Let $(M;g)$ be a closed manifold . fix a point $x$ in $M$ and denote by $G=\pi_1(M;x)$ now let $\alpha$ in G i have 2 questions :
1- can $\alpha$ be represented by a closed geodesic ?
2- can $\alpha $ be represented by a closed minimizing geodesic ?
Asked
Active
Viewed 255 times
2
-
i know 1 is true for free homotopy classes and i guess 2 is true also – Sep 26 '11 at 09:52
-
2 can't be true if the loop in $\pi_1$ can only be represented by a self-intersecting curve, which happens frequently. – Cheerful Parsnip Sep 26 '11 at 13:23