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 ==
* A solvable Lie group is trivially a solvmanifold. * Every nilpotent group is solvable, therefore, every nilmanifold is a solvmanifold. This class of examples includes ''n''-dimensional tori and the quotient of the 3-dimensional real Heisenberg group by its integral Heisenberg subgroup. * The Möbius band and the Klein bottle are solvmanifolds that are not nilmanifolds. * The mapping torus of an Anosov diffeomorphism of the ''n''-torus is a solvmanifold. For <math>n=2</math>, these manifolds belong to '''Sol''', one of the eight Thurston geometries.
== 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 <math>\mathfrak{g}</math> be a real Lie algebra. It is called a '''complete Lie algebra''' if each map
:<math>\operatorname{ad}(X)\colon \mathfrak{g} \to \mathfrak{g}, X \in \mathfrak{g}</math>
in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let ''G'' be a solvable Lie group whose Lie algebra <math>\mathfrak{g}</math> is complete. Then for any closed subgroup <math>\Gamma</math> of ''G'', the solvmanifold <math>G/\Gamma</math> is a '''complete solvmanifold'''.
== References == {{refbegin}} *{{Citation |author1-link=Louis Auslander |first1=Louis |last1=Auslander |title=An exposition of the structure of solvmanifolds. Part I: Algebraic theory |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-79/issue-2/An-exposition-of-the-structure-of-solvmanifolds-Part-I/bams/1183534430.pdf |journal=Bulletin of the American Mathematical Society |volume=79 |issue=2 |year=1973 |pages=227–261 |mr=0486307 |doi=10.1090/S0002-9904-1973-13134-9 |doi-access=free }} **{{Citation |author1-mask=1 |first1=Louis |last1=Auslander |title=Part II: $G$-induced flows |url=http://www.ams.org/bull/1973-79-02/S0002-9904-1973-13139-8 |journal=Bull. Amer. Math. Soc. |volume=79 |issue=2 |year=1973 |pages=262–285 |mr=0486308 |doi=10.1090/S0002-9904-1973-13139-8 |doi-access=free }} *{{Citation | last1=Cooper | first1=Daryl | last2=Scharlemann | first2=Martin |author2-link=Martin Scharlemann| department=Proceedings of 6th Gökova Geometry-Topology Conference | mr=1701636 | year=1999 | journal=Turkish Journal of Mathematics | issn=1300-0098 | volume=23 | issue=1 | title=The structure of a solvmanifold's Heegaard splittings | pages=1–18 | url=http://journals.tubitak.gov.tr/math/issues/mat-99-23-1/mat-23-1-1-98071.pdf }} * {{eom|first=V. V.|last=Gorbatsevich|title=Solv manifold}} {{refend}}
Category:Lie algebras Category:Structures on manifolds