvv2)(6)

= Eu1u2 + F (u1v2 + u2v1) +Gv1v2(7)

= (u1, v1)

(E FF G

)(u2

v2

)(8)

Therefore the angle can be expressed in terms of ui, vi and the first fundamental form,i.e. the angle θ can be computed on V .


13.2. Areas. We can also use the first fundamental form to measure areas from the pa-rameter space V . Let U ⊂ X be an open set and ~r : V → U be a parameterization of U .Let {f1, f2} be an orthonormal basis of R2 ⊃ V . The area element associated to (f1, f2) ismapped to the area element associated to (~ru, ~rv) in U by ~r.

The ratio of area of the two elements is given by |~ru ∧ ~rv|. We can therefore compute thearea of U as follows:

Area(U) =

∫V

|~r ∧ ~rv|dudv.

Now |~ru ∧ ~rv| = (~ru · ~ru)(~rv · ~rv)− (~ru · ~rv)2 = EG− F 2, so

Area(U) =

∫V

√EG− F 2dudv =

∫V

√det(g)dudv, g =

(E FF G

)The area computed should of course be invariant under changes of parameterization of U .

Let f : V → V be such a change, given by f(x, y) = (u(x, y), v(x, y)). Then

~rx = ~ruux + ~rvvx, ~ry = ~ruuy + ~rvvy

and

~rx ∧ ~ry = (uxvy − vxuy)~ru ∧ ~rv.

Therefore ∫V

|~rx ∧ ~ry| =∫V

|~ru ∧ ~rv||uxvy − vxuy|dxdy =

∫V

|~ru ∧ ~rv|dudv

and the area is independent of the choice of parameterization of U .

Example 13.2. Area of the sphere of radius a. Recall that for one parameterization of thesphere, we have

V = (−π, π)× (0, π), E = a2 sin2 v, G = a2, F = 0.

We cover the whole sphere minus a segment of zero area. Hence we can compute the areaby integrating over V !

Area(S2a) =

∫(−π,π)×(0,π)

√a4 sin2 vdudv = 2πa2

∫ π

0

| sin v|dv = 4πa2.

40 N. MOSHAYEDI

14. isometric surfaces

Isometries are transformation preserving lengths. Practically, we define them as follows

Definition 14.1. Isometry between SurfacesLet X, X be two smooth surfaces in R3. An isometry f : X → X is a diffeomorphism

such that for each curve c : I → X,

L(c) = L(f(c)),

i.e. f maps curves in X to curves in X of the same length. X and X are then said tobe isometric.

Theorem 39. Let U ⊂ X and U ⊂ X be two open sets. Then U is isometric to U iffthere exists parameterizations

~r : V → U, ~r : V → U ,

V ⊂ R2, with the same first fundamental form.

Two surfaces X and X are isometric if they can be covered by pairs of isometric open setsas in the theorem, and if the associated maps f agree on overlaps.

Example 14.1. Consider the cone X = {(x, y, z) ∈ R3 | x2 + y2 = a2z2, z > 0}. We canparameterize U = X \ {(x, y, z) ∈ R2 | x > 0} by V = (0,∞)× (0, 2π) using

~r(u, v) = a(u cos v~e1 + u sin v~e2) + u~e3,

where

~ru = a(cos v~e1 + sin v~e2) + ~e3

~rv = a(−u sin v~e1 + u cos v~e2)

and

E = 1 + a2, F = 0, G = a2u2.

So the first fundamental form is (1 + a2)du2 + a2u2dv2. Consider now the planar sector

X = {(x, y, z) ∈ R3 | z = 0, 0 < θ < β}, for β =√

a2

1+a22π, θ = Arg y

x. We see that (r, θ)

provide a parameterization of X. The first fundamental form is that of the plane in polarcoordinates: dr2 + r2dθ2. But we can parameterize X by V as follows. Set r =

√1 + a2u

and θ =√

a2

1+a2v. The first fundamental form becomes (1 + a2)du2 + a2u2dv2. Hence X and

X are isometric.


