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 be a surface equipped with a Riemannian metric. We will fix a point in and from now on always denotes the geodesic ball of radius centered at , and its boundary, which is called the geodesic sphere. In geodesic polar coordinates, let the area element of be locally given by
where is the Jacobian (with respect to polar coordinates). For our purpose it is more convenient to regard as a one-parameter family of functions in the variable . It is well-known that satisfies the Jacobi equation (here )
where is the Gaussian curvature (in polar coordinates). Indeed, if we fix a geodesic polar coordinates, and is the arc-length parametrized geodesic with initial “direction” starting from , then we can define a parallel orthonormal frame along . Then is a Jacobi field and so
From this (1) follows.
The first variation formula says (here is the arclength parameter)
Here is the geodesic curvature of the geodesic circle . (Indeed, the differential version is already true for the geodesic circle.) This implies
This is exactly the Gauss-Bonnet theorem (for a geodesic ball), which is usually written as