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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 Yamabe invariant ： ウィキペディア英語版 Yamabe invariant In mathematics, in the field of differential geometry, the Yamabe invariant (also referred to as the sigma constant) is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe. == Definition == Let $M$ be a compact smooth manifold (without boundary) of dimension $n\backslash geq\; 2$. The normalized Einstein–Hilbert functional $\backslash mathcal$ assigns to each Riemannian metric $g$ on $M$ a real number as follows: : $\backslash mathcal(g)\; =\; \backslash frac\}\},$ where $R\_g$ is the scalar curvature of $g$ and $dV\_g$ is the volume density associated to the metric $g$. The exponent in the denominator is chosen so that the functional is scale-invariant: for every positive real constant $c$, it satisfies $\backslash mathcal(cg)\; =\; \backslash mathcal(g)$. We may think of $\backslash mathcal(g)$ as measuring the average scalar curvature of $g$ over $M$. It was conjectured by Yamabe that every conformal class of metrics contains a metric of constant scalar curvature (the so-called Yamabe problem); it was proven by Yamabe, Trudinger, Aubin, and Schoen that a minimum value of $\backslash mathcal(g)$ is attained in each conformal class of metrics, and in particular this minimum is achieved by a metric of constant scalar curvature. We define : $Y(g)\; =\; \backslash inf\_\; \backslash mathcal(e^\; g),$ where the infimum is taken over the smooth real-valued functions $f$ on $M$. This infimum is finite (not $-\backslash infty$): Hölder's inequality implies $Y(g)\; \backslash geq\; -\backslash left(\backslash textstyle\backslash int\_M\; |R\_g|^\; \backslash ,dV\_g\backslash right)^$. The number $Y(g)$ is sometimes called the conformal Yamabe energy of $g$ (and is constant on conformal classes). A comparison argument due to Aubin shows that for any metric $g$, $Y(g)$ is bounded above by $\backslash mathcal(g\_0)$, where $g\_0$ is the standard metric on the $n$-sphere $S^n$. It follows that if we define : $\backslash sigma(M)\; =\; \backslash sup\_\; Y(g),$ where the supremum is taken over all metrics on $M$, then $\backslash sigma(M)\; \backslash leq\; \backslash mathcal(g\_0)$ (and is in particular finite). The real number $\backslash sigma(M)$ is called the Yamabe invariant of $M$. 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「Yamabe invariant」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.