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Differentiable manifold
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 , the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.). Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature, almost flat spaces arise as quotients of nilmanifolds, and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow. In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green–Tao) and ergodic theory (see, e. g., Host–Kra).

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 If the subgroup acts cocompactly (via right multiplication) on N, then the quotient manifold will be a compact nilmanifold. As Mal'cev has shown, every compact
nilmanifold is obtained this way. Such a subgroup 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. 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 be a lattice in a simply connected nilpotent Lie group N, as above. Endow N with a left invariant (Riemannian) metric. Then the subgroup acts by isometries on N via left multiplication. Thus the quotient is a compact space locally isometric to N. Note: this space is naturally diffeomorphic to
Compact nilmanifolds also arise as principal bundles. For example, consider a 2 step nilpotent Lie group N which admits a lattice (see above). Let 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 is a lattice in Z. Hence, is a p dimensional compact torus. Since Z is central in N, the group G acts on the compact nilmanifold with quotient space 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. 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.

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 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 of the nilmanifold can be represented by the element in the fundamental domain. Here denotes the floor function of x, and 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.

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.

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.