Complete manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, there are straight paths extending infinitely in all directions.

Formally, a manifold ${\displaystyle M}$ is (geodesically) complete if for any maximal geodesic ${\displaystyle \ell :I\to M}$, it holds that ${\displaystyle I=(-\infty ,\infty )}$.^[1] A geodesic is maximal if its domain cannot be extended.

Equivalently, ${\displaystyle M}$ is (geodesically) complete if for all points ${\displaystyle p\in M}$, the exponential map at ${\displaystyle p}$ is defined on ${\displaystyle T_{p}M}$, the entire tangent space at ${\displaystyle p}$.^[1]