## Non-compact and complete manifold

**Thm:** A normal geodesic is called shortest if and only if is shortest. A complete Riemannian manifold is non-compact if and only if for any point , there is a shortest geodesic such that

*Proof:* If there is a shortest geodesic , then is a sequence of points with no converging subsequence, because , . So is non-compact.

If is complete and non-compact, from the Hopf-Rinow theorem, there exists , such that as . Suppose is the shortest normal geodesic connecting and , . Then such that , going to subsequence if necessary. Consider , if it is not shortest for some , then there exists . Since for any fixed ,

when is large enough, we get contradiction. So is shortest.

### Like this:

Like Loading...

*Related*

or leave a trackback:

Trackback URL.