Article · Wikipedia archive · Last revised Jul 9, 2026

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. 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.

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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.

Examples

Properties

  • 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.
  • The fundamental group of an arbitrary solvmanifold is polycyclic.
  • 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 g {\displaystyle {\mathfrak {g}}} be a real Lie algebra. It is called a complete Lie algebra if each map

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

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

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