Solvmanifold

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In mathematics, a solvmanifold is the quotient space of a solvable Lie group by a closed subgroup. (Some definitions require that the Lie group be connected and simply-connected, or that the quotient be compact.)

Because every nilpotent group is solvable, every nilmanifold is a solvmanifold; however, there are solvmanifolds that are not nilmanifolds.

Let \mathfrak{g} be a real Lie algebra. It is called a complete Lie algebra Let G / Γ be a solvmanifold, and let \mathfrak{g} be the Lie algebra of G. Then G / Γ is a complete solvmanifold if \mathfrak{g} is a complete Lie algebra, that is, if each map

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

in its adjoint representation has only real eigenvalues.