Splitting theorem

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This article describes a result related to the Ricci curvature of a Riemannian manifold

Statement

Any complete Riemannian manifold whose Ricci curvature is everywhere nonnegative can be expressed as a direct product of a Euclidean space and a Riemannian manifold which does not contain any line. (A line here is a geodesic every finite segment of which realizes the distance between its endpoints).

Relation with other results

Bonnet-Myers theorem

Further information: Bonnet-Myers theorem

The Bonnet-Myers theorem states a very similar result for a manifold which has positive Ricci curvature everywhere, bounded below by a positive constant. Note that the Bonnet-Myers theorem, which asserts that any such manifold is compact, implies that it does not contain any line. Thus, the splitting theorem is closely related to the Bonnet-Myers theorem, the difference being that it allows zero curvature.