mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a solvmanifold is a

homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...

of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

subgroup. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.) A special class of solvmanifolds,

nilmanifold In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, the q ...

s, 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

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 Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...

and the

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

are solvmanifolds that are not nilmanifolds. * The

mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the boun ...

of an

Anosov diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...

of the ''n''-torus is a solvmanifold. For

n=2

, 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 George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy o ...

and proved by

Louis Auslander Louis Auslander (July 12, 1928 – February 25, 1997) was a Jewish American mathematician. He had wide-ranging interests both in pure and applied mathematics and worked on Finsler geometry, geometry of solvmanifolds and nilmanifolds, locally aff ...

and Richard Tolimieri. * The

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

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 group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

s 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

\mathfrak

be a real

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

. It is called a complete Lie algebra if each map :

\operatorname(X)\colon \mathfrak \to \mathfrak, X \in \mathfrak

in its adjoint representation is hyperbolic, i.e., it has only real eigenvalues. Let ''G'' be a solvable Lie group whose Lie algebra

\mathfrak

is complete. Then for any closed subgroup

\Gamma

of ''G'', the solvmanifold

G/\Gamma

is a complete solvmanifold.

References

* ** * * {{refend Lie algebras Structures on manifolds