# Splitting theorem

The splitting theorem is a classical theorem in Riemannian geometry. It states that if a complete Riemannian manifold M with Ricci curvature

${\displaystyle {\rm {Ric}}(M)\geq 0}$

has a straight line, i.e., a geodesic γ such that

${\displaystyle d(\gamma (u),\gamma (v))=|u-v|}$

for all

${\displaystyle u,v\in \mathbb {R} ,}$

then it is isometric to a product space

${\displaystyle \mathbb {R} \times L,}$

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

${\displaystyle {\rm {Ric}}(L)\geq 0.}$

## 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

1. ^ Cohn-Vossen, S. (1936). "Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken". Матем. сб. 1. 43 (2): 139–164.
2. ^ Toponogov, V. A. (1959). "Riemannian spaces containing straight lines". Dokl. Akad. Nauk SSSR (in Russian). 127: 977–979.
3. ^ 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.
4. ^ Galloway, Gregory J. (1989). "The Lorentzian splitting theorem without the completeness assumption". J. Differential Geom. 29 (2): 373–387. doi:10.4310/jdg/1214442881.
5. ^ 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.