216.239.88.118 at 03:09, 31 October 2013

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← Older revision		Revision as of 05:09, 31 October 2013
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	I'~~m Luther but I~~ by ~~no means really favored~~ that ~~title~~. ~~Delaware~~ is the ~~only place I~~'~~ve been residing~~ in. ~~To perform handball~~ is the ~~thing she enjoys~~ most of ~~all~~. ~~I am~~ a ~~production~~ and ~~distribution officer~~.<br><br>~~Also visit my site :: extended car warranty~~ (~~[http:~~//~~Llollz~~.~~com~~/~~blogs~~/~~post~~/~~5238 just click the up coming internet site~~])		{{Other uses\|Affine manifold (disambiguation)}}

			In [[differential geometry]], an '''affine manifold''' is a [[differentiable manifold]] equipped with a [[flat connection\|flat]], [[torsion tensor\|torsion-free]] [[affine connection\|connection]].

			Equivalently, it is a manifold that is (if connected) [[covering\|covered]] by an open subset of <math>{\Bbb R}^n</math>, with [[Covering space#Monodromy action\|monodromy]] acting by [[affine transformation]]s. This equivalence is an easy corollary of [[Cartan–Ambrose–Hicks theorem]].

			Equivalently, it is a manifold equipped with an atlas—called '''the affine structure'''—with all transition functions between [[chart]]s affine (that is, have constant jacobian matrix);<ref>Bishop, R.L.; Goldberg, S.I. (1968), pp. 223–224.</ref> two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an '''affine manifold''' and the charts which are affinely related to those of the affine structure are called '''affine charts'''. In each affine coordinate domain the coordinate [[vector field]]s form a [[parallelization (mathematics)\|parallelization]] of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between [[linear]] [[connection (mathematics)\|connection]] (also called [[affine connection]]) and a [[web (differential geometry)\|web]].

			==Formal definition==
			An '''affine manifold''' <math>M\,</math> is a real [[manifold]] with charts <math>\psi_i\colon U_i\to{\Bbb R}^n</math> such that <math>\psi_i\circ\psi_j^{-1}\in {\rm Aff}({\Bbb R}^n)</math> for all <math>i, j\, ,</math> where <math>{\rm Aff}({\Bbb R}^n)</math> denotes the [[Lie group]] of affine transformations.

			An affine manifold is called '''complete''' if its [[universal covering]] is [[homeomorphic]] to <math>{\Bbb R}^n</math>.

			In the case of a compact affine manifold <math>M</math>, let <math>G</math> be the [[fundamental group]] of <math>M</math> and <math>\tilde M</math> be its [[universal cover]]. One can show that each <math>n</math>-dimensional affine manifold comes with a developing map <math>D\colon {\tilde M}\to{\Bbb R}^n</math>, and a [[homomorphism]] <math>\varphi\colon G\to {\rm Aff}({\Bbb R}^n)</math>, such that <math>D</math> is an [[immersion (mathematics)\|immersion]] and equivariant with respect to <math>\varphi</math>.

			A [[fundamental group]] of a compact complete flat affine manifold is called '''an affine [[crystallographic group]]'''. Classification of affine crystallographic groups is a difficult problem, far from being solved. The [[Space group\|Riemannian crystallographic groups]] (also known as [[Bieberbach group]]s) were classified by [[Ludwig Bieberbach]], answering a question posed by [[David Hilbert]]. In his work on [[Hilbert's eighteenth problem\|Hilbert's 18-th problem]], [[Space group#Bieberbach's theorems\|Bieberbach proved]] that any Riemannian crystallographic group contains an abelian subgroup of finite index.

			==Important longstanding conjectures==
			Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases.

			The most important of them are

			*[[Markus conjecture]] (1961) stating that a compact affine manifold is complete if and only if it has constant volume.<ref>Hirsch M. and Thurston W., ''Foliated bundles, invariant measures, and flat manifolds,'' Ann. Math. (2) 101, (1975) 369–390.</ref> Known in dimension 3.

			*[[Auslander conjecture]] (1964)<ref>Auslander L., ''The structure of locally complete affine manifolds,'' Topology 3 (1964), 131–139.</ref><ref>Fried D. and Goldman W., ''Three dimensional affine crystallographic groups'', Adv. Math. 47 (1983), 1–49.</ref> stating that any affine crystallographic group contains a [[polycyclic group\|polycyclic subgroup]] of finite [[Index of a subgroup\|index]]. Known in dimensions up to 6,<ref>H. Abels, G. A. Margulis and G. A. Soifer, ''On the Zariski closure of the linear part of a properly discontinuous group of affine transformations,'' J. Differential Geom., 60 (2002), 315344.</ref> and when the holonomy of the flat connection preserves a [[Lorentz metric]].<ref>William M. Goldman and Yoshinobu Kamishima, ''The fundamental group of a compact flat Lorentz space form is virtually polycyclic'', J. Differential Geom. Volume 19, Number 1 (1984)</ref> Since every virtually polycyclic crystallographic group preserves a volume form, Auslander conjecture implies the "only if" part of the Markus conjecture.<ref>Herbert Abels, ''Properly Discontinuous Groups of Affine Transformations: A Survey'', Geom. Dedicata, 87,. 309–333 (2001)</ref>

			*[[Chern conjecture (affine geometry)\|Chern conjecture]] (1955) The [[Euler class]] of an affine manifold vanishes.<ref>Konstant B., Sullivan D., ''The Euler characteristic of an affine space form is zero,'' Bull. Amer. Math. Soc. 81 (1975), no. 5, 937–938.</ref>

			==Notes==
			{{reflist}}

			==References==
			* {{citation \| last1=Nomizu\|first1=K.\|authorlink=Katsumi Nomizu\|last2=Sasaki\|first2=S. \| title = Affine Differential Geometry\| publisher=[[Cambridge University Press]] \| year=1994\|isbn=978-0-521-44177-3}}
			*{{cite book \| last = Sharpe \| first = R. W. \| title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program \| publisher = Springer \| location = New York \| year=1997 \| isbn=0-387-94732-9}}
			* {{citation \| last1=Bishop\|first1=R.L.\|last2=Goldberg\|first2=S.I. \| title = Tensor Analysis on Manifolds\| publisher=The Macmillan Company \| year=1968\|edition=First Dover 1980\|isbn=0-486-64039-6}}

			{{DEFAULTSORT:Affine Manifold}}
			[[Category:Group theory]]
			[[Category:Affine geometry]]
			[[Category:Structures on manifolds]]
			[[Category:Differential geometry]]
			[[Category:Manifolds]]

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