[Tu Differential Geometry] 17. The Gauss-Bonnet Theorem

This article is one of Manifold, Differential Geometry, Fibre Bundle.

17.1 Geodesic Curvature

In the same way the curvature of a curve in Euclidean space is defined using the uaual derivative, we can define the geodesic curvature of a curve in a Riemannian manifold using the covariant derivative along the curve.

Consider a unit-speed curve $\gamma(s):[a,b]\rightarrow M$ is a Riemannian manifold $M$. The Riemannian connection of $M$ induces a covariant derivative $D/ds$ along the curve. Let $T(s)=\gamma'(s)$. The curve $\gamma(s)$ is a geodesic if and only if $DT/ds$ vanishes identically. Thus, the magnitude $\lVert DT/ds\rVert$ gives a measure of the extent to which $\gamma(s)$ fails to be a geodesic. It is called the geodesic curvature $\tilde{\kappa}_g$ of the curve $\gamma(s)$: $$\tilde{\kappa}_g=\lVert \frac{DT}{ds}\rVert.$$ So defined, the geodesic curvature is always nonnegative.

It follows directly from the definition that a unit-speed curve is a geodesic if and only if its geodesic curvature is zero.

17.2 The Angle Function Along a Curve

At a point $p$ of an oriented Riemannian manifold $M$, the angle $\zeta$ between two vectors $u$ and $v$ in the tangent space $T_pM$ is given by the formula $$\cos \zeta =\frac{\langle u,v\rangle}{\lVert u\rVert \lVert v\rVert}.$$ In general, the angle $\zeta$ is defined only up to an integer multple of $2\pi$.

Suppose now that $(U,e_1,e_2)$ is a framed open set on which there is a positively oriented orthonormal frame $e_1,e_2$. If $c:[a,b]\rightarrow U$ is a $C^\infty$ curve, let $zeta_-(t)$ be the angle from $e_{1,c(t)}$ to $c'(t)$ in $T_{c(t)}M$. Because $\zeta_-(t)$ is defined only up to an integer multiple of $2\pi$, $\zeta_-$ is a function from $[a,b]$ to $\mathbb{R}/2\pi \mathbb{Z}$. It is a $C^\infty$ function, since it is locally a branch of $\cos^{-1}\langle c'(t),e_{1,c(t)}\rangle /\lVert c'(t)\rVert$. Let $\zeta_0$ be a real number such that $$\cos \zeta_0=\frac{\langle c'(a),e_{1,c(a)}\rangle}{\lVert c'(a)\rVert}.$$ Since $\mathbb{R}$ is a covering space of $\mathbb{R}/2\pi\mathbb{Z}$ and the interval $[a,b]$ is simply connected, we know from the theory of covering spaces that there is a unique $C^\infty$ map $\zeta:[a,b]\rightarrow \mathbb{R}$ with a specified initial value $\zeta(a)=\zeta_0$ that covers $\zeta_-$: 

We call $\zeta:[a,b]\rightarrow \mathbb{R}$ the angle function with initial value $\zeta_0$ along the curve $c$ relative to the frame $e_1,e_2$. Since an angle function along a curve is uniquely determined by its initial value, any two angle functions along the curve $c$ relative to $e_1,e_2$ differ by a constant integer multiple of $2\pi$.

17.3 Signed Geodesic Curvature on an Oriented Surface

Mimicking the definition of the signed curvature for a plane curve, we can give the geodesic curvature a sign on an oriented Riemannian $2$-manifold. Given a unit-speed curve $\gamma(s):[a,b]\rightarrow M$ on an oriented Riemannian $2$-manifold $M$, choose a unit vector field $\mathbf{n}$ along the curve so that $T$, $\mathbf{n}$ is positively oriented and orthonormal. Since $\langle T,T\rangle\equiv 1$ on the curve, $$\frac{d}{ds}\langle T,T\rangle=0.$$ By Theorem 13.2, $$2\left\langle \frac{DT}{ds},T\right\rangle=0.$$ Since the tangent space of $M$ at $\gamma(s)$ is $2$-dimensional and $DT/ds$ is perpendicular to $T$, $DT/ds$ must be a scalar multiple of $\mathbf{n}$.

