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In the [[mathematics|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


<math> K\ge \delta\,.</math>
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Let ''pqr'' be a [[geodesic triangle]] in ''M'', such that the geodesic ''pq'' is minimal and if &delta; &ge; ''0'', the length of the side ''pr'' is less than <math>\pi / \sqrt \delta</math>.
Let ''p''&prime;''q''&prime;''r''&prime; be a geodesic triangle in the [[space form]] ''M''<sub>&delta;</sub> such that the length of sides ''p&prime;q&prime;'' and ''p&prime;r&prime;''is equal to that of ''pq'' and ''pr'' respectively and the angle at ''p&prime;'' is equal to that at ''p''.  
Then
 
:<math>d(q,r) \le d(q',r').\,</math>
 
When the sectional curvature is bounded from above, a corollary to the [[Rauch comparison theorem]] yields an analogous statement, but with the reverse inequality.
 
==See also==
* [[Rauch comparison theorem]]
 
==References==
* {{citation | last=Chavel|first=Isaac | title=Riemannian Geometry; A Modern Introduction | edition=second | publisher=Cambridge University Press | year=2006 }}
* {{citation | last=Berger|first=Marcel | authorlink=Marcel Berger|title= A Panoramic View of Riemannian Geometry | publisher=Springer-Verlag | year=2004 | id = ISBN 3-540-65317-1}}
* {{Citation | last1=Cheeger | first1=Jeff | last2=Ebin | first2=David G. | title=Comparison theorems in Riemannian geometry | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-4417-5 | id={{MathSciNet | id = 2394158}} | year=2008}}
 
==External links==
* [http://tzamfirescu.tricube.de/TZamfirescu-183.pdf Pambuccian V., Zamfirescu T. "Paolo Pizzetti: The forgotten originator of triangle comparison geometry". ''Hist Math'' 38:8 (2011)]
 
[[Category:Theorems in Riemannian geometry]]
[[Category:Geometric inequalities]]
 
 
{{differential-geometry-stub}}

Latest revision as of 05:17, 4 November 2014


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