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

Properties

Completeness

Let be a real Lie algebra. It is called a complete Lie algebra if each map in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let G be a solvable Lie group whose Lie algebra is complete. Then for any closed subgroup \Gamma of G, the solvmanifold G/\Gamma is a complete solvmanifold.