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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 Nilmanifold ： ウィキペディア英語版 Nilmanifold In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space $N/H$, the quotient of a nilpotent Lie group ''N'' modulo a closed subgroup ''H''. This notion was introduced by A. Mal'cev in 1951. In the Riemannian category, there is also a good notion of a nilmanifold. A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it. The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization: every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric (see Wilson 〔E. Wilson, "Isometry groups on homogeneous nilmanifolds", Geometriae Dedicata 12 (1982) 337–346〕). Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature,〔Milnor, John ''Curvatures of left invariant metrics on Lie groups.'' Advances in Math. 21 (1976), no. 3, 293–329.〕 almost flat spaces arise as quotients of nilmanifolds,〔Gromov, M. ''Almost flat manifolds.'' J. Differential Geom. 13 (1978), no. 2, 231–241.〕 and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow.〔Chow, Bennett; Knopf, Dan, ''The Ricci flow: an introduction.'' Mathematical Surveys and Monographs, 110. American Mathematical Society, Providence, RI, 2004. xii+325 pp. ISBN 0-8218-3515-7〕 In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green–Tao 〔Ben Green and Terence Tao, (Linear equations in primes ), 22 April 2008.〕) and ergodic theory (see, e.g., Host–Kra 〔Bernard Host and Bryna Kra, (Nonconventional ergodic averages and nilmanifolds ), Ann. of Math. (2) 161 (2005), no. 1, 397–488.〕). == Compact nilmanifolds == A compact nilmanifold is a nilmanifold which is compact. One way to construct such spaces is to start with a simply connected nilpotent Lie group ''N'' and a discrete subgroup $\backslash Gamma$. If the subgroup $\backslash Gamma$ acts cocompactly (via right multiplication) on ''N'', then the quotient manifold $N/\; \backslash Gamma$ will be a compact nilmanifold. As Mal'cev has shown, every compact nilmanifold is obtained this way.〔A. I. Mal'cev, ''On a class of homogeneous spaces'', AMS Translation No. 39 (1951).〕 Such a subgroup $\backslash Gamma$ as above is called a lattice in ''N''. It is well known that a nilpotent Lie group admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also Raghunathan.〔Raghunathan, Chapter II, ''Discrete Subgroups of Lie Groups'', M. S. Raghunathan〕 A compact Riemannian nilmanifold is a compact Riemannian manifold which is locally isometric to a nilpotent Lie group with left-invariant metric. These spaces are constructed as follows. Let $\backslash Gamma$ be a lattice in a simply connected nilpotent Lie group ''N'', as above. Endow ''N'' with a left-invariant (Riemannian) metric. Then the subgroup $\backslash Gamma$ acts by isometries on ''N'' via left-multiplication. Thus the quotient $\backslash Gamma\; \backslash backslash\; N$ is a compact space locally isometric to ''N''. Note: this space is naturally diffeomorphic to $N\; /\; \backslash Gamma$. Compact nilmanifolds also arise as principal bundles. For example, consider a 2-step nilpotent Lie group ''N'' which admits a lattice (see above). Let $Z=()$ be the commutator subgroup of ''N''. Denote by p the dimension of ''Z'' and by q the codimension of ''Z''; i.e. the dimension of ''N'' is p+q. It is known (see Raghunathan) that $Z\; \backslash cap\; \backslash Gamma$ is a lattice in ''Z''. Hence, $G\; =\; Z/(Z\; \backslash cap\; \backslash Gamma\; )$ is a ''p''-dimensional compact torus. Since ''Z'' is central in ''N'', the group G acts on the compact nilmanifold $P\; =\; N/\; \backslash Gamma$ with quotient space $M=P/G$. This base manifold ''M'' is a ''q''-dimensional compact torus. It has been shown that ever principal torus bundle over a torus is of this form, see.〔Palais, R. S.; Stewart, T. E. ''Torus bundles over a torus.'' Proc. Amer. Math. Soc. 12 1961 26–29.〕 More generally, a compact nilmanifold is torus bundle, over a torus bundle, over...over a torus. As mentioned above, almost flat manifolds are intimately compact nilmanifolds. See that article for more information. 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「Nilmanifold」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.