Let $M$ be a compact Riemann manifold without boundary. Please is this true that each homotopy class of closed curves contains a geodesic?



geodesics on a compact manifold
Presumably this is a reference request (since the statement is sort of well-known).


See Theorem IV.5.1 in


*Chavel, Isaac*, Riemannian geometry. A modern introduction, Cambridge Studies in Advanced Mathematics 98. Cambridge: Cambridge University Press (ISBN 0-521-61954-8/pbk; 0-521-85368-0/hbk). xvi, 471 p. (2006). [ZBL1099.53001](https://zbmath.org/?q=an:1099.53001).


