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

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

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

$$L(c):={\int }_0^1{\parallel \dot{c}(t)\parallel }_{c(t)}𝑑t$$

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
Canonical name	GeodesicCompleteness
Date of creation	2013-06-03 13:04:01
Last modified on	2013-06-03 13:04:01
Owner	jacou (1000048)
Last modified by	unlord (1)
Numerical id	14
Author	jacou (1)
Entry type	Definition
Classification	msc 53C22