Nilmanifold

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In mathematics, a nilmanifold is quotient space of a nilpotent Lie group modulo a closed subgroup, or, equivalently, a homogeneous space with a nilpotent Lie group acting transitively on it.

This notion was introduced by A. Mal'cev in 1951.

Another way to realize a nilmanifold is to start with a simply connected nilpotent Lie group, and construct the quotient space by a discrete subgroup Γ. As Mal'cev has shown, every compact nilmanifold is obtained this way.^[1]

Nilmanifolds are important geometrical objects, but are increasingly also being seen as having a role to play in arithmetic combinatorics (see, for example, ^[2]) and ergodic theory (such as in ^[3]).

Contents 1 Compact Riemannian nilmanifolds 2 Complex nilmanifolds 3 Properties 4 Examples 4.1 Nilpotent Lie groups 4.2 Abelian Lie groups 5 Generalizations 6 References

[edit] Compact Riemannian nilmanifolds

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 (see Raghunathan^[4] for details).

Take a simply connected nilpotent Lie group N which admits a lattice. 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. The lattice in the Lie algebra gives rise to a discrete subgroup Γ. We endow N with a left-invariant (Riemannian) metric. Now Γ can be view as a discrete group of isometries acting on N by left mutiplication, since we endowed N with a left-invariant metric.

To construct the compact nilmanifold we quotient N by the group action of Γ and obtain .

In the literature, the term nilmanifold usually denotes a compact nilmanifold, not necessarily Riemannian.

[edit] 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 which squares to -Id_g. This operator is called a complex structure if its eigenspaces, corresponding to eigenvalues , are subalgebras in . 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.^[5]

[edit] Properties

Compact nilmanifolds (except a torus) are never homotopy formal.^[6] This implies immediately that compact nilmanifolds (except a torus) cannot admit a Kähler structure (see also ^[7]).

Topologically, all nilmanifolds can be obtained as interated torus bundles over a torus. This is easily seen from a filtration by ascending central series. ^[8]

[edit] Examples

[edit] 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 coefficents. 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 of the nilmanifold can be represented by the element in the fundamental domain. Here 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^[9].

[edit] 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 . Another familiar example might be the compact 2-torus or Euclidean space under addition.

[edit] 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.

[edit] References

1. ^ A. I. Mal'cev, On a class of homogeneous spaces, AMS Translation No. 39 (1951).
2. ^ Ben Green and Terence Tao, Linear equations in primes, 22 April 2008.
3. ^ Bernard Host and Bryna Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), no. 1, 397–488.
4. ^ Chapter II, Discrete Subgroups of Lie Groups, M. S. Raghunathan
5. ^ Keizo Hasegawa Complex and Kähler structures on Compact Solvmanifolds, J. Symplectic Geom. Volume 3, Number 4 (2005), 749-767.
6. ^ Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65--71.
7. ^ C. Benson, C.S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27(4) (1988) 513--518.
8. ^ Sönke Rollenske, Geometry of nilmanifolds with left-invariant complex structure and deformations in the large, 40 pages, arXiv:0901.3120.]
9. ^ Ben Green and Terence Tao, Linear equations in primes, 22 April 2008.