A (pseudo-)Riemannian manifold $(X,g)$ is called *geodesically complete* (or just *complete*, for short) if each of its geodesics extends indefinitely, hence if the geodesic exponential map at every point $x \in X$ is defined on the full tangent space at that point, $\exp \colon T_x X \longrightarrow X$.

The Euclidean spaces $\mathbb{R}^n$ and the (round or squashed) n-spheres $S^n$ are geodesically complete. But any open ball of finite radius inside $\mathbb{R}^n$ is not.

See also

- Wikipedia,
*Riemannian manifold – Geodesic completeness*

Last revised on April 13, 2019 at 14:08:34. See the history of this page for a list of all contributions to it.