In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it.
, the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.
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.
One way to construct such spaces is to start with a simply connected nilpotent Lie group N and a discrete subgroup
acts cocompactly (via right multiplication) on N, then the quotient manifold
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.
[8] A compact Riemannian nilmanifold is a compact Riemannian manifold which is locally isometric to a nilpotent Lie group with left-invariant metric.
be a lattice in a simply connected nilpotent Lie group N, as above.
For example, consider a 2-step nilpotent Lie group N which admits a lattice (see above).
Since Z is central in N, the group G acts on the compact nilmanifold
As mentioned above, almost flat manifolds are intimately compact nilmanifolds.
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
This operator is called a complex structure if its eigenspaces, corresponding to eigenvalues
In this case, I defines a left-invariant complex structure on the corresponding Lie group.
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.
[11] This implies immediately that compact nilmanifolds (except a torus) cannot admit a Kähler structure (see also [12]).
This is easily seen from a filtration by ascending central series.
The most familiar nilpotent Lie groups are matrix groups whose diagonal entries are 1 and whose lower diagonal entries are all zeros.
This nilpotent Lie group is also special in that it admits a compact quotient.
One possible fundamental domain is (isomorphic to) [0,1]3 with the faces identified in a suitable way.
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] A simpler example would be any abelian Lie group.
For example, one can take the group of real numbers under addition, and the discrete, cocompact subgroup consisting of the integers.
Another familiar example might be the compact 2-torus or Euclidean space under addition.
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.