Hopf-Rinow theorem

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)