Homotopy-Class of Curves in Complete Connected Riemannian Manifold contains Minimal Geodesic of Class

Theorem

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

Let $H_{p q}$ homotopy class of segments from $p \in M$ to $q \in M$.

Let $\gamma_{p q}$ be a geodesic from $p \in M$ to $q \in M$ such that $\gamma_{p q} \in H_{p q}$.

Let $y_{p q} \in H_{p q}$ be an admissible curve.

Let $L_g$ be the Riemannian length.

Then in every $H_{p q}$ there is a $\gamma_{pq}$ which is minimizing among all $y_{p q} \in H_{p q}$:

$\forall p, q \in M : \exists \gamma_{pq} \in H_{p q} : \forall y_{p q} \in H_{p q} : \map {L_g} {\gamma_{p q} } \le \map {L_g} {y_{p q} }$

Proof

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Sources

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