Splitting theorem

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

${\rm Ric} (M) \ge 0$

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

d (γ(u),γ(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

${\rm Ric} (L) \ge 0.$

The theorem was proved by Jeff Cheeger and Detlef Gromoll, based on an earlier result of Victor Andreevich Toponogov, which required non-negative sectional curvature.

[edit] References

Jeff Cheeger; Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, Journal of Differential Geometry 6 (1971/72), 119–128. MR 0303460
V. A. Toponogov, Riemann spaces with curvature bounded below (Russian), Uspehi Mat. Nauk 14 (1959), no. 1 (85), 87–130. MR 0103510