Definition 17.1. The signed geodesic curvature at a point $\gamma(s)$ of a unit-speed curve in an oriented Riemannian $2$-manifold is the number $\kappa_g(s)$ such that $$\frac{DT}{ds}=\kappa_g \mathbf{n}.$$ The signed geodesic curvature $\kappa_g$ can also be computed as $$\kappa_g=\left\langle \frac{DT}{ds},\mathbf{n}\right\rangle.$$

Let $U$ be an open subset of the oriented Riemannian $2$-manifold $M$ such that there is an orthonormal frame $e_1,e_2$ on $U$. We assume that the frame $e_1,e_2$ is positively oriented. Suppose $\gamma:[a,b]\rightarrow U$ is a unit-speed curve. Let $\zeta:[a,b]\rightarrow U$ be an angle function along the curve $\gamma$ relative to $e_1,e_2$. Thus, $\zeta(s)$ is the angle that the velocity vector $T(s)$ makes relative to $e_1$ at $\gamma(s)$ (Figure 17.1). In terms of the angle $\zeta$, $$\begin{align}T &=& (\cos \zeta)e_1 &+& (\sin \zeta)e_2,\\ \mathbf{n} &=& -(\sin \zeta)e_1 &+& (\cos \zeta)e_2,\end{align}$$ where $e_1,e_2$ are evaluated at $c(t)$.

Proposition 17.2. Let $\omega^1_2$ be the connection form of an affine connection on a Riemannian $2$-manifold relative to the positively oriented orthonormal frame $e_1,e_2$ on $U$. Then the signed geodesic curvature of the unit-speed curve $\gamma$ is given by $$\kappa_g=\frac{d\zeta}{ds}-\omega^1_2(T).$$

Proof. Differentiating (1) with respect to $s$ gives $$\frac{DT}{ds}=\left( \frac{d}{ds}\cos \zeta \right) e_1 + (\cos \zeta)\frac{De_1}{ds}+\left( \frac{d}{ds}\sin \zeta\right) e_2+(\sin \zeta)\frac{De_2}{ds}.$$ In this sum, $e_i$ really means $e_{i,c(t)}$ and $De_{i,c(t)}/ds=\nabla_Te_i$ by Theorem 13.1 (iii), so that by Proposition 11.4, $$\frac{De_1}{ds}=\nabla_Te_1=\omega^2_1(T)e_2=-\omega^1_2(T)e_2,$$ $$\frac{De_2}{ds}=\nabla_Te_2=\omega^1_2(T)e_1.$$ Hence, $$\frac{DT}{ds}=-(\sin \zeta)\frac{d\zeta}{ds}e_1-(\cos \zeta)\omega^1_2(T)e_2+(\cos \zeta)\frac{d\zeta}{ds}e_2+(\sin \zeta)\omega^1_2(T)e_1=\left( \frac{d\zeta}{ds}-\omega^1_2(T)\right) \mathbf{n}.$$ So $$\kappa_g =\frac{d\zeta}{ds}-\omega^1_2(T).$$

Since $\kappa_g$ is a $C^\infty$ function on the interval $[a,b]$, it can be integrated. The integral $\int_a^b\kappa_gds$ is called the total geodesic curvature of the unit-speed curve $\gamma:[a,b]\rightarrow M$.

