So, I'm working with determining that the geodesics on an ellipsoid (at the moment an ellipsoid of revolution) are closed. I have read on the wiki page for geodesics on the ellipsoid that they are all closed, I understand the meridian and parallels are closed but what about the others. I understand that every 2-dimensional sphere has 3 closed geodesics, but I also read that any deformed sphere has infinetly many closed geodesics as well but I don't see how one can say this. I also read that any Reimannian manifold who's rational cohomology ring is generate by at least two elements has infinetly many closed geodesics. However I have not don't THAT much algebraic topology so I was wondering if there is a way to prove/explain this another way.
Also, the wiki page also says that a trixial ellipsoid only has 3 closed geodesics, is this because of this theorem of closed geodesics or because we know that every other geodesic is open.