In
mathematics, a solvmanifold is a
homogeneous space of a
connectedsolvable Lie group. It may also be characterized as a quotient of a connected solvable
Lie group by a
closedsubgroup. (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.
A solvmanifold is diffeomorphic to the total space of a
vector bundle over some compact solvmanifold. This statement was conjectured by
George Mostow and proved by
Louis Auslander and Richard 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.
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 of G, the solvmanifold is a complete solvmanifold.