Let $M$ be a compact Riemann manifold without boundary. Please is this true that each homotopy class of closed curves contains a geodesic?
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8$\begingroup$ Each non-trivial free homotopy class. $\endgroup$– Alexandre EremenkoCommented Apr 5, 2023 at 15:01
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2$\begingroup$ And even the trivial class (constant maps are closed geodesics). $\endgroup$– Moishe KohanCommented Apr 5, 2023 at 15:17
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$\begingroup$ This is more appropriate for MathStackExchange. $\endgroup$– Moishe KohanCommented Apr 5, 2023 at 16:56
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1 Answer
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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.
answered Apr 5, 2023 at 15:42