Complete Riemannian space

A Riemannian space with its internal distance function that is complete as a metric space with metric .

Let be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a) is complete; b) for each point the exponential mapping is defined on all of (where is the tangent space to at ); and c) every closed set that is bounded with respect to the distance is compact (the Hopf–Rinow theorem). Consequences are: Any two points in a complete Riemannian space can be joined in by a geodesic of length ; any geodesic is indefinitely extendable.

There is a generalization [2] of this theorem to the case of a space with a non-symmetric distance function.

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	S.E. Cohn-Vossen, "Existenz kürzester Wege" Compos. Math. , 3 (1936) pp. 441–452

Comments

Let be a point of the Riemannian manifold . Then is called geodesically complete at if is defined on all of , and is geodesically complete if this is the case at all . For to be complete (or, equivalently, geodesically complete) it suffices that is geodesically complete at one point.

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)