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Hopf-Rinow theorem

From Encyclopedia of Mathematics
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If is a connected Riemannian space with distance function and a Levi-Civita connection, then the following assertions are equivalent:

1) is complete;

2) for every point the exponential mapping is defined on the whole tangent space ;

3) every closed set that is bounded with respect to is compact.

Corollary:

Any two points can be joined in by a geodesic of length . This was established by H. Hopf and W. Rinow [1].

A generalization of the Hopf–Rinow theorem (see [4]) is: If and are two points in , then either there exists a curve joining them in a shortest way or there exists a geodesic emanating from with the following properties: 1) is homeomorphic to ; 2) if a sequence of points on does not have limit points on , then it does not have limit points in , that is, is closed in ; 3) contains the shortest connection between any two points on ; 4) for every point ; and 5) the length of is finite and does not exceed . Here the function is not necessarily symmetric, and every point can be joined in a shortest possible (not necessarily unique) way to any point in a certain neighbourhood .

Corollary:

If there are no bounded rays in , then every bounded set in is compact.

References

[1] H. Hopf, W. Rinow, "Ueber den Begriff der vollständigen differentialgeometrischen Flächen" Comm. Math. Helv. , 3 (1931) pp. 209–225
[2] G. de Rham, "Sur la réducibilité d'un espace de Riemann" Comm. Math. Helv. , 26 (1952) pp. 328–344
[3] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[4] S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian)


Comments

Let . The manifold is called geodesically complete at if is defined on all of . The manifold is geodesically complete if this holds for all . The Hopf–Rinow theorem also includes the statement that geodesic completeness is equivalent to geodesic completeness at one .

A geodesic joining and and of minimal length is called a minimizing geodesic.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Hopf–Rinow theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hopf%E2%80%93Rinow_theorem&oldid=22591