Suppose $M$ is a connected, compact orientable 2-dimensional Riemannian manifold, with positive Gaussian curvature. I'd like to show that two non-self-intersecting closed geodesics must intersect each other.
I tried to use Gauss-Bonnet Theorem to prove this, but I wasn't successful. Can somebody help me ? Thank you.
Fine. It's a sphere, by Synge. The (smooth) Jordan Curve Theorem says that a closed curve bounds two hemispheres, not necessarily of equal volume but of equal total curvature. If your two geodesics are not identical, there is an overlap of each "hemisphere" for one geodesic with a hemisphere for the other, making for positive total curvature on the intersections. So these are not disjoint and the boundary curves intersect each other.
Put another way, if the geodesics do not intersect the sphere is divided into two disjoint disks and an intermediate annulus, like the Earth above 45 degrees North Latitude, below 45 degrees South Latitude, and a ring in between that includes the Equator. Gauss Bonnet says that the two disks, together, have total curvature equal to the entirety of the sphere. So the supposed annulus has zero total curvature and is one-dimensional, that is the two geodesics are identical.