# Intersecting geodesics in a positive curvature manifold

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.

• $M$ is connected? – Emanuele Paolini Dec 15 '13 at 21:01
• Yes. I will edit it. Thank you. – thetruth Dec 15 '13 at 21:01
• Compactness says there is a positive lower bound on curvature. Toponogov. Part of it is that your manifold is a sphere. – Will Jagy Dec 15 '13 at 21:14
• Actually, Gauss-Bonnet and some topology works. The fact that you have a sphere is attributed to Synge. Where did you get this problem? It is a jump in difficulty from your previous one on geodesics. – Will Jagy Dec 15 '13 at 21:27
• en.wikipedia.org/wiki/Synge%27s_theorem – Will Jagy Dec 15 '13 at 21:37

• Thank you very much Will. Your answer is clear and quite elegant, though I'm not supposed to know Synge's Theorem - even if it says on wikipedia that is a classic result. Is there another (not high-tech) way to prove that $M$ must be a sphere ? Thank you so much again :) – thetruth Dec 15 '13 at 21:55