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 [math]\displaystyle{ N/H }[/math], the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.[1]
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[2]).
Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature,[3] almost flat spaces arise as quotients of nilmanifolds,[4] and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow.[5]
In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green–Tao[6]) and ergodic theory (see, e.g., Host–Kra[7]).
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 [math]\displaystyle{ \Gamma }[/math]. If the subgroup [math]\displaystyle{ \Gamma }[/math] acts cocompactly (via right multiplication) on N, then the quotient manifold [math]\displaystyle{ N/ \Gamma }[/math] will be a compact nilmanifold. As Mal'cev has shown, every compact nilmanifold is obtained this way.[1]
Such a subgroup [math]\displaystyle{ \Gamma }[/math] 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 Mal'cev's criterion. Not all nilpotent Lie groups admit lattices; for more details, see also M. S. Raghunathan.[8]
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]\displaystyle{ \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]\displaystyle{ \Gamma }[/math] acts by isometries on N via left-multiplication. Thus the quotient [math]\displaystyle{ \Gamma \backslash N }[/math] is a compact space locally isometric to N. Note: this space is naturally diffeomorphic to [math]\displaystyle{ N / \Gamma }[/math].
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]\displaystyle{ 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]\displaystyle{ Z \cap \Gamma }[/math] is a lattice in Z. Hence, [math]\displaystyle{ 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]\displaystyle{ P = N/ \Gamma }[/math] with quotient space [math]\displaystyle{ M=P/G }[/math]. This base manifold M is a q-dimensional compact torus. It has been shown that every principal torus bundle over a torus is of this form, see.[9] More generally, a compact nilmanifold is a 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.
Complex nilmanifolds
Historically, a complex nilmanifold meant a quotient of a complex nilpotent Lie group over 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.
An almost complex structure on a real Lie algebra g is an endomorphism [math]\displaystyle{ I:\; g \rightarrow g }[/math] which squares to −Idg. This operator is called a complex structure if its eigenspaces, corresponding to eigenvalues [math]\displaystyle{ \pm \sqrt{-1} }[/math], are subalgebras in [math]\displaystyle{ g \otimes {\mathbb 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. It is easy to see that every connected complex homogeneous manifold equipped with a free, transitive, holomorphic action by a real Lie group is obtained this way.
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.
Complex nilmanifolds are usually not homogeneous, as complex varieties.
In complex dimension 2, the only complex nilmanifolds are a complex torus and a Kodaira surface.[10]
Properties
Compact nilmanifolds (except a torus) are never homotopy formal.[11] This implies immediately that compact nilmanifolds (except a torus) cannot admit a Kähler structure (see also [12]).
Topologically, all nilmanifolds can be obtained as iterated torus bundles over a torus. This is easily seen from a filtration by ascending central series.[13]
Examples
Nilpotent Lie groups
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.
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]\displaystyle{ \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]3 with the faces identified in a suitable way. This is because an element [math]\displaystyle{ \begin{pmatrix} 1 & x & z \\ & 1 & y \\ & & 1\end{pmatrix}\Gamma }[/math] of the nilmanifold can be represented by the element [math]\displaystyle{ \begin{pmatrix} 1 & \{x\} & \{z-x \lfloor y \rfloor \} \\ & 1 & \{y\} \\ & & 1\end{pmatrix} }[/math] in the fundamental domain. Here [math]\displaystyle{ \lfloor x \rfloor }[/math] denotes the floor function of x, and [math]\displaystyle{ \{ x \} }[/math] the 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.[6]
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]\displaystyle{ \R/\Z }[/math]. Another familiar example might be the compact 2-torus or Euclidean space under addition.
Generalizations
A parallel construction based on solvable Lie groups produces a class of spaces called solvmanifolds. An important example of a solvmanifolds are Inoue surfaces, known in complex geometry.
References
- ↑ 1.0 1.1 Mal'cev, Anatoly Ivanovich (1951). "On a class of homogeneous spaces.". American Mathematical Society Translations (39).
- ↑ Wilson, Edward N. (1982). "Isometry groups on homogeneous nilmanifolds". Geometriae Dedicata 12 (3): 337–346. doi:10.1007/BF00147318.
- ↑ Milnor, John (1976). "Curvatures of left invariant metrics on Lie groups". Advances in Mathematics 21 (3): 293–329. doi:10.1016/S0001-8708(76)80002-3.
- ↑ Gromov, Mikhail (1978). "Almost flat manifolds". Journal of Differential Geometry 13 (2): 231–241. doi:10.4310/jdg/1214434488.
- ↑ 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
- ↑ 6.0 6.1 Green, Benjamin; Tao, Terence (2010). "Linear equations in primes". Annals of Mathematics 171 (3): 1753–1850. doi:10.4007/annals.2010.171.1753.
- ↑ Host, Bernard; Kra, Bryna (2005). "Nonconventional ergodic averages and nilmanifolds". Annals of Mathematics. (2) 161 (1): 397–488. doi:10.4007/annals.2005.161.397.
- ↑ Raghunathan, M. S. (1972). Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 68. New York-Heidelberg: Springer-Verlag. ISBN 978-3-642-86428-5. "Chapter II"
- ↑ Palais, R. S.; Stewart, T. E. Torus bundles over a torus. Proc. Amer. Math. Soc. 12 1961 26–29.
- ↑ Keizo Hasegawa (2005). "Complex and Kähler structures on Compact Solvmanifolds". Journal of Symplectic Geometry 3 (4): 749–767. doi:10.4310/JSG.2005.v3.n4.a9. https://www.projecteuclid.org/journalArticle/Download?urlid=jsg%2F1154467635.
- ↑ Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71.
- ↑ Benson, Chal; Gordon, Carolyn S. (1988). "Kähler and symplectic structures on nilmanifolds". Topology 27 (4): 513–518. doi:10.1016/0040-9383(88)90029-8.
- ↑ Sönke Rollenske, 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
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