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.
\pi/\sqrt\delta
d(q,r)\led(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 .