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

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 T_pX, we denote kvk_p = 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, ∂

∂x^ji_p.

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 (g_ij(p))ⁿ_i,j=1 is positive definite.

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

n

X

i,j=1

gijdxⁱdx^j.

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)k_γ(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

A set of vector fields {ei : 1 ≤ i ≤ n} on an n-dimensional Riemannian manifold X is called an orthonormal frame if he_i, e_j)i = δ_ij 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 θ_i(ej) = δij.

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

dµ = θ₁∧ · · · ∧ θ_n

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 {θ⁰_i} is the dual coframe to another positively oriented orthonormal frame {e⁰_i}, then θ⁰₁∧ · · · ∧ θ_n⁰ = θ1∧ · · · ∧ θ_n.

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

dθ1 = 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

dω₁₂= −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 ∈ T_pH², with p = (x, y), then hv, wi_p = v₁w₁+ v₂w₂

y² .

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=

√E_v

√

G du +

√G_u

√ E dv.

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

(2.1) K = − 1

√ EG

( √E_v

√ G

!

v

+

√G_u

√ 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.

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 Σ_g = #^g_i=1T² of g ≥ 1 two dimensional tori is

χ(#^g_i=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) ∼= π1(Σg).

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 Σ_g. 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 Σ_g. 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 = Σ_g\ {y}. Then the union of 2g circles is a deformation retract of U. Hence π₁(U, x₀) ∼= π₁(∨^2g_i=1S¹, ∗) = ∗^2g_i=1Z is a free group with 2g generators. Let α1, β₁, · · · , α_g, β_g 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.

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) → π1(Σg, 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

[α⁰_i, β_i⁰],

where α_i⁰ = δ⁻αiδ and β_j⁰ = δ⁻βjδ 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

[α⁰_i, β⁰_i] = δ⁻¹

g

Y

i=1

[αi, βi]

! δ.

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

i=1[α_i, β_i].

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, · · · , a_g, b_g|[a₁, b₁] · · · [a_g, b_g] = 1i, i.e. it is generated by a finite set {a₁, b₁, · · · , a_g, b_g} with the relationQg

i=1[a_i, b_i] = 1. Here the commutator [a_i, b_i] is defined to be a_ib_ia⁻¹_i b⁻¹_i .

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 α_j 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:

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.