# Rauch comparison theorem

In Riemannian geometry, the **Rauch comparison theorem**, named after Harry Rauch who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. This theorem is formulated using Jacobi fields to measure the variation in geodesics.

## Statement of the TheoremEdit

This section needs attention from an expert in Mathematics. (February 2015) |

Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along , and let be normal Jacobi fields along and such that and . Suppose that the sectional curvatures of and satisfy whenever is a 2-plane containing and is a 2-plane containing . Then for all .

## See alsoEdit

## ReferencesEdit

- do Carmo, M.P.
*Riemannian Geometry*, Birkhäuser, 1992. - Lee, J. M.,
*Riemannian Manifolds: An Introduction to Curvature*, Springer, 1997.

This Differential geometry related article is a stub. You can help Wikipedia by expanding it. |