Geodesics on Compact Manifolds

Question

Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$?

I know that this is certainly not true if $M$ is not compact, but I couldn't find a counterexample for the compact case.

Can anybody help me out?

Thanks,

S.

Notice that your manifold has to have a metric for the question to make sense (you could get by with a projective class of connections on $M$...) — Mariano Suárez-Álvarez, Nov 10 '10 at 05:24
Indeed, I should have written "Riemannian manifold", I guess. — Sam, Nov 10 '10 at 06:01

score 10 · Accepted Answer · answered Nov 10 '10 at 05:08

10

Yes. This is part of the classical Hopf-Rinow theorem, q.v.

answered Nov 10 '10 at 05:08

Thanks Mariano. I don't know Hopf-Rinow yet - I'm still about 70 pages away from it! =) – Sam Nov 10 '10 at 06:00

score 1 · Answer 2 · answered Dec 07 '10 at 20:37

1

Just minimize the energy of the path in the homotopy class of paths connecting p and q.

answered Dec 07 '10 at 20:37

OrbiculaR

What's the missing step here? Presumably there's some fact about the space of paths from p to q in a compact manifold that doesn't hold in general that I just don't know about. – Aaron Mazel-Gee Dec 08 '10 at 05:25
2

If the manifold is not complete, then you have a problem. Take for instance R^2 with the origin removed, p=(-1,0) and q=(1,0). Then there is no geodesic between them. On the other hand, if the Riemannian metric makes the manifold complete, you've won! – OrbiculaR Dec 08 '10 at 15:39

2 Answers2