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.