Corollary 17.3. Let $M$ be an oriented Riemannian $2$-manifold. Assume that the image of the unit-speed curve $\gamma:[a,b]\rightarrow M$ is a $1$-dimensional submanifold $C$ with boundary. If $C$ lies in an open set $U$ with positively oriented orthonormal frame $e_1,e_2$ and connection form $\omega_2^1$, then its total geodesic curvature is $$\int_a^b \kappa_gds=\zeta(b)-\zeta(a)-\int_C\omega^1_2.$$

Proof. Note that $\gamma^{-1}:C\rightarrow [a,b]$ is acoordinate map on $C$, so that $\int_C\omega^1_2=\int_a^b\gamma^*\omega^1_2.$$Let$s$be the coordinate on$[a,b]$. Then$$\begin{align}\gamma^*\omega^1_2=f(s)ds\end{align}$$for some$C^\infty$function$f(s)$. To find$f(s)$, apply both sides of (3) to$d/ds$:$$f(s)=(\gamma^*\omega^1_2)\left( \frac{d}{ds}\right)=\omega^1_2\left( \gamma_*\frac{d}{ds}\right)=\omega^1_2(\gamma'(s))=\omega^1_2(T).$$Hence,$\gamma^*\omega^1_2=\omega^1_2(T)ds$. By Proposition 17.2,$$\int_a^b \kappa_gds=\int_a^b \frac{d\zeta}{ds}ds-\int_a^b\omega_2^1(T)ds=\zeta(b)-\zeta(a)-\int_a^b\omega_2^1(T)ds=\zeta(b)-\zeta(a)-\int_a^b\gamma^*\omega^1_2=\zeta(b)-\zeta(a)-\int_C\omega_2^1.$$

17.4 Gauss-Bonnet Formula for a Polygon

jump angle and interior angle...

Theorem 17.4 (Hopf Umlaufsatz). Let $(U,e_1,e_2)$ be a framed open set on an oriented Riemannian $2$-manifold, and $\gamma:[a,b]\rightarrow U$ a positively oriented piecewise smooth simple closed curve. Then the total change in the angle of $T(s)=\gamma'(s)$ around $\gamma$ is $$\sum^m_{i=1}\Delta \zeta_i+\sum^m_{i=1}\epsilon_i=2\pi.$$

Theorem 17.5 (Gauss-Bonnet formula for a polygon). Under the hypotheses above, $$\int_a^b \kappa_gds=2\pi -\sum^m_{i=1}\epsilon_i-\int_RK\mbox{vol}.$$

Proof. The integral $\int_a^b \kappa_gds$ is the sum of the total geodesic curvature on each edge of the simple closed curve. As before, we denote by $\Delta\zeta_i$ the change in the angle $\zeta$ along the $i$th edge of $\gamma$. By Corollary 17.3, $$\begin{align} \int_a^b \kappa_gds=\sum_{i=1}^m\int_{s_{i-1}}^{s_i}\kappa_g ds =\sum_i \Delta \zeta_i-\int_C\omega^1_2.\end{align}$$ By the Hopf Umlausatz, $$\begin{align}\sum \Delta\zeta_i=2\pi-\sum \epsilon_i.\end{align}$$

Recall from Section 12 that on a framed open set $(U,e_1,e_2)$ of a Riemannian $2$-manifold with orthonormal frame $e_1,e_2$ and dual frame $\theta^1,\theta^2$, if $[\Omega^i_j]$ is the curvature matrix relative to the frame $e_1,e_2$, then $$\Omega^1_2=d\omega^1_2=K\theta^1\wedge\theta^2.$$ Therefore, $$\begin{align}\int_C\omega^1_2=\int_{\partial R}\omega^1_2=\int_Rd\omega^1_2=\int_RK\theta^1\wedge\theta^2=\int_RK\mbox{vol}.\end{align}$$ Combining (4), (5), and (6) gives the Gauss-Bonnet formula for a polygon.

17.5 Triangles on a Riemannian $2$-Manifold

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17.6 Gauss-Bonnet Theorem for a Surface

Theorem 17.9 (Gauss-Bonnet theorem). For a compact oriented Riemannian $2$-manifold $M$, $$\int_M K\mbox{vol}=2\pi\chi(M).$$

Proof. Triangulization, use $\chi=V-E+F$.

17.7 Gauss-Bonnet Theorem for a Hypersurface in $\mathbb{R}^{2n+1}$

$$\int_MK\mbox{vol}_M=\frac{\mbox{vol}(S^{2n})}{2}\chi(M).$$