The Gauss–Bonnet Theorem

Abstract

In this chapter, we prove our first major local-to-global theorem in Riemannian geometry: the Gauss–Bonnet theorem. The grandfather of all such theorems in Riemannian geometry, it asserts the equality of two very differently defined quantities on a compact Riemannian 2-manifold: the integral of the Gaussian curvature, which is determined by the local geometry, and $2\pi$ times the Euler characteristic, which is a global topological invariant.

Problems

• 9-1. Let (M, g) be an oriented Riemannian 2-manifold with nonpositive Gaussian curvature everywhere. Prove that there are no geodesic polygons with exactly 0, 1, or 2 ordinary vertices. Give examples of all three if the curvature hypothesis is not satisfied.

• 9-2. Let (M, g) be a Riemannian 2-manifold. If $\gamma$ is a geodesic polygon in M with n vertices, the angle excess of ${\varvec{\gamma }}$ is defined as

  $$\begin{aligned} E(\gamma ) = \bigg (\sum _{i=1}^n \theta _i\bigg ) - (n-2)\pi , \end{aligned}$$

  where $\theta _1,\dots ,\theta _n$ are the interior angles of $\gamma$. Show that if M has constant Gaussian curvature K, then every geodesic polygon has angle excess equal to K times the area of the region bounded by the polygon.

• 9-3. Given $h\in (-R, R)$, let $C_h$ be the circle in $\mathbb S^2(R)\subseteq \mathbb {R}^3$ where $z=h$ (where we label the standard coordinates in $\mathbb {R}^3$ as (x, y, z)), and let $\Omega$ be the subset of $\mathbb S^2(R)$ where $z>h$. Compute the signed curvature of $C_h$ and verify the Gauss–Bonnet formula in this case.

• 9-4. Let $T\subseteq \mathbb {R}^3$ be the torus of revolution obtained by revolving the circle ${(r-2)^2} + z^2 =1$ around the z-axis (see p. 19). Compute the Gaussian curvature of T and verify the Gauss–Bonnet theorem in this case.

• 9-5. This problem outlines a proof that every compact smooth 2-manifold has a smooth triangulation.

  • (a) Show that it suffices to prove that there exist finitely many convex geodesic polygons whose interiors cover M, and each of which lies in a uniformly normal convex geodesic ball. (A curved polygon is called convex if the union of the polygon and its interior is a geodesically convex subset of M.)

  • (b) Using Theorem 6.17, show that there exist finitely many points $v_1$, ..., $v_k$ and a positive number $\varepsilon$ such that the geodesic balls $B_{3\varepsilon }(v_i)$ are geodesically convex and uniformly normal, and the balls $B_{\varepsilon }(v_i)$ cover M.

  • (c) For each i, show that there is a convex geodesic polygon in $B_{3\varepsilon }(v_i)$ whose interior contains $B_{\varepsilon }(v_i)$. [Hint: Let the vertices be sufficiently nearby points on the circle of radius $2\varepsilon$ around $v_i$.]

  • (d) Prove the result.

    (Used on p. 276.)

• 9-6. Let $M\subseteq \mathbb {R}^3$ be a compact, embedded, 2-dimensional Riemannian submanifold. Show that M cannot have $K\le 0$ everywhere. [Hint: Look at a point where the distance from the origin takes a maximum.]

• 9-7. Suppose M is either the 2-sphere of radius R or the hyperbolic plane of radius R for some $R>0$. Show that similar triangles in M are congruent. More precisely, if $\gamma _1$ and $\gamma _2$ are geodesic triangles in M such that corresponding side lengths are proportional and corresponding interior angles are equal, then there exists an isometry of M taking $\gamma _1$ to $\gamma _2$.

• 9-8. Use the Gauss–Bonnet theorem to prove that every compact connected Lie group of dimension 2 is isomorphic to the direct product group $\mathbb S^1\times \mathbb S^1$. [Hint: See Problem 8-17.]

• 9-9.

  • (a) Show that there is an upper bound for the areas of geodesic triangles in the hyperbolic plane $\mathbb H^2(R)$, and compute the least upper bound.

  • (b) Two distinct maximal geodesics in the hyperbolic plane $\mathbb H^2$ are said to be asymptotically parallel if they have unit-speed parametrizations $\gamma _1,\gamma _2: \mathbb {R}\mathrel {\rightarrow }\mathbb H^2$ such that $d_{\breve{g}}(\gamma _1(t),\gamma _2(2))$ remains bounded as $t\mathrel {\rightarrow }+\infty$ or as $t\mathrel {\rightarrow }-\infty$. An ideal triangle in $\mathbb H^2$ is a region whose boundary consists of three distinct maximal geodesics, any two of which are asymptotically parallel to each other. Show that all ideal triangles have the same finite area, and compute it. Be careful to justify any limits.

• 9-10. The Gauss–Bonnet theorem for surfaces with boundary: Suppose (M, g) is a compact Riemannian 2-manifold with boundary, endowed with a smooth triangulation such that the intersection of each curved triangle with $\partial M$, if not empty, is either a single vertex or a single edge. Then

  $$\begin{aligned} \int _M K\,dA + \int _{\partial M} \kappa _N\, ds = 2\pi \chi (M), \end{aligned}$$

  where $\kappa _N$ is the signed geodesic curvature of $\partial M$ with respect to the inward-pointing normal N.

• 9-11. Suppose g is a Riemannian metric on the cylinder $\mathbb S^1 \times [0, 1]$ such that both boundary curves are totally geodesic. Prove that the Gaussian curvature of g either is identically zero or attains both positive and negative values. Give examples of both possibilities.

• 9-12. Prove the plane curve classification theorem (Theorem 1.5). [Hint: Show that every smooth unit-speed plane curve $\gamma (t) = (x(t), y(t))$ satisfies the second-order ODE $\gamma ''(t) = \kappa _N(t)N(t)$, where N is the unit normal vector field given by $N(t) = (-y'(t), x'(t))$.] (Used on p. 4.)

• 9-13. Use the four-dimensional Chern–Gauss–Bonnet formula (9.14) to prove that a compact 4-dimensional Einstein manifold must have positive Euler characteristic unless it is flat.