Nilmanifold

Summary

A nilmanifold is a homogeneous space $N/H$ of a connected nilpotent Lie group $N$ by a closed subgroup $H$. The most studied ones are the compact quotients $N/\Gamma$ of a simply connected nilpotent Lie group by a lattice $\Gamma$. The torus is the abelian case, and the Heisenberg manifold is the first nonabelian one. Compact nilmanifolds give many examples in geometry, such as symplectic manifolds that are not Kähler.