# 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 Malcev, who proved 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.

## Properties

• A solvmanifold is diffeomorphic to the total space of a vector bundle over some compact solvmanifold. This statement was conjectured by G. Mostow and proved by L. Auslander and R. Tolimieri.
• A compact solvmanifold is determined up to diffeomorphism by its fundamental group.
• Fundamental groups of compact solvmanifolds may be characterized as group extensions of free abelian groups of finite rank by finitely generated torsion-free nilpotent groups.
• Every solvmanifold is aspherical. Among all compact homogeneous spaces, solvmanifolds may be characterized by the properties of being aspherical and having a solvable fundamental group.

## Odd section

Let $\mathfrak{g}$ be a real Lie algebra. It is called a complete Lie algebra if each map

ad$(X): \mathfrak{g} \to \mathfrak{g}, X \in \mathfrak{g}$

in its adjoint representation is hyperbolic, i.e. has real eigenvalues. Let G be a solvable Lie group whose Lie algebra $\mathfrak{g}$ is complete. Then for any closed subgroup Γ of G, the solvmanifold G/Γ is a complete solvmanifold.