quality:corruption
Hopf–Rinow theorem

The Riemannian manifold M {\displaystyle M} with its Levi-Civita connection is geodesically complete if the domain of every maximal geodesic is (− ∞, ∞) {\displaystyle (-\infty,\infty)}. The plane R 2 {\displaystyle \mathbb {R} ^{2}} is geodesically complete. On the other hand, the punctured plane R 2 ∖ { (0, 0) } {\displaystyle \mathbb {R} ^{2}\smallsetminus \{(0,0)\}} with the restriction of the Riemannian metric from R 2 {\displaystyle \mathbb {R} ^{2}} is not geodesically complete as the maximal geodesic with initial conditions p = (1, 1) {\displaystyle p=(1,1)}, v = (1, 1) {\displaystyle v=(1,1)} does not have domain R {\displaystyle \mathbb {R} }.

The Hopf–Rinow theorem characterizes geodesically complete manifolds.

Theorem: Let (M, g) {\displaystyle (M,g)} be a connected Riemannian manifold. The following are equivalent: