Existence of Closed Geodesics

Theorem

Let $\struct {M, g}$ be a compact connected Riemannian manifold.

Every nontrivial free homotopy class in $M$ is represented by a closed geodesic that has minimum length among all admissible loops in the given free homotopy class.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous): $\S 6$: Geodesics and Distance. Closed Geodesics