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) \ge 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) \ge 0.

History

For the surfaces, the theorem was proved by Stephan 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]

↑ S. Cohn-Vossen, “Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken”, Матем. сб., 1(43):2 (1936), 139–164
↑ Toponogov, V. A. Riemannian spaces containing straight lines. (Russian) Dokl. Akad. Nauk SSSR 127 1959 977–979.
↑ Eschenburg, J.-H. The splitting theorem for space-times with strong energy condition. J. Differential Geom. 27 (1988), no. 3, 477–491.
↑ Galloway, Gregory J.(1-MIAM) The Lorentzian splitting theorem without the completeness assumption. J. Differential Geom. 29 (1989), no. 2, 373–387.
↑ Newman, Richard P. A. C. A proof of the splitting conjecture of S.-T. Yau. J. Differential Geom. 31 (1990), no. 1, 163–184.

Jeff Cheeger; Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, Journal of Differential Geometry 6 (1971/72), 119–128. MR 0303460
V. A. Toponogov, Riemann spaces with curvature bounded below (Russian), Uspehi Mat. Nauk 14 (1959), no. 1 (85), 87–130. MR 0103510