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 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 Mδ 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
- 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
- Pambuccian V., Zamfirescu T. "Paolo Pizzetti: The forgotten originator of triangle comparison geometry". Hist Math 38:8 (2011)
|This Differential geometry related article is a stub. You can help Wikipedia by expanding it.|