15. The Second Fundamental Form

The first fundamental form describes intrinsic properties of a surface, i.e. propertiesmeasurable by an inhabitant of the surface that would know about the 3-dimensional worldin which the surface is embedded. This is why there is a corresponding notion for abstractsurfaces, the Riemannian metric.

The second fundamental form gives information about how the surface is embedded in R3

and is not intrinsic. Consider a local parameterization ~r(u, v) of a surface X. Recall thatthe unit normal is given by

~n(u, v) =~ru ∧ ~rv|~ru ∧ ~rv|

.

We take the surface, and displace it by an amount t along ~n. At least for t small enough,we obtain a 1-parameter family of surfaces

~R(u, v, t) = ~r(u, v)− t~n(u, v)

with

~Ru = ~ru − t~nu, ~Rv = ~rv − t~nv.We have a family of first fundamental forms

I~r(t) = E(t)du2 + 2F (t)dudv +G(t)dv2.

We compute

1

2

∂

∂tI~r(t) |t=0= −(~ru · ~nudu2 + (~ru · ~nv + ~rv · ~nu)dudv + ~rv · ~nvdv2).

The right hand side is the second fundamental form. Just like the first one, it is a quadraticform on the tangent space of X.

Definition 15.1. Second Fundamental FormThe second fundamental form of a parameterized surface is the quadratic form

II~r := Ldu2 + 2Mdudv +Ndv2,

where

L = ~ruu · ~n, M = ~ruv · ~n, N = ~rvv · ~n.


(1) The plane

~r(u, v) = ~a+ u~b+ v~c.

We get ~ruu = ~ruv = ~rvv = 0 and hence II~r = 0.(2) The sphere of radius a. We have ~r = a~n, so

~ru · ~nu = a−1~ru · ~ru...and so on. Therefore II~r = 1

aI~r.

42 N. MOSHAYEDI

Proposition 29. If ~r : V → U is a paremeterization and II~r = 0 on V , then U iscontained in a plane in R3.

To understand better what the second fundamental form is meaning, consider a surfacedefined by the graph of a function f : R2 → R:

~r(x, y) = x~e1 + y~e2 + f(x, y)~e3

with

~rx = ~e1 + fx~e3, ~ry = ~e2 + fy~e3, ~rxx = fxx~e3, ~rxy = fxy~e3, ~ryy = fyy~e3.

At a critical point of f , fx = fy = 0, so ~n = ~e3. II~r then coincides with the Hessian of fat the critical point:

II~r =

(L MM N

)=

(fxx fxyfxy fyy

).

Consider now any surface X ⊂ R3. At p ∈ X we can locally parameterize X by its tangentplane. X is then locally around p the graph of a function f : TpX → R. The fact that thetangent plane is tangent to X implies that p is a critical point of f .

We can use one knowledge of the theory of critical points of functions of 2 variables.

• If fxxfyy − f 2xy = det(II~r) > 0, then p is a local maximum if II~r is negative definite

and a local minimum if II~r is positive definite

• If det(II~r) < 0, then p is a saddle point. The tangent plane lies on both sides of thesurface.


Proposition 30. Any closed surface X ⊂ R3 has points at which II~r is positive definite(for a suitable choice of the unit normal).

Proposition 31. Under a orientation preserving change of parameterization f : V →V , the second fundamental form satisfies the same transformation formula as the firstfundamental form: (

L M

M N

)=

(ux vxuy vy

)(L MM N

)(ux uyvx vy

).

Definition 15.2. Principal CurvaturesLet ~r : V → U ⊂ R3 be a local parameterization of X such that ‖~rx‖ = ‖~ry‖ = 1,

~rx ·~ry = 0 at p ∈ U , i.e. {~rx, ~ry} form an orthonormal basis of TpX. Then the eigenvaluesk1, k2 of II~r at p are called the principal curvatures of X at p. The eigenvectors are theprincipal directions.

Remark 15.1. For any parameterization ~r, ~r satisfying the orthonormality condition above,(ux vxuy vy

)is an orthogonal matrix. II~r and II~r therefore have the same eigenvalues, so the

principal curvatures are independent of the parameterization.

Example 15.2. Recall a local parameterization of a torus of inner radius a − b and outerradius a+ b.

