Structural proof theory

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

$\mathrm{R}\mathrm{i}\mathrm{c}(M)\ge 0$

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

$\mathrm{R}\mathrm{i}\mathrm{c}(L)\ge 0.$

History

For the surfaces, the theorem was proved by Stephan Cohn-Vossen. Victor Andreevich Toponogov generalized it to manifolds with non-negative sectional curvature. 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.

References

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