Geodesics and Completeness

A curve $\gamma :[a,b]\to M$ in a Riemannian manifold $(M,g)$ is called a geodesic if it is a critical point of the length functional,

Riemannian Exponential Map

Let $(M,g)$ be a semi-Riemannian manifold, and fix a point $p\in M$. The Riemannian exponential map ${\exp }_p:T_{p}M\to M$ sends a tangent vector $v$ to the point reached by following the geodesic starting at $p$ with initial velocity $v$, for a unit time.