On a connected Riemannian manifold, geodesic completeness, metric completeness and compact closed bounded sets are all equivalent conditions, and under them any two points are joined by a length-minimizing geodesic. Named for Heinz Hopf and Willi Rinow, it is a foundational existence result in Riemannian geometry.
Facts
StatementOn a connected Riemannian manifold with no boundary, geodesic completeness, metric completeness and compactness of every closed bounded subset are all equivalent, and any two points can then be joined by a length-minimizing geodesic. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Hopf-Rinow Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, theorem topic sentence
The Hopf-Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds.
lead paragraph, attribution and publication sentence
It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931.
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