Gauss Bonnet Theorem

GBT for simple closed curves

Theorem Let \(\gamma(s)\) be a unit-speed simple closed curve on a surface patch \(\sigma\) of length \(L\), and assume that \(\gamma\) is positively orientaed. Then,

\[\int_0^L \kappa_g ds = 2\pi - \int_{int(\gamma)}K dA_\sigma\]

where \(\kappa_g\) is the geodesic curvature, \(K\) is the Gaussian curvature.

Example

Theorem Suppose that \(\sigma\) has \(K\leq 0\) everywhere. Prove that there are no simple closed geodesics on \(\sigma\).

proof. Suppose exists some simple closed geodesic \(\gamma\), so that \(\kappa_g = 0\), thus

\[\int_{int(\gamma)} KdA_\sigma = 2\pi\]

If \(K\leq 0\), then the integral will be negative, hence not possible.

GB for Curvilinear polygons

A curvilinear polygon in \(\mathbb R^2\) is a continuous map \(\pi:\mathbb R\rightarrow\mathbb R^2\) s.t. for some real number \(T\) and some points \(0 = t_0 < t_1 < \cdots < t_n = T\), \(\pi(t) = \pi(t')\) IFF \(t'-t = kT\), \(\pi\) is piece-wise smooth for each open interval \((t_i, t_{i+1})\). The one-sided derivatives exists and are non-zero and not parallel. The points are called vertices, and the intervals are edges.

Theorem Let \(\gamma\) be positively oriented unit-speed curvilinear polygon with \(n\) edges on a surface \(\sigma\), and \(a_1,..,a_n\) ne the interior angles at its vertices. Then,

\[\int_0^L \kappa_g ds = \sum_{i=1}^n a_i - (n-2)\pi - \int_{int(\gamma)} K dA_\sigma\]

Corollary

If \(\gamma\) is a curvilinear polygon with \(n\) edges each of which is an arc of a geodesic, then the internal edges \(a\)'s of the polygon satisfy the equation,

\[\sum_{i=1}^n a_i = (n-2)\pi - \int_{int(\gamma)} K dA_\sigma\]

Example: Surface of Revolution

For \(u_1 < u_2\) be constant, let \(\gamma_1,\gamma_2\) be the two parallels at \(u_1, u_2\) on \(\sigma\), Let the region \(U\) be the surface between the two parallel circles or length \(L_1, L_2\).

For surface of revolution, the normal is

\[\mathbf N = (-g'\cos v, -g'\sin v, f')\]

Note that both \(\gamma_1,\gamma_2\) are circles of radius \(f(u_i)\), centered at \((0, 0, g(u_i))\), consider \(\gamma_1\),

\[\gamma_1'(v) = f(u_1)(-\sin v, \cos v, 0), \gamma_1'' = f(u_1) (-\cos v, -\sin v, 0)\]

Thus, we have the geodesic curvature

\[\kappa_g = \frac{\gamma''\cdot(\mathbf N\times \gamma')}{\|\gamma'\|^{3/2} } = \frac{f'(u_1)}{f(u_1)}\]

Then, we have

\[\int_0^{L1} \kappa_g ds = \int_0^{2\pi f(u_1)} \frac{f'(u_1)}{f(u_1)} ds = 2\pi f'(u_1), \int_0^{L2} \kappa_g ds = 2\pi f'(u_2)\]

Then, note that the Gaussian curvature for surface of revolution is \(K = -f''/f\), so that

\[\iint_U K dA_\sigma = \int_0^{2\pi}\int_{f(u_1)}^{f(u_2)} f''/f dudv = 2\pi (f'(u_1) - f'(u_2))\]

Thus, we have that

\[\int_0^{L1} \kappa_g ds - \int_0^{L2} \kappa_g ds = \iint_U K dA_\sigma \]

This follows GB for a curvilinear polygon with vertices \(\sigma(u_1, v_0), \sigma(u_2, v_0)\)

GB for Compact Surfaces

A triangulation of a surface \(\Sigma\) is a collection of curvilinear polygons, each of which is contained, together with its interior, in one of the \(\sigma_i(U_i)\) being a region in one of the surface patch in the atlas. Such that, - each point of \(\Sigma\) is in at least on curvilinear polygon. - each pair of curvilinear polygon are either disjoint, or intersection is a common edge or a common vertex. - each edge is an edge of exactly two polygons.

Euler Number

For a triangulation of a compact surface with finitely many polygons, the Euler number is

\[\chi = V-E+F\]

Theorem for any triangulation of \(\Sigma\),

\[\int_\Sigma K dA = 2\pi \chi\]

Corollary Euler number if independent of choice of triangulation.

Genus

A genus is obtained for gluing \(g\) tori together, where \(g\) is the number of holes. When \(g=0\), we have a sphere, \(g=1\) is a torus, and \(g=2\) we get a "8" shaped surfaces.

Theorem \(\chi(T_g) = 2 -2 g\) where \(T_g\) is genus \(g\).

Corollary \(\int_{T_g} KdA = 4\pi (1-g)\)

Theorem If a compact surface \(\Sigma\) is diffeomorphic to the torus, then \(\int_S KdA = 0\).

proof. Take \(g=1\) and apply the theorem. Also, since \(\Sigma\) is compact, \(K > 0\) for some points \(p\in\Sigma\).

Theorem If \(\Sigma\) is compact with \(K>0\) everywhere. Then, \(S\) is diffeomorphic to a sphere.

proof. \(\int_\Sigma KdA > 0\implies 1 - g > 0\), \(g\in\mathbb N\) hence the only choice of \(g\) is 0.

Note that the converse is not necessarily true, since we can have \(K=0\) for some points.