Geodesics and Completeness

A curve in a Riemannian manifold is called a geodesic if it is a critical point of the length functional,

Riemannian Exponential Map

Let be a semi-Riemannian manifold, and fix a point . The Riemannian exponential map sends a tangent vector to the point reached by following the geodesic starting at with initial velocity , for a unit time.