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_{p}X, we denote
kvk_{p} = g(p)(v, v).

Let (x^{1}, · · · , x^{n}; U ) be a local coordinate system on X. Then we have local vector fields

∂

∂x^{i}, 1 ≤ i ≤ n

, on U. We denote

gij(p) = h ∂

∂x^{i}, ∂

∂x^{j}i_{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
ds^{2} =

n

X

i,j=1

gijdx^{i}dx^{j}.

When X is a Riemannian surface with a local coordinate system (u, v), the Riemannian
metric is often written as ds^{2} = Edu^{2}+ 2F dudv + Gdv^{2}.

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|_{[t}_{k−1}_{,t}_{k}_{]} is a smooth curve on X for
each 1 ≤ k ≤ n.

Given a curve^{1} 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 {θ^{i} : 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 = θ_{1}∧ · · · ∧ 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 {θ^{0}_{i}} is the dual coframe to another positively oriented orthonormal
frame {e^{0}_{i}}, then θ^{0}_{1}∧ · · · ∧ θ_{n}^{0} = θ1∧ · · · ∧ θ_{n}.

2. Geometry of Riemannian surfaces

Theorem 2.1. Let X be a Riemannian surface. Suppose that {e_{1}, e_{2}} is an orthonormal
frame with dual coframe {θ1, θ2}. Then there exists a unique one-form ω_{12} = −ω21 such
that

dθ1 = ω12∧ θ_{2}, dθ2= ω21∧ θ_{1}.

Definition 2.1. We call ω_{12} the connection one-form associated with the orthonormal
frame {e1, e2}

Suppose that {e^{0}_{1}, e^{0}_{2}} is another orthonormal frame on X with the same orientation as
{e_{1}, e_{2}}, then

e^{0}_{1} = f e_{1}+ ge_{2}, e^{0}_{2} = −ge_{1}+ f e_{2}

for some smooth functions f, g on X. Let ω^{0}_{12} be the connection one form associated with
{e^{0}_{1}, e^{0}_{2}}. We leave to the reader to show that

ω12= ω^{0}_{12}− τ,
where τ = f dg − gdf.

Theorem 2.2. Let X be a Riemannian surface. Assume that {e_{1}, e_{2}} is an orthonormal
frame on X and ω_{12}is its associated connection one-form. Then there is a smooth function
K : X → R such that

dω_{12}= −Kθ_{1}∧ θ_{2}

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 = dω_{12}^{0} .
Then −K^{0}θ_{1}^{0} ∧ θ_{2}^{0} = −Kθ_{1}∧ θ^{0}_{2}. On the other hand, θ_{1}^{0} ∧ θ_{2}^{0} = θ_{1}∧ θ_{2}, we see that K^{0} = K.

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

Example 2.1. Let H^{2} = {(x, y) ∈ R^{2} : y > 0} be the Poincare upper half plane. Then H^{2}
is a smooth manifold. We equip H^{2} with the Riemannian metric

ds^{2} = dx^{2}+ dy^{2}
y^{2} .

In other words, if v = (v_{1}, v_{2})_{p}, w = (w_{1}, w_{2})_{p} ∈ T_{p}H^{2}, with p = (x, y), then
hv, wi_{p} = v_{1}w_{1}+ v_{2}w_{2}

y^{2} .

Let us consider the vector fields e1 = y_{∂x}^{∂} e2= y_{∂y}^{∂} where p = (x, y) ∈ H^{2}. Then {e1, e2} is
a moving frame on H^{2}. The dual coframe to this frame is given by θ1 = dx/y and θ2= dy/y.

Then dθ_{1} = dx ∧ dy/y^{2} and dθ_{2} = 0. We find ω_{12}= dx/y = θ_{1}. We see that
dω_{12}= dx ∧ dy

y^{2} = θ_{1}∧ θ_{2}.

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

−1.

Let us assume that the Riemannian metric ds^{2}on the surface X has the following expres-
sion ds^{2} = Edu^{2}+ Gdv^{2}, where (u, v) is a local coordinate system on X. Then θ_{1}=√

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
ω_{12}=

√E_{v}

√

G du +

√G_{u}

√ E dv.

This formula implies that the Gaussian curvature of the Riemannian surface (X, ds^{2}) 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 ds^{2} = λ^{2}(du^{2}+
dv^{2}) is

K = −1

λ^{2}∆ ln λ.

Proof. We have E = G = λ^{2}. 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 S^{2} is 2.

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

Example 2.3. The Euler characteristic of the closed unit disk D^{2} is 1.

The closed unit disk D^{2} is the disjoint union of open disk whose Euler characteristic is 1
and the unit circle whose Euler characteristic is 0. Hence χ(D^{2}) = 1.

Example 2.4. The Euler characteristic of the two dimensional torus T^{2} is χ(T^{2}) = 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 surfaces^{2}. Then

(2.2) χ(X#Y ) = χ(X) + χ(Y ) − 2.

Proof. We know that X = (X \ D^{2})` D^{2}. Then χ(X \ D^{2}) = 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 T^{2}#T^{2}
of 2 two dimensional tori: χ(T^{2}#T^{2}) = −2. Inductively, the Euler characteristic of the
connected sum Σ_{g} = #^{g}_{i=1}T^{2} of g ≥ 1 two dimensional tori is

χ(#^{g}_{i=1}T^{2}) = 2 − 2g.

We set Σ_{0}= S^{2}.

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 S^{2}. Since S^{2} is simply connected, π_{1}(S^{2}) = 0.

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

π1(T^{2}) has the following presentation: π1(T^{2}) = 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 x_{0} (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 π_{1}(U, x_{0}) ∼=
π_{1}(∨^{2g}_{i=1}S^{1}, ∗) = ∗^{2g}_{i=1}Z is a free group with 2g generators. Let α1, β_{1}, · · · , α_{g}, β_{g} be the
generator of π_{1}(U, x_{0}). On the other hand, S^{1} 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
ϕ_{1}: π_{1}(U ∩ V, x_{1}) → π_{1}(U, x_{1}) is given by

ϕ([c]) =

g

Y

i=1

[α^{0}_{i}, β_{i}^{0}],

where α_{i}^{0} = δ^{−}αiδ and β_{j}^{0} = 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

[α^{0}_{i}, β^{0}_{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

π_{1}(X) = ha_{1}, b_{1}, · · · , a_{g}, b_{g}|[a_{1}, b_{1}] · · · [a_{g}, b_{g}] = 1i,
i.e. it is generated by a finite set {a_{1}, b_{1}, · · · , 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_{i}b_{i}a^{−1}_{i} b^{−1}_{i} .

Since the first homology group H_{1}(X, Z) is the abelianization of π1(X), we obtain:

Corollary 2.4. The homology group H_{1}(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 π_{1}(X) → H_{1}(X, Z) = π1(X)/[π_{1}(X), π_{1}(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 S^{2}. The Gaussian curvature of
S^{2} is 1, the total curvature of S^{2} is the surface area 4π of the unit sphere S^{2} and hence

Z

S^{2}

Kdµ = 2πχ(S^{2}).

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.