# Splitting theorem

*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.