Nilmanifold Explained

N/H , the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.

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

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

Such a subgroup

\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 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

\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 \Gamma 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 / \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 = [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/ \Gamma with quotient space M = P/G . 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

I: g → 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 ⊗ { 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) cannotadmit a Kähler structure (see also ^[12]).

Topologically, all nilmanifolds can be obtainedas 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

\Gamma 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.

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

\R/\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. Edward N.. Wilson. Isometry groups on homogeneous nilmanifolds. . 12 . 1982. 3. 337–346. 10.1007/BF00147318. 0661539. 10338.dmlcz/147061. 123611727. free.
2. Milnor. John. John Milnor. Curvatures of left invariant metrics on Lie groups. . 21. 1976. 3. 293–329. 0425012. 10.1016/S0001-8708(76)80002-3. free.
3. Gromov. Mikhail. Mikhail Leonidovich Gromov. Almost flat manifolds. . 13 . 1978. 2. 231–241. 0540942. 10.4310/jdg/1214434488. free.
4. Chow, Bennett; Knopf, Dan, The Ricci flow: an introduction. Mathematical Surveys and Monographs, 110. American Mathematical Society, Providence, RI, 2004. xii+325 pp.
5. Green . Benjamin . Ben Green (mathematician). Tao . Terence . Terence Tao. Linear equations in primes . . 171 . 3 . 2010 . 10.4007/annals.2010.171.1753 . 1753–1850 . math.NT/0606088. 2680398. 119596965 .
6. Bernard. Host. Bryna. Kra. Bryna Kra. Nonconventional ergodic averages and nilmanifolds. . (2). 161 . 2005. 1. 397–488. 2150389. 10.4007/annals.2005.161.397. free.
7. Mal'cev . Anatoly Ivanovich . Anatoly Maltsev . 1951 . On a class of homogeneous spaces. . American Mathematical Society Translations . 39 .
8. Book: Raghunathan, M. S.. M. S. Raghunathan

   . M. S. Raghunathan. Chapter II. Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 68. Springer-Verlag. New York-Heidelberg. 1972. 0507234. 978-3-642-86428-5 .

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. Journal of Symplectic Geometry. 3. 4. 2005. 749–767. 10.4310/JSG.2005.v3.n4.a9. 2235860. 1120.53043. 6955295. 0804.4223.
11. Keizo Hasegawa, Minimal models of nilmanifolds, Proc. Amer. Math. Soc. 106 (1989), no. 1, 65–71.
12. Benson . Chal . Gordon . Carolyn S. . Carolyn S. Gordon . Kähler and symplectic structures on nilmanifolds . . 27 . 4 . 1988 . 10.1016/0040-9383(88)90029-8 . 513–518 . 0976592 . free .
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