Geodesically complete

In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which every maximal (inextendible) geodesic is defined on ${\displaystyle \mathbb {R} }$.

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete.

Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, the spheres ${\displaystyle \mathbb {S} ^{n}}$ and the tori ${\displaystyle \mathbb {T} ^{n}}$ (with their natural Riemannian metrics) are all complete manifolds.

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