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.
It is called a complete Lie algebra if each map in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues.