Splitting theorem


The splitting theorem is a classical theorem in Riemannian geometry.
It states that if a complete Riemannian manifold M with Ricci curvature
has a straight line, i.e., a geodesic γ such that
for all
then it is isometric to a product space
where is a Riemannian manifold with

History

For surfaces, the theorem was proved by Stefan Cohn-Vossen.
Victor Andreevich Toponogov generalized it to manifolds with non-negative sectional curvature.
Jeff Cheeger and Detlef Gromoll proved that non-negative Ricci curvature is sufficient.
Later the splitting theorem was extended to Lorentzian manifolds with nonnegative Ricci curvature in the time-like directions.