Complete Riemannian space

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

Let $M$ be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a) $M$ is complete; b) for each point $p\in M$ the exponential mapping $\exp_p$ is defined on all of $M_p$ (where $M_p$ is the tangent space to $M$ at $p$); and c) every closed set $A\subset M$ that is bounded with respect to the distance $\rho$ is compact (the Hopf–Rinow theorem). Consequences are: Any two points $p,q$ in a complete Riemannian space $M$ can be joined in $M$ by a geodesic of length $\rho(p,q)$; 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 $p$ be a point of the Riemannian manifold $M$. Then $M$ is called geodesically complete at $p$ if $\exp_p$ is defined on all of $T_pM$, and $M$ is geodesically complete if this is the case at all $p$. For $M$ to be complete (or, equivalently, geodesically complete) it suffices that $M$ is geodesically complete at one point.

References

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