# Complete equals geodesically complete

## Contents

## Statement

Let be a Riemannian manifold. Then, the following are equivalent:

- is geodesically complete: in other words, geodesics can be extended indefinitely in both directions, or equivalently, the exponential map at a point is defined on the whole tangent space at the point
- is geodesically complete at
*one point*: i.e. there exists such that the exponential map is defined on the whole of - is complete as a metric space, where the distance between two points is defined as the infimum of lengths of all curves between the two points.