Nilmanifold

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.[7]

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 Malcev'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.

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]

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]

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.