翻訳と辞書 Words near each other ・ Isotrias huemeri ・ Isotrias hybridana ・ Isotrias joannisana ・ Isotrias martelliana ・ Isotrias penedana ・ Isotrias rectifasciana ・ Isotrias stramentana ・ Isotricha intestinalis ・ Isotrilophus ・ Isotron ・ Isotropic bands ・ Isotropic coordinates ・ Isotropic etching ・ Isotropic formulations ・ Isotropic line ・ Isotropic manifold ・ Isotropic quadratic form ・ Isotropic radiation ・ Isotropic radiator ・ Isotropic solid ・ Isotropis ・ Isotropy ・ Isotta Brothers ・ Isotta degli Atti ・ Isotta Fraschini ・ Isotta Fraschini Asso XI ・ Isotta Fraschini D65 ・ Isotta Fraschini D80 ・ Isotta Fraschini Delta ・ Isotta Fraschini T8 and T12 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 Isotropic manifold ： ウィキペディア英語版 Isotropic manifold In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold $(M,g)$ is isotropic if for any point $p\backslash in\; M$ and unit vectors $v,w\backslash in\; T\_pM$, there is an isometry $\backslash varphi$ of $M$ with $\backslash varphi(p)=p$ and $\backslash varphi\_\backslash ast(v)=w$. Every isotropic manifold is homogeneous, i.e. for any $p,q\backslash in\; M$ there is an isometry $\backslash varphi$ of $M$ with $\backslash varphi(p)=q.$ This can be seen by considering a geodesic $\backslash gamma:()\backslash to\; M$ from $p$ to $q$ and taking the isometry which fixes $\backslash gamma(1)$ and maps $\backslash gamma\text{'}(1)$ to $-\backslash gamma\text{'}(1).$ ==Examples== The simply-connected space forms (the n-sphere, hyperbolic space, and $\backslash mathbb^n$) are isotropic. It is not true in general that any constant curvature manifold is isotropic; for example, the flat torus $T=\backslash mathbb^2/\backslash mathbb^2$ is not isotropic. This can be seen by noting that any isometry of $T$ which fixes a point $p\backslash in\; T$ must lift to an isometry of $\backslash mathbb^2$ which fixes a point and preserves $\backslash mathbb^2$; thus the group of isometries of $T$ which fix $p$ is discrete. Moreover, it can be seen that no oriented surface with constant curvature and negative Euler characteristic is isotropic. Moreover, there are isotropic manifolds which do not have constant curvature, such as the complex projective space $\backslash mathbb^n$ ($n>1$) equipped with the Fubini-Study metric. Further examples of isotropic manifolds are given by the rank one symmetric spaces, including the projective spaces $\backslash mathbb^n$, $\backslash mathbb^n$, $\backslash mathbb^n$, and $\backslash mathbb^2$, as well as their noncompact hyperbolic analogues. A manifold can be homogeneous but not isotropic, such as $\backslash mathbb\backslash times\; S^2$ with the product metric. 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「Isotropic 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.