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