# 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) |

**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