Toponogov's Theorem

In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region of low curvature.

Let M be an m-dimensional Riemannian manifold with sectional curvature K satisfying

Let pqr be a geodesic triangle in M, such that the geodesic pq is minimal and if δ ≥ 0, the length of the side pr is less than . Let p′q′r′ be a geodesic triangle in the space form M_δ such that the length of sides p′q′ and p′r′is equal to that of pq and pr respectively and the angle at p′ is equal to that at p. Then

When the sectional curvature is bounded from above, a corollary to the Rauch comparison theorem yields an analogous statement, but with the reverse inequality.