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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Theorem of the three geodesics ： ウィキペディア英語版 Theorem of the three geodesics In differential geometry it is a "famous theorem"〔.〕 that every Riemannian manifold with the topology of a sphere has three closed geodesics that form simple closed curves without self-intersections. This result has been called the theorem of the three geodesics.〔 The result can also be extended to quasigeodesics on a convex polyhedron. ==History and proof== This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an ellipsoid, and from the study of the geodesics on an ellipsoid, the shortest paths for ships to travel. In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators.〔.〕 In 1905, Henri Poincaré conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics,〔.〕 and in 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the conjecture, which was later found to be flawed〔.〕 One proof of this conjecture examines the homology of the space of smooth curves on the sphere, and uses the curve-shortening flow to find a simple closed geodesic that represents each of the three nontrivial homology classes of this space.〔.〕 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「Theorem of the three geodesics」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.