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Hopf-Rinow Theorem


Let M be a Riemannian manifold, and let the topological metric on M be defined by letting the distance between two points be the infimum of the lengths of curves joining the two points. The Hopf-Rinow theorem then states that the following are equivalent:

1. M is geodesically complete, i.e., all geodesics are defined for all time.

2. M is geodesically complete at some point p, i.e., all geodesics through p are defined for all time.

3. M satisfies the Heine-Borel property, i.e., every closed bounded set is compact.

4. M is metrically complete.


See also

Heine-Borel Theorem, Riemannian Manifold

This entry contributed by John Derwent

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References

Petersen, P. Riemannian Geometry. New York: Springer Verlag, p. 125, 1998.

Referenced on Wolfram|Alpha

Hopf-Rinow Theorem

Cite this as:

Derwent, John. "Hopf-Rinow Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Hopf-RinowTheorem.html

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