Complete equals geodesically complete

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Let (M,g) be a Riemannian manifold. Then, the following are equivalent:

  1. M 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
  2. M is geodesically complete at one point: i.e. there exists p \in M such that the exponential map \exp_p is defined on the whole of T_p(M)
  3. M 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.

Facts used


Complete implies geodesically complete