en>Josve05a: clean up, added orphan tag using AWB

2013-03-27T16:51:19Z

clean up, added orphan tag using AWB

← Older revision		Revision as of 18:51, 27 March 2013
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	~~I would like to introduce myself to you~~, ~~I am Jayson Simcox but I~~ [~~http://165~~.~~132~~.39.~~93/xe/visitors/372912 online psychics~~] ~~don't like when people use my complete title~~. ~~Mississippi~~ is ~~where his house~~ is. ~~My day job is an invoicing officer but I~~'~~ve already utilized for an additional 1~~. The ~~favorite hobby for him~~ and ~~his kids~~ is to ~~play lacross~~ and ~~he'll~~ be ~~starting something else along with clairvoyants~~ [[~~http://203~~.~~250~~.78.~~160~~/~~zbxe~~/~~?document_srl=1792908 203~~.~~250~~.~~78.160~~]] it.<br><br>~~my webpage online reader~~ - [http://~~ustanford~~.~~com~~/~~index~~.php?do=/~~profile~~-~~38218~~/~~info/ ustanford.com~~] -		In [[mathematics]], a '''solvmanifold''' is a [[homogeneous space]] of a [[connected (topology)\|connected]] [[solvable Lie group]]. It may also be characterized as a quotient of a connected solvable Lie group by a [[closed (topology)\|closed]] [[subgroup]]. (Some authors also require that the Lie group be simply-connected, or that the quotient be compact.)
			A special class of solvmanifolds, [[nilmanifold]]s, was introduced by [[Malcev]], who proved 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 [[torus\|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 ''n''=2, these manifolds belong to '''Sol''', one of the eight [[Thurston geometry\|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\|G. Mostow]] and proved by [[Louis Auslander\|L. Auslander]] and R. Tolimieri.

			* The [[fundamental group]] of an arbitrary solvmanifold is [[polycyclic group\|polycyclic]].

			* A compact solvmanifold is determined up to diffeomorphism by its fundamental group.

			* Fundamental groups of compact solvmanifolds may be characterized as [[group extension]]s of [[free abelian group]]s of finite rank by finitely generated torsion-free nilpotent groups.

			* Every solvmanifold is [[aspherical space\|aspherical]]. Among all compact homogeneous spaces, solvmanifolds may be characterized by the properties of being aspherical and having a solvable fundamental group.

			== Odd section ==

			Let <math>\mathfrak{g}</math> be a real [[Lie algebra]]. It is called a '''complete Lie algebra''' if each map

			:ad<math>(X): \mathfrak{g} \to \mathfrak{g}, X \in \mathfrak{g}</math>

			in its [[adjoint representation of a Lie algebra\|adjoint representation]] is hyperbolic, i.e. has real [[eigenvalue]]s. Let ''G'' be a solvable Lie group whose Lie algebra <math>\mathfrak{g}</math> is complete. Then for any closed subgroup ''Γ'' of ''G'', the solvmanifold ''G''/''Γ'' is a '''complete solvmanifold'''.

			== References ==

			* [[Louis Auslander\|L. Auslander]], ''An exposition of the structure of solvmanifolds'' [http://www.ams.org/bull/1973-79-02/S0002-9904-1973-13134-9 I], [http://www.ams.org/bull/1973-79-02/S0002-9904-1973-13139-8 II], Bull. Amer. Math. Soc., 79:2 (1973), pp. 227–261, 262–285
			*{{Citation \| last1=Cooper \| first1=Daryl \| last2=Scharlemann \| first2=Martin \| title=Proceedings of 6th Gökova Geometry-Topology Conference \| url=http://mistug.tubitak.gov.tr/bdyim/toc.php?dergi=mat&yilsayi=1999/1 \| mr=1701636 \| year=1999 \| journal=Turkish Journal of Mathematics \| issn=1300-0098 \| volume=23 \| issue=1 \| chapter=The structure of a solvmanifold's Heegaard splittings \| pages=1–18}}

			* {{eom\|author=V.V. Gorbatsevich\|id=S/s086100}}

			[[Category:Lie algebras]]
			[[Category:Structures on manifolds]]

en>777sms at 03:44, 14 April 2012

2012-04-14T03:44:37Z

New page

I would like to introduce myself to you, I am Jayson Simcox but I [http://165.132.39.93/xe/visitors/372912 online psychics] don't like when people use my complete title. Mississippi is where his house is. My day job is an invoicing officer but I've already utilized for an additional 1. The favorite hobby for him and his kids is to play lacross and he'll be starting something else along with clairvoyants [[http://203.250.78.160/zbxe/?document_srl=1792908 203.250.78.160]] it.<br><br>my webpage online reader - [http://ustanford.com/index.php?do=/profile-38218/info/ ustanford.com] -

Poisson ring - Revision history

en>Josve05a: clean up, added orphan tag using AWB

en>777sms at 03:44, 14 April 2012