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| In [[mathematics]], a '''nilmanifold''' is a [[differentiable manifold]] which has a transitive [[nilpotent group|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]] <math>N/H</math>, the quotient of a nilpotent [[Lie group]] ''N'' modulo a [[closed (topology)|closed]] [[subgroup]] ''H''. This notion was introduced by [[Anatoly Maltsev|A. Mal'cev]] in 1951.
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| 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 <ref>E. Wilson, "Isometry groups on homogeneous nilmanifolds", Geometriae Dedicata 12 (1982) 337–346</ref>).
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| Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature,<ref>Milnor, John ''Curvatures of left invariant metrics on Lie groups.'' Advances in Math. 21 (1976), no. 3, 293–329.</ref> [[Almost flat manifold|almost flat spaces]] arise as quotients of nilmanifolds,<ref>Gromov, M. ''Almost flat manifolds.'' J. Differential Geom. 13 (1978), no. 2, 231–241.</ref> and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow.<ref>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</ref>
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| In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in [[arithmetic combinatorics]] (see Green–Tao <ref>Ben Green and Terence Tao, [http://arxiv.org/pdf/math.NT/0606088 Linear equations in primes], 22 April 2008.</ref>) and [[ergodic theory]] (see, e.g., Host–Kra <ref>Bernard Host and [[Bryna Kra]], [http://www.math.northwestern.edu/~kra/papers/convnil.pdf Nonconventional ergodic averages and nilmanifolds], Ann. of Math. (2) 161 (2005), no. 1, 397–488.</ref>).
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| == Compact nilmanifolds ==
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| 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]] <math> \Gamma </math>. If the subgroup <math> \Gamma </math> acts cocompactly (via right multiplication) on ''N'', then the quotient manifold <math>N/ \Gamma </math> will be a compact nilmanifold. As Mal'cev has shown, every compact
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| nilmanifold is obtained this way.<ref>A. I. Mal'cev, ''On a class of homogeneous spaces'', AMS Translation No. '''39''' (1951).</ref>
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| Such a subgroup <math> \Gamma </math> as above is called a [[lattice (group theory)|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.<ref>Raghunathan, Chapter II, ''Discrete Subgroups of Lie Groups'', M. S. Raghunathan</ref>
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| 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 <math> \Gamma </math> be a lattice in a simply connected nilpotent Lie group ''N'', as above. Endow ''N'' with a left-invariant (Riemannian) metric. Then the subgroup <math>\Gamma</math> acts by isometries on ''N'' via left-multiplication. Thus the quotient <math>\Gamma \backslash N</math> is a compact space locally isometric to ''N''. Note: this space is naturally diffeomorphic to <math>N / \Gamma </math>.
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| Compact nilmanifolds also arise as [[principal bundles]]. For example, consider a 2-step [[nilpotent Lie group]] ''N'' which admits a lattice (see above). Let <math>Z=[N,N]</math> 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 <math>Z \cap \Gamma</math> is a lattice in ''Z''. Hence, <math>G = Z/(Z \cap \Gamma )</math> is a ''p''-dimensional compact torus. Since ''Z'' is central in ''N'', the group G acts on the compact nilmanifold <math>P = N/ \Gamma</math> with quotient space <math>M=P/G</math>. 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.<ref>Palais, R. S.; Stewart, T. E. ''Torus bundles over a torus.'' Proc. Amer. Math. Soc. 12 1961 26–29.</ref> More generally, a compact nilmanifold is torus bundle, over a torus bundle, over...over a torus.
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| As mentioned above, [[almost flat manifold]]s are intimately compact nilmanifolds. See that article for more information.
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| ==Complex nilmanifolds==
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| Historically, a '''complex nilmanifold''' meant a quotient of a complex nilpotent Lie group over
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| a [[cocompact lattice]]. An example of such a nilmanifold is an [[Iwasawa manifold]]. From the 1980s, another (more general) notion of a complex nilmanifold gradually replaced this one.
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| An '''almost complex structure''' on a real Lie algebra ''g'' is an endomorphism <math>I:\; g \rightarrow g</math> which squares to
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| −Id<sub>''g''</sub>. This operator is called '''a complex structure''' if its eigenspaces, corresponding to eigenvalues
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| <math>\pm \sqrt{-1}</math>, are subalgebras in <math>g \otimes {\Bbb C}</math>. In this case, ''I'' defines a left-invariant complex structure on the corresponding Lie group. Such a manifold (''G'',''I'') is called '''a complex group manifold'''.
