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

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

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

Let $L_g$ be the Riemannian length.

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


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