geodesic completeness

A Riemannian metric on a manifold M is said to be geodesically completePlanetmathPlanetmathPlanetmath iff its geodesic flow is a complete flow, i.e. iff for every point pM and every tangent vector vTpM at p the solution to the geodesic equation


with initial conditionMathworldPlanetmath γ(0)=p, γ˙(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


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


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