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 comparison 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 $$ K\ge \delta\,.$$ Let pqr be a geodesic triangle, i.e. a triangle whose sides are geodesics, in M, such that the geodesic pq is minimal and if &delta; > 0, the length of the side pr is less than $$\pi / \sqrt \delta$$. Let p&prime;q&prime;r&prime; be a geodesic triangle in the model space M&delta;, i.e. the simply connected space of constant curvature &delta;, such that the lengths of sides p&prime;q&prime; and p&prime;r&prime; are equal to that of pq and pr respectively and the angle at p&prime; is equal to that at p. Then


 * $$d(q,r) \le d(q',r').\,$$

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