Hopf-Rinow theorem

If $M$ is a connected Riemannian space with Riemannian metric $\rho$ and a Levi-Civita connection, then the following assertions are equivalent:

1) $M$ is a complete Riemannian space;

2) for every point $p\in M$ the exponential mapping $\exp_p$ is defined on the whole tangent space $M_p$;

3) every closed set $A\subset M$ that is bounded with respect to $\rho$ is compact.

Corollary:

Any two points $p,q\in M$ can be joined in $M$ by a geodesic of length $\rho(p,q)$. This was established by H. Hopf and W. Rinow [1].

A generalization of the Hopf–Rinow theorem (see [4]) is: If $p$ and $q$ are two points in $M$, then either there exists a curve joining them in a shortest way or there exists a geodesic $L$ emanating from $p$ with the following properties: 1) $L$ is homeomorphic to $0\leq t<1$; 2) if a sequence of points on $L$ does not have limit points on $L$, then it does not have limit points in $M$, that is, $L$ is closed in $M$; 3) $L$ contains the shortest connection between any two points on $L$; 4) $\rho(p,x)+\rho(x,q)=\rho(p,q)$ for every point $x\in L$; and 5) the length of $L$ is finite and does not exceed $\rho(p,q)$. Here the function $\rho(p,q)$ 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 $U_p$.

Corollary:

If there are no bounded rays in $M$, then every bounded set in $M$ 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 $p\in M$. The manifold $M$ is called geodesically complete at $p$ if $\exp_p$ is defined on all of $T_pM$. The manifold $M$ is geodesically complete if this holds for all $p$. The Hopf–Rinow theorem also includes the statement that geodesic completeness is equivalent to geodesic completeness at one $p\in M$.

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

References

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