Geodesically complete manifold

A complete manifold (or geodesically complete manifold) M is a pseudo-Riemannian manifold for which, starting at any point $p$, you can follow a "straight" line indefinitely along any direction. More formally, the exponential map for Riemannian manifolds at point $p$ is defined on the whole $T_{p}M$.

The main result concerning complete manifolds is the Hopf-Rinow theorem.