~r(u, v)(a+ b cosu)(cos v~e1 + sin v~e2) + b sinu~e3.

~ru, ~rv do not form an orthonormal basis of TpX for p = ~r(0, 0). However, by definingx = bu, y = (a+ b)v and performing the change of parameterization, we find

~rx = − sinx

b

(cos

y

a+ b~e1 + sin

y

a+ b~e2

)+ cos

x

b~e3

~ry =a+ b cosu

a+ b

(− sin

y

a+ b~e1 + cos

y

a+ b~e2

)and ~rx(0, 0) = ~e3 and ~ry(0, 0) = ~e2. Hence we get that {~rx, ~ry} is an orthonormal basis of

TpX.We can easily compute

~rxx(0, 0) = −1

b~e1, ~rxy(0, 0) = 0, ~ryy(0, 0) = − 1

a+ b~e1.

Moreover ~n = ~e3 × ~e2 = −~e1. Hence

II~r(0, 0) =1

bdx2 +

1

a+ bdy2

and

k1 =1

b, k2 =

1

a+ b.

44 N. MOSHAYEDI

b and a+ b are the radii of the circles in X going through p!

Remark 15.2. The principal curvatures are not intrinsic. The cylinder is isometric to the flatrectangle.

16. The Gaussian curvature

Definition 16.1. Gaussian CurvatureThe Gaussian curvature of a surface X ⊂ R3 is the function K : X → R given at each

point p ∈ X by

K(p) = k1(p) · k2(p),

where k1(p) and k2(p) are the principal curvatures at p.

Remark 16.1. We saw that the principal curvatures are independent of the parameterizationof X. The Gaussian curvature is therefore independent of the parameterization.

Proposition 32. We have

K(p) =det(II~r)

det(I~r)=LN −M2

EG− F 2

for ~r any local parameterization of X around p.


(1) For a plane we have II~r = 0 and therefore K = 0.

(2) For a sphere with radius a we have II~r = a−1I~r and therefore K = a−2.

the Gaussian curvature is defined using the second fundamental form, which is not intrinsicto the surface (i.e. depends on the embedding X → R3). Nevertheless, it turns out thatK depends only on I~r and is therefore intrinsic. This is the content of Gauss’s egregiumtheorem.


Tho prove it we need some tools. Consider a local parameterization ~r : V → U ⊂ X ⊂ R3

and a smooth family of tangent vectors ~a(u, v) = f~ru + g~rv ∈ T~r(u,v)X for f, g : V → R somesmooth functions.

After differentiation with respect to u or v we do not necessarily get a vector in T~r(u,v)X.What we can do is predict on the tangential component.

Definition 16.2. Tangential DerivativeLet ~a be as above. The tangential derivative of ~a with respect to u is

∇u~a = ~au − (~n · ~a)~n = ~au + (~nu · ~a)~n,

where ~n is the unit normal.

Lemma 40. ∇u~a can be expressed in terms of the components E,F,G, their derivativesand the derivatives of f and g.

Theorem 41. Theorema Egregium (Gauss 1828)The Gaussian curvature is invariant under isometries of surfaces. I.e. if U ⊂ X,

U ⊂ X, f : U → U an isometry and p ∈ U , then

K(p) = K(f(p)).

Proof. Let us compute

∇v∇u~a = ~avu − (~n · ~avu)~n+∇v((~nu · ~a)~n) = ~avu − (~n · ~avu)~n+ (~nu · ~a)~nv,

since ~nv is tangent. Exchanging u and v, we get

∇v∇u~a−∇u∇v~a = (~nu · ~a)~nv − (~nv · ~a)~nu = (~nu × ~nv)× ~a.But ~nu and ~nv are tangent, so ~nu × ~nv = λ~n. From the Lemma 40, λ is intrinsic, i.e.

expressible in terms of E,F,G and their derivatives. Now we compute

λ~n · (~ru × ~rv) = (~nu × ~nv) · (~ru × ~rv) = (~nu · ~nu)(~nv · ~rv)− (~nu · ~rv)(~nv · ~ru) = LN −M2.

Now since ~n · (~ru × ~rv) =√EG− F 2, we get

LN −M2 = λ√EG− F 2.

