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

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


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