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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 space form ''M''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 . ==References== * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Toponogov's theorem」の詳細全文を読む スポンサード リンク
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