Complete equals geodesically complete
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.