翻訳と辞書 Words near each other ・ Riespach ・ Riess ・ Riess spiral ・ Riessersee ・ Rieste ・ Rieste railway station ・ Riester ・ Riesweiler ・ Riesz ・ Riesz function ・ Riemann's minimal surface ・ Riemannian ・ Riemannian circle ・ Riemannian connection on a surface ・ Riemannian geometry ・ Riemannian manifold ・ Riemannian Penrose inequality ・ Riemannian submanifold ・ Riemannian submersion ・ Riemannian theory ・ Riemann–Hilbert correspondence ・ Riemann–Hilbert problem ・ Riemann–Hurwitz formula ・ Riemann–Lebesgue lemma ・ Riemann–Liouville integral ・ Riemann–Roch theorem ・ Riemann–Roch theorem for smooth manifolds ・ Riemann–Roch theorem for surfaces ・ Riemann–Siegel formula ・ Riemann–Siegel theta function Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riemannian manifold ： ウィキペディア英語版 Riemannian manifold In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (''M'',''g'') is a real smooth manifold ''M'' equipped with an inner product $g\_p$ on the tangent space $T\_pM$ at each point $p$ that varies smoothly from point to point in the sense that if ''X'' and ''Y'' are vector fields on ''M'', then $p\; \backslash mapsto\; g\_p(X(p),Y(p))$ is a smooth function. The family $g\_p$ of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry. A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. ==Introduction== In 1828, Carl Friedrich Gauss proved his Theorema Egregium (''remarkable theorem'' in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. ''See'' differential geometry of surfaces. Bernhard Riemann extended Gauss's theory to higher-dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces. Albert Einstein used the theory of Riemannian manifolds to develop his general theory of relativity. In particular, his equations for gravitation are constraints on the curvature of 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「Riemannian manifold」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.