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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 Pseudoholomorphic curve ： ウィキペディア英語版 Pseudoholomorphic curve In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or ''J''-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory. ==Definition== Let $X$ be an almost complex manifold with almost complex structure $J$. Let $C$ be a smooth Riemann surface (also called a complex curve) with complex structure $j$. A pseudoholomorphic curve in $X$ is a map $f\; :\; C\; \backslash to\; X$ that satisfies the Cauchy–Riemann equation :$\backslash bar\; \backslash partial\_\; f\; :=\; \backslash frac(df\; +\; J\; \backslash circ\; df\; \backslash circ\; j)\; =\; 0.$ Since $J^2\; =\; -1$, this condition is equivalent to :$J\; \backslash circ\; df\; =\; df\; \backslash circ\; j,$ which simply means that the differential $df$ is complex-linear, that is, $J$ maps each tangent space :$T\_xf(C)\backslash subseteq\; T\_xX$ to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term $\backslash nu$ and to study maps satisfying the perturbed Cauchy–Riemann equation :$\backslash bar\; \backslash partial\_\; f\; =\; \backslash nu.$ A pseudoholomorphic curve satisfying this equation can be called, more specifically, a $(j,\; J,\; \backslash nu)$-holomorphic curve. The perturbation $\backslash nu$ is sometimes assumed to be generated by a Hamiltonian (particularly in Floer theory), but in general it need not be. A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of $X$, so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov–Witten invariants, for example, we consider only closed domains $C$ of fixed genus $g$ and we introduce $n$ marked points (or punctures) on $C$. As soon as the punctured Euler characteristic $2\; -\; 2\; g\; -\; n$ is negative, there are only finitely many holomorphic reparametrizations of $C$ that preserve the marked points. The domain curve $C$ is an element of the Deligne–Mumford moduli space of curves. 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「Pseudoholomorphic curve」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.