HenceK depends only on the first fundamental form I~r. As I~r is invariant under isometries,K is invariant as well.

�

17. The Gauss-Bonnet theorem

Recall the rotation index theorem: A curve is simple if its rotation index

nc =1

2π

∫ L

0

Kc(t)dt = ±1.

46 N. MOSHAYEDI

The integral of the curvature over a curve provides topological information about it. TheGauss-Bonnet theorem is an analog for surfaces: The Euler characteristic of a surface canbe computed by integrating the Gaussian curvature.

Definition 17.1. Geodesic CurvatureLet c : I → X ⊂ R3 be a smooth curve in X parameterized by arc-length. The geodesic

curvature Kg of c is defined by

Kg(t) = c(t) · (~n× c),where ~n is the unit normal to X.

Theorem 42. Gauss-Bonnet ILet ~r : V → U ⊂ X be a local parameterization of X. Let c : I → U be a smooth

simple curve in U parametereized by arc-length, enclosing a region R. Then∫I

Kgdt = 2π −∫R

KdA,

where Kg is the geodesic curvature, t ∈ I, K is the Gaussian curvature of X and dAis the area element on X.

Theorem 43. Gauss-Bonnet IILet X be a smooth orientable closed surface with Riemannian metric. Then∫

X

KdA = 2πχ(X),

where χ(X) is the Euler characteristic of X.

Remark 17.1. We can extend the theorem to curves not necessarily contained in the domainof a single parameterization, provided the unit normal can be globally defined.


Corollary 4. The sum of the angles of a curvilinear triangle R is

π +

∫R

KdA+

∫I

Kgdt.


(1) Let X be a plane. Then we know that K = 0. A straight line in X has Kg = 0.Hence the sums of the angle of an ordinary triangle is π.

(2) Let X be a unit sphere. Then we know K = 1. A segment of great circles has Kg = 0.For instance

c(s) = (cos s, sin s, 0) = ~n

c(s) = (− sin s, cos s, 0)

c(s) = −c(s)

Hence we get Kg = c · (~n × c) = 0. Therefore if ∆ is a triangle on X whose sidesare segments of great circles, we have

α + β + γ = π +

∫∆

KdA = π + Area(∆).

18. Morse Functions

The Gauss-Bonnet theorem is important because it relates topological data (the Eulercharacteristic) to geometrical data (the Gaussian curvature). Here, we see a theorem relatingthe Euler characteristic to differential data on the surface, namely a smooth function. Thisresult is a basic application of Morse theory, a set of powerful techniques to study thetopology of manifolds from the knowledge of a smooth function.

Let X be a closed surface. Let f : X → R be a smooth function. For instance, take X tobe a surface in R3 and f to be the height along a particular axis of R3.

48 N. MOSHAYEDI

As X is compact, f has a maximum and a minimum, but it may also have other criticalpoints. Let ~r : V → U ⊂ X be a local parameterization of X. f determines a function fV

on V by fV = f ◦ ~r.

Definition 18.1. Critical PointWe say that f has a critical point at p ∈ U , p = ~r(a), if

fVu (a) = fVv (a) = 0.

This condition is independent of the choice of parameterization. Indeed, if ϕ(x, y) =(u(x, y), v(x, y)), then we have

fV′

x = fVu ux + fVv vx, fV′

y = fVu uy + fVv vy.

Hence the definition above is unambiguous. Recall that the Hessian matrix at a criticalpoint is given by the matrix of second derivative(

fVuu fVuvfVvu fVvv

).

Under ϕ, it transforms as follows(ux uyvx vy

)(fV′

xx fV′

xy

fV′

yx fV′

yy

)(ux vxuy vy

)=

(fVuu fVuvfVvu fVvv

)Taking the determinant

(fVuufVvv − (fVuv)

2) = (uxvy − uyvx)2(fV′

xx fV ′

yy − (fV′

xy )2).

Therefore, the fact that the determinant of the Hessian is positive, negative or vanishes isindependent of the choice of parameterization of X.

Definition 18.2. Degenerate PointA critical point p ∈ X of a smooth function f : X → R is degenerate if the determinant

of the Hessian vanishes at p.

