翻訳と辞書 Words near each other ・ Poimenesperus ・ Poimeniko ・ Poinachi ・ Poinar ・ Poinard ・ Poincaré ・ Poincaré (crater) ・ Poincaré complex ・ Poincaré conjecture ・ Poincaré disk model ・ Poincaré duality ・ Poincaré group ・ Poincaré half-plane model ・ Poincaré inequality ・ Poincaré map ・ Poincaré metric ・ Poincaré model ・ Poincaré plot ・ Poincaré recurrence theorem ・ Poincaré residue ・ Poincaré Seminars ・ Poincaré separation theorem ・ Poincaré series ・ Poincaré series (modular form) ・ Poincaré space ・ Poincaré sphere ・ Poincaré–Bendixson theorem ・ Poincaré–Birkhoff theorem ・ Poincaré–Birkhoff–Witt theorem ・ Poincaré–Hopf theorem 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 Poincaré metric ： ウィキペディア英語版 Poincaré metric In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below. ==Overview of metrics on Riemann surfaces== A metric on the complex plane may be generally expressed in the form :$ds^2=\backslash lambda^2(z,\backslash overline)\backslash ,\; dz\backslash ,d\backslash overline$ where λ is a real, positive function of $z$ and $\backslash overline$. The length of a curve γ in the complex plane is thus given by :$l(\backslash gamma)=\backslash int\_\backslash gamma\; \backslash lambda(z,\backslash overline)\backslash ,\; |dz|$ The area of a subset of the complex plane is given by :$\backslash text(M)=\backslash int\_M\; \backslash lambda^2\; (z,\backslash overline)\backslash ,\backslash frac\backslash ,dz\; \backslash wedge\; d\backslash overline$ where $\backslash wedge$ is the exterior product used to construct the volume form. The determinant of the metric is equal to $\backslash lambda^4$, so the square root of the determinant is $\backslash lambda^2$. The Euclidean volume form on the plane is $dx\backslash wedge\; dy$ and so one has :$dz\; \backslash wedge\; d\backslash overline=(dx+i\backslash ,dy)\backslash wedge\; (dx-i\; \backslash ,\; dy)=\; -2i\backslash ,dx\backslash wedge\; dy.$ A function $\backslash Phi(z,\backslash overline)$ is said to be the potential of the metric if :$4\backslash frac$ \frac)=\lambda^2(z,\overline). The Laplace–Beltrami operator is given by :$\backslash Delta\; =\; \backslash frac$ \frac \frac \left( \frac + \frac \right). The Gaussian curvature of the metric is given by :$K=-\backslash Delta\; \backslash log\; \backslash lambda.\backslash ,$ This curvature is one-half of the Ricci scalar curvature. Isometries preserve angles and arc-lengths. On Riemann surfaces, isometries are identical to changes of coordinate: that is, both the Laplace–Beltrami operator and the curvature are invariant under isometries. Thus, for example, let ''S'' be a Riemann surface with metric $\backslash lambda^2(z,\backslash overline)\backslash ,\; dz\; \backslash ,\; d\backslash overline$ and ''T'' be a Riemann surface with metric $\backslash mu^2(w,\backslash overline)\backslash ,\; dw\backslash ,d\backslash overline$. Then a map :$f:S\backslash to\; T\backslash ,$ with $f=w(z)$ is an isometry if and only if it is conformal and if :$\backslash mu^2(w,\backslash overline)\; \backslash ;$ \frac \frac } = \lambda^2 (z, \overline ) . Here, the requirement that the map is conformal is nothing more than the statement :$w(z,\backslash overline)=w(z),$ that is, :$\backslash frac\{\backslash partial\; \backslash overline\{z\}\}\; w(z)\; =\; 0.$ 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「Poincaré metric」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.