Geodesic manifold

In mathematics, 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 at point p, is defined on T_pM, the entire tangent space at p.

Equivalently, consider a maximal geodesic ${\displaystyle \ell \colon I\to M}$. Here ${\displaystyle I}$ is an open interval of ${\displaystyle \mathbb {R} }$, and, because geodesics are parameterized with "constant speed", it is uniquely defined up to transversality. Because ${\displaystyle I}$ is maximal, ${\displaystyle \ell }$ maps the ends of ${\displaystyle I}$ to points of ∂M, and the length of ${\displaystyle I}$ measures the distance between those points. A manifold is geodesically complete if for any such geodesic ${\displaystyle \ell }$, we have that ${\displaystyle I=(-\infty ,\infty )}$.