solvmanifold

In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifolds, nilmanifolds, was introduced by Anatoly Maltsev, who proved the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated. Examples A solvable Lie group is trivially a solvmanifold. Every nilpotent group is solvable, therefore, every nilmanifold is a solvmanifold. This class of examples includes n-dimensional tori and the quotient of the 3-dimensional real Heisenberg group by its integral Heisenberg subgroup. The Möbius band and the Klein bottle are solvmanifolds that are not nilmanifolds. The mapping torus of an Anosov diffeomorphism of the n-torus is a solvmanifold. For , these manifolds belong to Sol, one of the eight Thurston geometries. Properties