Definition 18.3. Morse FunctionA Morse function on X is a smooth function whose critical points are all non-

degenerate.

Theorem 44. Let f be a Morse function on a smooth closed surface X. Then the Eulerchartcteristic χ(X) is the sum of the numbers of maximums and minimums of f minusthe number of saddle points.

19. Geodesics

Geodesics on a surface X are the analogs of straight lines in the plane. Lines have twomain properties:

(1) A line segment between two points is the shortest path between them. Similarly, ageodesic segment between two points on X is the shortest path between them locally.


(2) Lines are the straightest paths: the tangent vector is constant.

Definition 19.1. GeodesicLet c : I → X be a smooth curve parameterized by arc-length in a surface X ⊂ R3.

Then c is called a geodesic of X if c(t) is normal to X at all t ∈ I.

Remark 19.1. Notice the following:

(1) This is equivalent to saying that the tangential component of c vanishes, hencegeodesics are the straightest curves on X.

(2) Recall that the geodesic curvature is Kg = c · (~n× c), so the geodesic curvature of ageodesic vanishes.


(1) Suppose that c lies in a plane X. Then, c lies in X as well. Therefore c is normalonly if it vanishes. Hence the geodesics in a plane are straight line segments.

(2) We saw that if c is a segment of great circles in a sphere, then c is normal. Hencesegments of great circles are geodesics on the sphere.

(3) Assume that X admits a plane of symmetry P . Then P ∩X is a geodesic. Indeed,if c(I) = P ∩X, then c(I) is invariant under symmetry. c is invariant as well, so liesin P . But c is orthogonal to c, which also lies in P . Hence c is normal to X.

Proposition 33. Let γ = (u, v) : I → V be the preimage of a curve c : I → U ⊂ Xthrough a parameterization ~r : V → U ⊂ X, i.e. c = ~r ◦ γ. Then c is a geodesic if andonly if

d

dt(Eu+ F v) =

1

2(Euu

2 + 2Fuuv +Guv2)

d

dt(Fu+Gv) =

1

2(Evu

2 + 2Fvuv +Gvv2)

Remark 19.2. The proposition shows that one can check whether a curve is a geodesicusing only the first fundamental form. Therefore geodesics are intrinsic objects. A geodesicis mapped to a geodesic by isometries and we can define geodesics on abstract Riemannsurfaces by using the proposition 33.


(1) Let X be a plane, parameterized by Cartesian coordinates. Then E = G = 1 andF = 0. The geodesic equation are

u = 0, v = 0.

Therefore u = a1t + b1 and v = a2t + b2 for a1, a2, b1, b2 ∈ R. Hence the geodesicsare straight lines.

(2) Let X be a cylinder, with

~r(u, v) = a(cos v~e1 + sin v~e2) + u~e3

and the first fundamental form du2 + a2dv2. X is isometric to a plane, so thegeodesics are of the form

50 N. MOSHAYEDI

u = a1t+ b1, v = a2t+ b2.

Therefore the geodesics are helices:

c(t) = a(cos(a1t+ b1)~e1 + sin(a2t+ b2)~e2) + (a1t+ b1)~e3.

Proposition 34. Let X be a surface in R3 and (p, ξ) ∈ TpX. Then there is a uniquegeodesic c on X such that c(0) = p and c = ξ, i.e. c passes through p in the direction ξ.

Application: Let p be a point on the sphere X. Then for any (p, ξ) ∈ TpX, there is agreat circle through p in the direction ξ. Hence all the geodesics on the sphere are segmentsof great circles.

Proposition 35. Let X be a surface in R3, ~r : V → U ⊂ X be a local parameterizationand p, q ∈ U two points. Assume that c : [a, b] → U is geodesic with c(a) = p andc(b) = q. Then among all the curves from a to b, c minimizes locally the length.

Remark 19.3. A geodesic is not necessarily the shortest path between two points, but thereverse is true.

Institut fur Mathematik, Universitat Zurich, Winterthurerstrasse 190 CH-8057 ZurichE-mail address, N. Moshayedi: [email protected], [email protected]

geometry and topology€¦ · geometry and topology nima moshayedi notes, february 1, 2015 contents

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