Im trying to prove that if we have a compact 2-dimensional orientable Riemannian manifold with positive gauss curvature then any two non-self intersecting closed geodesics must intersect each other.First a thing that im a bit confused when it says non-self intersecting closed geodesics does it mean that they dont intersect each other and themselves?
Well i started with using the Gauss-Bonet Theorem and we can conclude that the manifold is going to have a positive euler charateristic but from here i dont know what to do, so any tips are appreciated.
Your assumption about the Gauss curvature tells you something about the Euler Characteristic of the surface, and in turn about the topology of the surface.
Now assume the claim is false. Cut the surface along the two geodesics, apply Gauss-Bonnet and make use of the fact that you know it's topology when you look at the remains.
(You will want to look up or calculate the Euler Characteristic of an annulus).