Splitting theorem

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

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

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

d(\gamma (u),\gamma (v))=|u-v|

for all

u,v\in \mathbb {R} ,

then it is isometric to a product space

\mathbb {R} \times L,

where $L$ is a Riemannian manifold with

{\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]

^ 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.