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 the first structural theorems. Properties of general solvmanifolds are similar, but somewhat more complicated.

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

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

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