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# A Riemannian manifold (X, g) is a manifold X together with a Riemannian metric g

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GEOMETRY OF COMPACT RIEMANNIAN SURFACE

JIA-MING (FRANK) LIOU

All the manifolds in this note is assumed to be connected, smooth and oriented.

1. Riemannian manifold

A Riemaninan metric g on a manifold X is an assignment to each point p ∈ X an inner product g(p) on each tangent space TpX for p ∈ X such that for each pair of vector fields V, W on X, the function p 7→ g(p)(V, W ) is smooth. A Riemannian manifold (X, g) is a manifold X together with a Riemannian metric g. We also denote g(p)(V, W ) by hV, W ip

for each pair of vector fields (V, W ) on X. If v is a tangent vector in TpX, we denote kvkp = g(p)(v, v).

Let (x1, · · · , xn; U ) be a local coordinate system on X. Then we have local vector fields

 ∂

∂xi, 1 ≤ i ≤ n



, on U. We denote

gij(p) = h ∂

∂xi, ∂

∂xjip.

It follows from the definition that the function p 7→ (gij(p)) is a smooth matrix valued functions on U and that for each p ∈ U, the n × n-matrix (gij(p))ni,j=1 is positive definite.

It is very common to denote the Riemannian metric g by ds2 =

n

X

i,j=1

gijdxidxj.

When X is a Riemannian surface with a local coordinate system (u, v), the Riemannian metric is often written as ds2 = Edu2+ 2F dudv + Gdv2.

Definition 1.1. A map γ : [a, b] → X is called a smooth curve on X if there is an open interval I containing [a, b] so that γ can be extended to a smooth map from I into X.

A piecewise smooth curve is a map γ : [a, b] → X with the property that there exists a partition a = t0 < t1 < · · · < tn = b of [a, b] so that γ|[tk−1,tk] is a smooth curve on X for each 1 ≤ k ≤ n.

Given a curve1 γ : [a, b] on a Riemannian manifold X, we define its arc-length by L(γ) =

Z b a

k ˙γ(t)kγ(t)dt.

Let (X, g) be a Riemannian manifold. For each p, q ∈ X, we define d(p, q) = inf

γ L(γ),

where inf runs over all piecewise smooth curves γ on X connecting p, q.

Theorem 1.1. The function d is a metric on X such that the metric topology on (X, d) coincides with the given topology on X.

1All the curves now are assumed to be piecewise smooth.

1

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A set of vector fields {ei : 1 ≤ i ≤ n} on an n-dimensional Riemannian manifold X is called an orthonormal frame if hei, ej)i = δij for 1 ≤ i, j ≤ n. A set of one-forms {θi : 1 ≤ i ≤ n} is said to be the dual coframe of an orthonormal frame {ei} if θi(ej) = δij.

Let {θi : 1 ≤ i ≤ n} be the dual coframe to a positively oriented orthonormal frame {ei} on X. The n-form

dµ = θ1∧ · · · ∧ θn

is called the volume-form on X. Notice that dµ is a symbol for the volume form, we do not think of it as a differential of some n − 1-form.

Theorem 1.2. The definition of volume form on X does not depends on the choice of orthonormal frame, i.e. if {θ0i} is the dual coframe to another positively oriented orthonormal frame {e0i}, then θ01∧ · · · ∧ θn0 = θ1∧ · · · ∧ θn.

2. Geometry of Riemannian surfaces

Theorem 2.1. Let X be a Riemannian surface. Suppose that {e1, e2} is an orthonormal frame with dual coframe {θ1, θ2}. Then there exists a unique one-form ω12 = −ω21 such that

1 = ω12∧ θ2, dθ2= ω21∧ θ1.

Definition 2.1. We call ω12 the connection one-form associated with the orthonormal frame {e1, e2}

Suppose that {e01, e02} is another orthonormal frame on X with the same orientation as {e1, e2}, then

e01 = f e1+ ge2, e02 = −ge1+ f e2

for some smooth functions f, g on X. Let ω012 be the connection one form associated with {e01, e02}. We leave to the reader to show that

ω12= ω012− τ, where τ = f dg − gdf.

Theorem 2.2. Let X be a Riemannian surface. Assume that {e1, e2} is an orthonormal frame on X and ω12is its associated connection one-form. Then there is a smooth function K : X → R such that

12= −Kθ1∧ θ2

and that the function K does not depend on the choice of orthonormal frame field.

Proof. Since f df + gdg = 0 and τ = f dg − gdf, we know that dτ = 0. Then dω12 = dω120 . Then −K0θ10 ∧ θ20 = −Kθ1∧ θ02. On the other hand, θ10 ∧ θ20 = θ1∧ θ2, we see that K0 = K.

