nLab
geodesic completeness

Contents

Contents

Idea

A (pseudo-)Riemannian manifold (X,g)(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 xXx \in X is defined on the full tangent space at that point, exp:T xXX\exp \colon T_x X \longrightarrow X.

Examples

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

References

See also

Last revised on April 13, 2019 at 10:08:34. See the history of this page for a list of all contributions to it.