[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 ith 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).