Hopf-Rinow Theorem/Corollary 2
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Theorem
Let $\struct {M, g}$ be a complete connected Riemannian Manifold.
Let $\gamma$ be a minimizing geodesic segment.
Then any two points of $M$ can be joined by some $\gamma$:
- $\forall p, q \in M : \exists \gamma : \paren {p \in \gamma} \land \paren{q \in \gamma}$
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Completeness