 The function K on X is called the Gaussian curvature of X.

Example 2.1. Let H2 = {(x, y) ∈ R2 : y > 0} be the Poincare upper half plane. Then H2 is a smooth manifold. We equip H2 with the Riemannian metric

ds2 = dx2+ dy2 y2 .

In other words, if v = (v1, v2)p, w = (w1, w2)p ∈ TpH2, with p = (x, y), then hv, wip = v1w1+ v2w2

y2 .

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Let us consider the vector fields e1 = y∂x e2= y∂y where p = (x, y) ∈ H2. Then {e1, e2} is a moving frame on H2. The dual coframe to this frame is given by θ1 = dx/y and θ2= dy/y.

Then dθ1 = dx ∧ dy/y2 and dθ2 = 0. We find ω12= dx/y = θ1. We see that dω12= dx ∧ dy

y2 = θ1∧ θ2.

This gives K = −1. Hence the Poincare upper half plane has constant Gaussian curvature

−1.

Let us assume that the Riemannian metric ds2on the surface X has the following expres- sion ds2 = Edu2+ Gdv2, where (u, v) is a local coordinate system on X. Then θ1=√

Edu and θ2 = √

Gdv form the dual coframe to the (local) moving frame {e1, e2} given by e1= 1

√ E

∂u and e2 = 1

√ G

∂v. The connection one form w.r.t. {e1, e2} is given by ω12=

√Ev

G du +

√Gu

√ E dv.

This formula implies that the Gaussian curvature of the Riemannian surface (X, ds2) is

(2.1) K = − 1

√ EG

( √Ev

√ G

!

v

+

√Gu

√ E

!

u

) . We leave to the reader to check the details.

Corollary 2.1. The Gaussian curvature of a Riemannian surface X with ds2 = λ2(du2+ dv2) is

K = −1

λ2∆ ln λ.

Proof. We have E = G = λ2. Using (2.1), we obtain the result.

 One can introduce the triangulation (or rectangular decomposition) of a compact Rie- mannian surface X: we can decompose into a finite collection of triangles (or rectangles).

Let v, e, f be the total number of vertices, edges, faces of the triangles.

Theorem 2.3. The number v − e + f is independent of the triangulation (rectangular decomposition) of X. It is called the Euler characteristic of X and denoted by χ(X).

We will see that this definition of Euler characteristic coincides with that defined by the dimension of de Rham cohomology of X.

Theorem 2.4. Two diffeomorphic surfaces have the same Euler characteristic.

Proof. Let f : X → Y be a diffeomorphism of compact oriented surfaces. If {∆i} is a triangulation of X, then {f (∆i)} is a triangulation of Y with the same v, e, f. 

The Euler characteristic of the disjoint union of X and Y is given by χ(Xa

Y ) = χ(X) + χ(Y ).

Example 2.2. The Euler characteristic of S2 is 2.

S2 can be identify with C ∪ {∞}. The Euler characteristic of C is 1 while the Euler characteristic of a point is also 1. Thus χ(S2) = 1 + 1 = 2.

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Example 2.3. The Euler characteristic of the closed unit disk D2 is 1.

The closed unit disk D2 is the disjoint union of open disk whose Euler characteristic is 1 and the unit circle whose Euler characteristic is 0. Hence χ(D2) = 1.

Example 2.4. The Euler characteristic of the two dimensional torus T2 is χ(T2) = 0.

Given two surfaces X and Y, we can construct a new surface X#Y called the connected sum of X and Y.

Theorem 2.5. Let X and Y be two compact oriented surfaces2. Then

(2.2) χ(X#Y ) = χ(X) + χ(Y ) − 2.

Proof. We know that X = (X \ D2)` D2. Then χ(X \ D2) = χ(X) − 1. Therefore χ(X#Y ) = (χ(X) − 1) + (χ(Y ) − 1) = χ(X) + χ(Y ) − 2.

 Formula 2.2 allows us to compute the Euler characteristic of the connected sum T2#T2 of 2 two dimensional tori: χ(T2#T2) = −2. Inductively, the Euler characteristic of the connected sum Σg = #gi=1T2 of g ≥ 1 two dimensional tori is

χ(#gi=1T2) = 2 − 2g.

We set Σ0= S2.

Now we state the classification theorem of compact oriented Riemann surface:

Theorem 2.6. If X is a compact oriented Riemannian surface, then X is diffeomorphic to Σg for some g. In this case, the number g is called the genus of X.

Since diffeomorphic surfaces are homeomorphic and the fundamental groups of homeo- morphic spaces are isomorphic, we obtain:

Corollary 2.2. If X is a compact Riemann surface of genus g ≥ 0, then π1(X) ∼= π1g).

