Splitting theorem
From Free net encyclopedia
The splitting theorem is a classical theorem in Riemannian geometry. It states that if a complete Riemannian manifold with Ricci curvature
- Ricc ≥ 0
has a straight line (i.e. a geodesic γ such that
- <math>d(\gamma(u),\gamma(v))=|u-v|</math>
for all
- <math>v,u\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
- Ricc ≥ 0.
The theorem was proved by Cheeger and Gromoll and based on earlier result of Toponogov.
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References
- Jeff Cheeger; Detlef Gromoll The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differential Geometry 6 (1971/72), 119--128.
- V. A. Toponogov, Riemann spaces with curvature bounded below. (Russian) Uspehi Mat. Nauk 14 1959 no. 1 (85), 87--130.