# geodesic completeness

A Riemannian metric on a manifold $M$ is said to be geodesically complete iff its geodesic flow is a complete flow, i.e. iff for every point $p\in M$ and every tangent vector $v\in T_{p}M$ at $p$ the solution to the geodesic equation

 $\nabla_{\dot{\gamma}}\dot{\gamma}=0$

with initial condition $\gamma(0)=p$, $\dot{\gamma}(0)=v$ is defined for all time. The Hopf-Rinow theorem asserts that a Riemannian metric is complete if and only if the corresponding metric on $M$ defined by

 $d(p,q)\colon=\inf\{L(c),\;c\colon[0,1]\to M,\;c(0)=p,\;c(1)=q\}$

is a complete metric (i.e. Cauchy sequences converge). Here $L(c)$ denote the length of the smooth curve $c$, i.e.

 $L(c)\colon=\int_{0}^{1}\|\dot{c}(t)\|_{c(t)}\,dt$

For a proof of the Hopf-Rinow theorem see Milnor’s monograph Morse Theory Princeton Annals of Math Studies 51 page 62.

Title geodesic completeness GeodesicCompleteness 2013-06-03 13:04:01 2013-06-03 13:04:01 jacou (1000048) unlord (1) 14 jacou (1) Definition msc 53C22