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 ${\displaystyle N/H}$, the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1949.^[1]

In the Riemannian category, there is also a good notion of a nilmanifold. 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.^[5]

In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green–Tao^[6]) and ergodic theory (see, e.g., Host–Kra^[7]).