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.