# nLab geodesic completeness

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## 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.