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| It is easy to see that every connected complex [[homogeneous space|homogeneous manifold]] equipped with a free, transitive, holomorphic action by a real Lie group is obtained this way.
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| Let ''G'' be a real, nilpotent Lie group. A '''complex nilmanifold''' is a quotient of a complex group manifold (''G'',''I''), equipped with a left-invariant complex structure, by a discrete, cocompact lattice, acting from the right.
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| Complex nilmanifolds are usually not homogeneous, as complex varieties.
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| In complex dimension 2, the only complex nilmanifolds are a complex torus and a [[Kodaira surface]].<ref>Keizo Hasegawa ''Complex and Kähler structures on Compact Solvmanifolds,'' J. Symplectic Geom. Volume 3, Number 4 (2005), 749–767.</ref>
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| ==Properties==
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| Compact nilmanifolds (except a torus) are never [[formal space|homotopy formal]].<ref>Keizo Hasegawa, ''Minimal models of nilmanifolds,'' Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71.</ref> This implies immediately that compact nilmanifolds (except a torus) cannot
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| admit a [[Kähler structure]] (see also <ref>C. Benson, C.S. Gordon, ''Kähler and symplectic structures on nilmanifolds'', Topology '''27'''(4) (1988) 513–518.</ref>).
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| Topologically, all nilmanifolds can be obtained
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| as iterated torus bundles over a torus. This is easily seen from a filtration by [[Upper central series|ascending central series]].<ref>Sönke Rollenske, [http://arxiv.org/abs/0901.3120 Geometry of nilmanifolds with left-invariant complex structure and deformations in the large], 40 pages, arXiv:0901.3120, Proc. London Math. Soc., 99, 425–460, 2009</ref>
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| == Examples ==
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| === Nilpotent Lie groups ===
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| From the above definition of homogeneous nilmanifolds, it is clear that any nilpotent Lie group with left-invariant metric is a homogeneous nilmanifold. The most familiar nilpotent Lie groups are matrix groups whose diagonal entries are 1 and whose lower diagonal entries are all zeros.
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| For example, the [[Heisenberg group]] is a 2-step nilpotent Lie group. This nilpotent Lie group is also special in that it admits a compact quotient. The group <math>\Gamma</math> would be the upper triangular matrices with integral coefficients. The resulting nilmanifold is 3-dimensional. One possible [[fundamental domain]] is (isomorphic to) [0,1]<sup>3</sup> with the faces identified in a suitable way. This is because an element <math>\begin{pmatrix} 1 & x & z \\ & 1 & y \\ & & 1\end{pmatrix}\Gamma</math> of the nilmanifold can be represented by the element <math>\begin{pmatrix} 1 & \{x\} & \{z-x \lfloor y \rfloor \} \\ & 1 & \{y\} \\ & & 1\end{pmatrix}</math> in the fundamental domain. Here <math>\lfloor x \rfloor</math> denotes the [[floor function]] of ''x'', and <math>\{ x \}</math> the [[Floor_function#Fractional_part|fractional part]]. The appearance of the floor function here is a clue to the relevance of nilmanifolds to additive combinatorics: the so-called bracket polynomials, or generalised polynomials, seem to be important in the development of higher-order Fourier analysis.<ref>Ben Green and Terence Tao, [http://arxiv.org/abs/math.NT/0606088 Linear equations in primes], Ann. of Math. Volume 171 (2010), Issue 3, 1753–1850</ref>
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| === Abelian Lie groups === | |
| A simpler example would be any abelian Lie group. This is because any such group is a nilpotent Lie group. For example, one can take the group of real numbers under addition, and the discrete, cocompact subgroup consisting of the integers. The resulting 1-step nilmanifold is the familiar circle <math>\mathbb{R}/\mathbb{Z}</math>. Another familiar example might be the compact 2-torus or Euclidean space under addition.
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| == Generalizations == | |
| A parallel construction based on [[solvable group|solvable]] Lie groups produces a class of spaces called '''solvmanifolds'''. An important example of a solvmanifolds are [[Inoue surface]]s, known in [[complex geometry]].
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| == References ==
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| <references/>
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| [[Category:Homogeneous spaces]]
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| [[Category:Smooth manifolds]]
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| [[Category:Lie groups]]
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