# 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.

## Contents

### 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 .

A generalization of the Hopf–Rinow theorem (see ) 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.

How to Cite This Entry:
Hopf-Rinow theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf-Rinow_theorem&oldid=37710
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article