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
History
For the surfaces, the theorem was proved by Stephan 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.
References
Splitting theorem Wikipedia(Text) CC BY-SA