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 ^[1]).

Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature^[2], almost flat spaces arise as quotients of nilmanifolds^[3], and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow^[4].

In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green-Tao ^[5]) and ergodic theory (see, e.g., Host-Kra ^[6]).

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 Γ. If the subgroup Γ acts cocompactly (via right multiplication) on N, then the quotient manifold N / Γ will be a compact nilmanifold. As Mal'cev has shown, every compact nilmanifold is obtained this way.^[7]

Such a subgroup Γ 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 group admit lattices; for more details, see also 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 Γ be a lattice in a simply connected nilpotent Lie group N, as above. Endow N with a left-invariant (Riemannian) metric. Then the subgroup Γ acts by isometries on N via left-multiplication. Thus the quotient $\Gamma \backslash N$ is a compact space locally isometric to N. Note: this space is naturally diffeomorphic to N / Γ.

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 = [N,N] 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 $Z \cap \Gamma$ is a lattice in Z. Hence, $G = Z/(Z \cap \Gamma )$ is a p-dimensional compact torus. Since Z is central in N, the group G acts on the compact nilmanifold P = N / Γ 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 ^[9]. 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.

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 $I:\; g \rightarrow g$ which squares to -Id_g. This operator is called a complex structure if its eigenspaces, corresponding to eigenvalues $\pm \sqrt{-1}$, are subalgebras in $g \otimes {\Bbb C}$. 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 Γ 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 $\begin{pmatrix} 1 & x & z \\ & 1 & y \\ & & 1\end{pmatrix}\Gamma$ of the nilmanifold can be represented by the element $\begin{pmatrix} 1 & \{x\} & \{z-x \lfloor y \rfloor \} \\ & 1 & \{y\} \\ & & 1\end{pmatrix}$ in the fundamental domain. Here $\lfloor x \rfloor$ denotes the floor function of x, and {x} 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.^[14]

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 $\mathbb{R}/\mathbb{Z}$. 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. ^ E. Wilson, "Isometry groups on homogeneous nilmanifolds", Geometriae Dedicata 12 (1982) 337-346
2. ^ Milnor, John Curvatures of left invariant metrics on Lie groups. Advances in Math. 21 (1976), no. 3, 293--329.
3. ^ Gromov, M. Almost flat manifolds. J. Differential Geom. 13 (1978), no. 2, 231--241.
4. ^ 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
5. ^ Ben Green and Terence Tao, Linear equations in primes, 22 April 2008.
6. ^ Bernard Host and Bryna Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), no. 1, 397–488.
7. ^ A. I. Mal'cev, On a class of homogeneous spaces, AMS Translation No. 39 (1951).
8. ^ Raghunathan, Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
9. ^ Palais, R. S.; Stewart, T. E. Torus bundles over a torus. Proc. Amer. Math. Soc. 12 1961 26--29.
10. ^ Keizo Hasegawa Complex and Kähler structures on Compact Solvmanifolds, J. Symplectic Geom. Volume 3, Number 4 (2005), 749-767.
11. ^ Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65--71.
12. ^ C. Benson, C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27(4) (1988) 513--518.
13. ^ 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
14. ^ Ben Green and Terence Tao, Linear equations in primes, Ann. of Math. Volume 171 (2010), Issue 3, 1753-1850