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Splitting theorem
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
- d(γ(u),γ(v)) = | u - v |
for all
)
then it is isometric to a product space
where L is a Riemannian manifold with
- Ricc ≥ 0.
The theorem was proved by Cheeger and Gromoll and based on earlier result of Toponogov .
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.
Last updated: 10-13-2005 01:26:27
10-26-2009 08:16:03
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The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details