A Riemannian metric on a manifold is said to be geodesically complete iff its geodesic flow is a complete flow, i.e. iff for every point and every tangent vector at the solution to the geodesic equation
with initial condition , is defined for all time. The Hopf-Rinow theorem asserts that a Riemannian metric is complete if and only if the corresponding metric on defined by
is a complete metric (i.e. Cauchy sequences converge). Here denote the length of the smooth curve , i.e.
For a proof of the Hopf-Rinow theorem see Milnor’s monograph Morse Theory Princeton Annals of Math Studies 51 page 62.