Jul 252020

Author: KKK

In this short note, we will give a simple proof of the Gauss-Bonnet theorem for a geodesic ball on a surface. The only prerequisite is the first variation formula and some knowledge of Jacobi field (second variation formula), in particular how its second derivative (or the second derivative of the Jacobian) is related to the curvature of the surface. This is different from most standard textbook proofs at the undergraduate level. (Of course, this is just a local version of the Gauss-Bonnet theorem and topology has not yet come into play.)

Let $M$ be a surface equipped with a Riemannian metric. We will fix a point $p$ in $M$ and from now on $B_r$ always denotes the geodesic ball of radius $r$ centered at $p$, and $\partial B_r$ its boundary, which is called the geodesic sphere. In geodesic polar coordinates, let the area element of $M$ be locally given by

$$\displaystyle \begin{array}{rl} \displaystyle dA=f(\theta, r) dr d\theta, =f_\theta(r) dr d\theta, \end{array} $$

where $f(\theta, r)$ is the Jacobian (with respect to polar coordinates). For our purpose it is more convenient to regard $f_\theta(r)$ as a one-parameter family of functions in the variable $r$. It is well-known that $f_\theta$ satisfies the Jacobi equation (here $’=\frac{d}{dr}$ )

$$\displaystyle \begin{array}{rl} \displaystyle {f_\theta}”(r)=-K(\theta, r) f_\theta(r),\quad f_\theta(0)=0,\quad {f_\theta}'(0)=1 \ \ \ \ \ (1)\end{array} $$

where $K=K(\theta, r)$ is the Gaussian curvature (in polar coordinates). Indeed, if we fix a geodesic polar coordinates, and $\gamma_\theta(t)$ is the arc-length parametrized geodesic with initial “direction” $\theta$ starting from $p$, then we can define a parallel orthonormal frame $e_1(t), e_2(t)=\gamma_\theta'(t)$ along $\gamma_\theta(t)$. Then $Y(t)=f_\theta(t)e_1(t)$ is a Jacobi field and so

$$\displaystyle \begin{array}{rl} \displaystyle Y”(t)={f_\theta}”(t) e_1 (t) =- R(Y(t), \gamma_\theta'(t))\gamma_\theta'(t) =& \displaystyle -K(\theta, t) Y(t)\\ =& \displaystyle -K(\theta, t) f_\theta (t) e_1(t). \end{array} $$

From this (1) follows.

The first variation formula says (here $s$ is the arclength parameter)

$$\displaystyle \begin{array}{rl} \displaystyle \frac{d}{dt} \left(\mathrm{Length}(\partial B_t)\right) =\frac{d}{dt} \left(\int_0^{2\pi}f_\theta( t) d\theta\right) =\int_{ \partial B_t}k_g ds =\int_0^{2\pi} k_g(\theta, t) f_\theta( t)d\theta. \end{array} $$

Here $k_g$ is the geodesic curvature of the geodesic circle $\partial B_t$. (Indeed, the differential version $\frac{f_\theta’}{f_\theta}=k_g$ is already true for the geodesic circle.) This implies

$$\displaystyle \begin{array}{rl} \displaystyle \int_{\partial B_t}k_g ds =\int_{0}^{2\pi} f_\theta'(t) d\theta. \end{array}$$

So by the fundamental theorem of calculus and (1), we have

$$\displaystyle \begin{array}{rl} \displaystyle \int_{\partial B_t}k_g ds =& \displaystyle \int_{0}^{2\pi}\left({f_\theta}'(0)+\int_0^t {f_\theta}”(r)dr\right)d\theta\\ =& \displaystyle \int_{0}^{2\pi}\left(1-\int_0^t K (\theta, r)f_\theta(r)dr\right)d\theta\\ =& \displaystyle 2 \pi-\int_{B_t} K dA. \end{array} $$

This is exactly the Gauss-Bonnet theorem (for a geodesic ball), which is usually written as

$$\displaystyle \begin{array}{rl} \displaystyle \int_{B_r}K dA+\int_{\partial B_r}k_g ds=2\pi. \end{array} $$

Nov 112013

最近好像事情不少, 没有能力来写啥深入的长篇大论.

微分几何最适合的入门书, 首推 Andrew Pressley 的 “Elementary Differential Geometry“!  2010年出版第二版, 不过笔误之类的小错误不少. 2012 年重印版本, 修正了很多.

这书的风格类似 David S. Dummit and Richard M. Foote, Abstract Algebra, 很多别的教科书忽略的细节, 这书都解释的很清晰. 不胜枚举的例子, 可以用来说明这一点. 诸如, 曲线的可允许参数变换, 自交点, 闭曲线都给出了严格定义, 反函数定理有专门一节的详细讨论.

本书还有一个特点是, 习题既有提示, 也有完整的解答, 分开的. 做出不来, 看提示; 还做不出就参考解答. 似乎, 还没有多少书是这样的.

就内容来说, 本书也非常丰富, 已经超出一般的入门教材. 最后三章分别论述 Hyperbolic geometry, Minimal surfaces, The Gauss-Bonnet theorem.

遗憾的是, 本书的观点是传统的.

其次, 是 Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces.

这个是经典, 不用多说, 上面 Andrew Pressley 的那本也被 Manfredo P. do Carmo 巨大的影响着. 阅读 Andrew Pressley 可以看的很清楚!

再来是, J. A. Thorpe, Elementary topics in Differential Geometry

4.  Wilhelm Klingenberg, A Course in  Differential Geometry, GTM51

5.  Wolfgang Kühnel, Differential Geometry: Curves – Surfaces – Manifolds, Second Edition