a question about undergraduate-level differential geometry(Gauss-Bonnet theorem)

Question

Let $S\subset R^3$ be a regular surface homeomorphic to a sphere. Let $\alpha\subset S $ be a simple closed geodesic in S,let A and B be a regions of S which have $\alpha$ as a common boundary. Let N:$S->S^2$ be the Gauss map of S. prove that N(A) and N(B) have the same area.

My thoughts: I think I need to use Gauss-Bonnet theorem,since the curve $\alpha$ can be mapped into a sphere,and the mapped the curve should be smooth(But I am not sure whether it is still geodesic),so use Gauss-Bonnet Theorem, we have $$\iint_{N(A)}Kds+\int_{0}^lk_{g}ds=2\pi X(s)$$,where X(s) is the Euler-poincare characteristic of a regular surface. if the geodesic curvature $k_{g}$=0,I can know that the area of N(A) should be $2\pi$,since the Gaussian curvature k=1 in the unit sphere,but I have no idea whether the geodesic curve in the original surface is still be geodesic in the sphere. Any help?

Answer 1

I suppose you need to assume that the Gauss curvature of $S$ is positive. In this case, $N$ is a diffeo from $S$ onto the sphere. Otherwise, you should define what area of $N(A)$ means.

In any case, if you use the Gauss-Bonnet theorem on $A$, then you obtain that $$ \int_A K=2\pi, $$ because the geodesic curvature of $\alpha$ vanishes, and the Euler characteristic of the disk $A$ is one.

On the other hand, $\int_A K$ is the spherical area of $A$ (that is, the area of $N(A)$ counting multiplicities). This can easily be seen using that $$ N_u\times N_v=K X_u\times X_v $$ for a local parameterization $X(u,v)$ of the surface. And integrating, $$ area(N(A))=\int_{N(A)} 1=\int_A |K|=2\pi. $$

Analogously, $area(N(B))=2\pi$.

Observe that, in general $N(\alpha)$ is not a geodesic.