nLab geodesic completeness

Idea

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, $\exp \colon T_x X \longrightarrow X$.

Examples

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.

Related concepts

• Clifford-Klein space form

  • spherical space form

• positive quaternion-Kähler manifold

References

See also

• Wikipedia, Riemannian manifold – Geodesic completeness