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