# 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.[1] Victor Andreevich Toponogov generalized it to manifolds with non-negative sectional curvature.[2] 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.[3][4][5]

## References

- Cohn-Vossen, S. (1936). "Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken".
*Матем. сб*. 1.**43**(2): 139–164. - Toponogov, V. A. (1959). "Riemannian spaces containing straight lines".
*Dokl. Akad. Nauk SSSR*(in Russian).**127**: 977–979. - Eschenburg, J.-H. (1988). "The splitting theorem for space-times with strong energy condition".
*J. Differential Geom*.**27**(3): 477–491. doi:10.4310/jdg/1214442005. - Galloway, Gregory J. (1989). "The Lorentzian splitting theorem without the completeness assumption".
*J. Differential Geom*.**29**(2): 373–387. doi:10.4310/jdg/1214442881. - Newman, Richard P. A. C. (1990). "A proof of the splitting conjecture of S.-T. Yau".
*J. Differential Geom*.**31**(1): 163–184. doi:10.4310/jdg/1214444093.

- Cheeger, Jeff; Gromoll, Detlef (1971). "The splitting theorem for manifolds of nonnegative Ricci curvature".
*Journal of Differential Geometry*.**6**(1): 119–128. doi:10.4310/jdg/1214430220. MR 0303460. - Toponogov, V. A. (1959). "Riemann spaces with curvature bounded below".
*Uspekhi Mat. Nauk*(in Russian).**14**(1): 87–130. MR 0103510.

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