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

:<math>{\rm Ric} (M) \ge 0 </math>

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

:<math>d(\gamma(u),\gamma(v))=|u-v|</math>

for all

:<math>u, v\in\mathbb{R},</math>

then it is isometric to a product space

:<math>\mathbb{R}\times L,</math>

where <math>L</math> is a Riemannian manifold with

:<math>{\rm Ric} (L) \ge 0.</math>

==History==
For the surfaces, the theorem was proved by [[Stephan Cohn-Vossen]].<ref>S. Cohn-Vossen, “Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken”, Матем. сб., 1(43):2 (1936), 139–164 </ref>
[[Victor Andreevich Toponogov]] generalized it to manifolds with non-negative [[sectional curvature]]. <ref>Toponogov, V. A. Riemannian spaces containing straight lines. (Russian) Dokl. Akad. Nauk SSSR 127 1959 977–979.</ref>
[[Jeff Cheeger]] and [[Detlef Gromoll]] proved that non-negative Ricci curvature is sufficient.

Later the splitting theorem was extended to [[Lorentzian manifold]]s with nonnegative Ricci curvature in the time-like directions.<ref>Eschenburg, J.-H.
The splitting theorem for space-times with strong energy condition.
J. Differential Geom. 27 (1988), no. 3, 477–491.</ref>
<ref>Galloway, Gregory J.(1-MIAM)
The Lorentzian splitting theorem without the completeness assumption.
J. Differential Geom. 29 (1989), no. 2, 373–387. </ref>
<ref>Newman, Richard P. A. C.
A proof of the splitting conjecture of S.-T. Yau.
J. Differential Geom. 31 (1990), no. 1, 163–184. </ref>

==References==
{{Reflist}}
*Jeff Cheeger; Detlef Gromoll, ''The splitting theorem for manifolds of nonnegative Ricci curvature'', [[Journal of Differential Geometry]] 6 (1971/72), 119–128. {{MathSciNet|id=0303460}}
*V. A. Toponogov, ''Riemann spaces with curvature bounded below'' (Russian), Uspehi Mat. Nauk 14 (1959), no. 1 (85), 87–130. {{MathSciNet|id=0103510}}

[[Category:Theorems in Riemannian geometry]]

Structural proof theory - Revision history

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