Why a Riemannian manifold minus one point is not complete?

Question

Could you give me a proof that a Riemannian manifold minus one point is ever complete?

Thanks!!

Answer 1

Think about a sequence of points in $n$ space that approaches the origin. Your manifold looks like $n$ space locally.

Answer 2

Sketch of reasinong: take any geodesic passing throuh that point, lets call it $p$. Choose a small geodesic segment on that geodeisc, containing $p$ in the interior, such that the segment is the unique minimizer between it's endpoints $u$ and $v$, say. Now remove $p$. Even if you may still find a geodesic joining $u$ and $v$ it can no longer be minimizing, since curves near the original geodesic segment will be shorter. So now you have two points which cannot be joined by a shortest geodesic, so $M$ is not (geodesically) complete. If you start out with metric completeness and use it's more straitforward, of course.