# 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, 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 *M*_{δ}, 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

When the sectional curvature is bounded from above, a corollary to the Rauch comparison theorem yields an analogous statement, but with the reverse inequality^{[citation needed]}.

## ReferencesEdit

- Chavel, Isaac (2006),
*Riemannian Geometry; A Modern Introduction*(second ed.), Cambridge University Press - Berger, Marcel (2004),
*A Panoramic View of Riemannian Geometry*, Springer-Verlag, ISBN 3-540-65317-1 - Cheeger, Jeff; Ebin, David G. (2008),
*Comparison theorems in Riemannian geometry*, AMS Chelsea Publishing, Providence, RI, ISBN 978-0-8218-4417-5, MR 2394158

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