nLab geodesic completeness




A (pseudo-)Riemannian manifold (X,g)(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 xXx \in X is defined on the full tangent space at that point, exp:T xXX\exp \colon T_x X \longrightarrow X.


The Euclidean spaces n\mathbb{R}^n and the (round or squashed) n-spheres S nS^n are geodesically complete. But any open ball of finite radius inside n\mathbb{R}^n is not.


See also

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