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 theoremasserts 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 TheoryPrinceton Annals of Math Studies 51 page 62.