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 pqr be a geodesic triangle, i.e. a triangle whose sides are geodesics, 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 model space Mδ, i.e. the simply connected space of constant curvature 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
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