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 ${\displaystyle L}$ is a Riemannian manifold with

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.

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