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.
Completeness.
Let formula_2 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 formula_2 is complete. Then for any closed subgroup formula_5 of "G", the solvmanifold formula_6 is a complete solvmanifold.