In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

 subgroup. An
Iwasawa manifold is a nilmanifold

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 N/H, the quotient of a nilpotent Lie group N modulo a closed...

, of real dimension 6.

As a complex manifold, such an Iwasawa manifold is an important example of
a compact complex manifold which does not admit any Kähler metric.