The splitting theorem is a classical theorem in Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

. It states that if a complete Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 M with Ricci curvature

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...

has a straight line, i.e., a geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

 γ such that


for all


then it is isometric to a product space


where is a Riemannian manifold with


The theorem was proved by Jeff Cheeger

Jeff Cheeger

Jeff Cheeger , is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and its applications to topology and analysis.-Biography:He graduated from Harvard University with a B.A....

 and Detlef Gromoll

Detlef Gromoll

Detlef Gromoll was a mathematician who studied distortions of shapes in three or more dimensions. In his soul theorem, published in 1972, he and Jeff Cheeger studied the properties of surfaces that have flat regions or curves like the outside of a sphere but not regions shaped liked saddles...

, based on an earlier result of Victor Andreevich Toponogov

Victor Andreevich Toponogov

Victor Andreevich Toponogov was a Russian mathematician, noted for his contributions to differential geometry and so-called Riemannian geometry "in the large"....

, which required non-negative sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

.