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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 Carnot–Carathéodory metric ： ウィキペディア英語版 Sub-Riemannian manifold In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal subspaces''. Sub-Riemannian manifolds (and so, ''a fortiori'', Riemannian manifolds) carry a natural intrinsic metric called the metric of Carnot–Carathéodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold). Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure. ==Definitions== By a ''distribution'' on $M$ we mean a subbundle of the tangent bundle of $M$. Given a distribution $H(M)\backslash subset\; T(M)$ a vector field in $H(M)\backslash subset\; T(M)$ is called horizontal. A curve $\backslash gamma$ on $M$ is called horizontal if $\backslash dot\backslash gamma(t)\backslash in\; H\_(M)$ for any $t$. A distribution on $H(M)$ is called completely non-integrable if for any $x\backslash in\; M$ we have that any tangent vector can be presented as a linear combination of vectors of the following types $A(x),\backslash \; ()(x),\backslash \; ]](x),\backslash dotsc\backslash in\; T\_x(M)$ where all vector fields $A,B,C,D,\; \backslash dots$ are horizontal. A sub-Riemannian manifold is a triple $(M,\; H,\; g)$, where $M$ is a differentiable manifold, $H$ is a ''completely non-integrable'' "horizontal" distribution and $g$ is a smooth section of positive-definite quadratic forms on $H$. Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as :$d(x,\; y)\; =\; \backslash inf\backslash int\_0^1\; \backslash sqrt\; \backslash ,\; dt,$ where infimum is taken along all ''horizontal curves'' $\backslash gamma:\; (1)\; \backslash to\; M$ such that $\backslash gamma(0)=x$, $\backslash gamma(1)=y$. 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「Sub-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.