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 , the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by A. Mal'cev in 1951.
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 ).
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 ).
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