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Splitting TheoremThe 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 -
for all - )
then it is isometric to a product space -
where 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.
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