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 ${\displaystyle N/H}$, the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.^[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 ${\displaystyle \Gamma }$. If the subgroup ${\displaystyle \Gamma }$ acts cocompactly (via right multiplication) on N, then the quotient manifold ${\displaystyle N/\Gamma }$ will be a compact nilmanifold. As Mal'cev has shown, every compact nilmanifold is obtained this way.^[1]

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

Compact nilmanifolds also arise as principal bundles. For example, consider a 2-step nilpotent Lie group N which admits a lattice (see above). Let ${\displaystyle Z=[N,N]}$ 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 ${\displaystyle Z\cap \Gamma }$ is a lattice in Z. Hence, ${\displaystyle G=Z/(Z\cap \Gamma )}$ is a p-dimensional compact torus. Since Z is central in N, the group G acts on the compact nilmanifold ${\displaystyle P=N/\Gamma }$ with quotient space ${\displaystyle M=P/G}$. 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 ${\displaystyle I:\;g\rightarrow g}$ which squares to −Id_g. This operator is called a complex structure if its eigenspaces, corresponding to eigenvalues ${\displaystyle \pm {\sqrt {-1}}}$, are subalgebras in ${\displaystyle g\otimes {\mathbb {C} }}$. 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 ${\displaystyle \Gamma }$ 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 ${\displaystyle {\begin{pmatrix}1&x&z\\&1&y\\&&1\end{pmatrix}}\Gamma }$ of the nilmanifold can be represented by the element ${\displaystyle {\begin{pmatrix}1&\{x\}&\{z-x\lfloor y\rfloor \}\\&1&\{y\}\\&&1\end{pmatrix}}}$ in the fundamental domain. Here ${\displaystyle \lfloor x\rfloor }$ denotes the floor function of x, and ${\displaystyle \{x\}}$ 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.^[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 ${\displaystyle \mathbb {R} /\mathbb {Z} }$. 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