Zahorski theorem

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

Examples

All compact manifolds and all homogeneous manifolds are geodesically complete.

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

A simple example of a non-complete manifold is given by the punctured plane $M:={\mathbb{ℝ}}^2\setminus \{0\}$ (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.

Path-connectedness, completeness and geodesic completeness

It can be shown that a finite dimensional path-connected Riemannian manifold is a complete metric space (with respect to the Riemannian distance) if and only if it is geodesically complete. This is the Hopf–Rinow theorem. This theorem does not hold for infinite dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.

References

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