Splitting theorem

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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)) = | uv |

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. MR0303460
  • V. A. Toponogov, Riemann spaces with curvature bounded below (Russian), Uspehi Mat. Nauk 14 (1959), no. 1 (85), 87–130. MR0103510