If X has genus 0, X is diffeomorphic to S2. Since S2 is simply connected, π1(S2) = 0.

If X has genus one, X is diffeomorphic to the two dimensional torus T2. The two di- mensional torus T2 is homeomorphic to the product of two unit circle S1× S1 and hence π1(T2) ∼= π1(S1) ⊕ π1(S1) ∼= Z2. The fundamental group π1(T2) is a free abelian group of rank 2; we can choose {a, b} as a generator. Since π1(T2) is abelian, [a, b] = aba−1b−1 = 1.

π1(T2) has the following presentation: π1(T2) = ha, b|[a, b] = 1i. This implies π1(X) = ha, b|[a, b] = 1i.

When X has genus > 1, then X is diffeomorphic to Σg. We can compute the fundamental group of Σg using Van-Kampen theorem. The surface Σg can be represented by 4g-polygon Pg with sides identified with pairs. Under the identification, the edges a1, b1, · · · , ag, bg

becomes circles on Σg. Any two of these circles only intersect at the based point x0 (when a based point is chosen). Let y be the center of Pg and V be the interior of Pg and U = Σg\ {y}. Then the union of 2g circles is a deformation retract of U. Hence π1(U, x0) ∼= π1(∨2gi=1S1, ∗) = ∗2gi=1Z is a free group with 2g generators. Let α1, β1, · · · , αg, βg be the generator of π1(U, x0). On the other hand, S1 is a deformation retract of U ∩ V, and hence

2In fact, this assumption can be changed to any spaces admitting triangulations.

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π1(U ∩ V, x1) ∼= Z is an infinite cyclic group where x1 is a based point of U ∩ V. Since V is homeomorphic to a disk, it is simply connected. The group homomorphism

ψ : π1(U, x0) → π1g, x0)

sending [αi] → [ai] and [βj] → [bj] is surjective (Van Kampen). If c is a loop at x1, then ϕ1: π1(U ∩ V, x1) → π1(U, x1) is given by

ϕ([c]) =

g

Y

i=1

0i, βi0],

where αi0 = δαiδ and βj0 = δβjδ and δ is the class of the path connecting x0 and x1. The kernel of ψ is the smallest subgroup containing the image of ϕ(π1(U ∩V, x1)) in π1(U ∩V, x0) via the identification δ: π1(U ∩ V, x1) → π1(U ∩ V, x0) sending [γ] to [δγδ] (Van Kampen).

Notice that

g

Y

i=1

0i, β0i] = δ−1

g

Y

i=1

i, βi]

! δ.

Hence the kernel of ψ is the smallest subgroup generated by a single relationQg

i=1i, βi].

We obtain that:

Corollary 2.3. Let X be a compact Riemann surface of genus g > 0. The fundamental group π1(X) has a presentation of the form

π1(X) = ha1, b1, · · · , ag, bg|[a1, b1] · · · [ag, bg] = 1i, i.e. it is generated by a finite set {a1, b1, · · · , ag, bg} with the relationQg

i=1[ai, bi] = 1. Here the commutator [ai, bi] is defined to be aibia−1i b−1i .

Since the first homology group H1(X, Z) is the abelianization of π1(X), we obtain:

Corollary 2.4. The homology group H1(X, Z) is the free abelian group of rank 2g generated by {α1, · · · , αg, β1, · · · , βg} where αj and βk are the image of aj and bk under the quotient map π1(X) → H1(X, Z) = π1(X)/[π1(X), π1(X)].

Since diffeomorphic surfaces have the same Euler characteristic, if X ∼= Σg for some g then χ(X) = 2 − 2g.

Corollary 2.5. Compact oriented surfaces have the same Euler characteristic if and only if they are diffeomorphic.

Proof. Let X and Y be two compact oriented surfaces. Then X ∼= Σg and Y ∼= Σh for some g, h. If χ(X) = χ(Y ), then g = h. Hence X ∼= Σg ∼= Y. Therefore X and Y are diffeomorphic.

The converse was proved before. 

3. Gauss-Bonnet Theorem

Now, we are going to study the relation between geometry and topology of compact oriented Riemannian surface. Let us look at the example of S2. The Gaussian curvature of S2 is 1, the total curvature of S2 is the surface area 4π of the unit sphere S2 and hence

Z

S2

Kdµ = 2πχ(S2).

In general, we can prove the following theorem:

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Theorem 3.1. (Gauss-Bonnet Theorem) Let X be a compact oriented Riemannian surface.

Then Z

X

Kdµ = 2πχ(X).

Proof. The proof will not be given here. 

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