Rauch comparison theorem

In Riemannian geometry the Rauch comparison theorem 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 large curvature, geodesics tend to converge, while for small (or negative) curvature, geodesics tend to spread. This theorem is formulated using Jacobi fields to measure the variation in geodesics.

References

*do Carmo, M.P. "Riemannian Geometry", Birkhäuser, 1992.