Hopf–Rinow theorem

In mathematics, the Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow.

The theorem is stated as follows: Let M be a Riemannian manifold. Then the following statements are equivalent:

1. The closed and bounded subsets of M are compact.
2. M is a complete metric space
3. M is geodesically complete; that is, for every p in M, the exponential map Failed to parse (Missing texvc executable; please see math/README to configure.): \exp_p

is defined on the entire tangent space Failed to parse (Missing texvc executable; please see math/README to configure.): T_pM

.

Furthermore, any one of the above implies that given any two points p and q in M, there exists a length minimizing geodesic connecting these two points (geodesics are in general extrema, and may or may not be minima).

Generalization

The Hopf-Rinow theorem is generalized to length-metric spaces the following way:

If a length-metric space Failed to parse (Missing texvc executable; please see math/README to configure.): (M,d)

is complete and locally compact then any two points in Failed to parse (Missing texvc executable; please see math/README to configure.): M
can be connected by minimizing geodesic and any bounded closed sets in Failed to parse (Missing texvc executable; please see math/README to configure.): M
is compact.

References

• Hopf, H., Rinow, W., Über den Begriff des vollständigen differentialgeometrischen Fläche, Comment. Math. Helv. 3 (1931), 209-225.
• Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2See section 1.4.fr:Théorème de Hopf-Rinow

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