View source for 3-O-alpha-D-mannopyranosyl-alpha-D-mannopyranose xylosylphosphotransferase

{{technical|date=June 2013}}
{{Orphan|date=April 2013}}

{{DISPLAYTITLE:(''G'', ''X'')-manifold}}

In mathematics, the concept of a '''{{math|(''G'', ''X'')}}-manifold''' generalizes several different kinds of [[manifold]]s, including [[Riemannian manifold]]s, [[affine manifold]]s, [[piecewise linear manifold]]s, etc.

==Definition==
Let {{math|''G''}} be a [[group (mathematics)|group]] [[Group action|acting]] on a manifold {{math|''X''}} via [[diffeomorphism]]s—i.e. for each <math>g\in G</math>, the map <math>x\mapsto gx</math> from {{math|''X''}} to itself is a diffeomorphism. A manifold {{math|''M''}} which satisfies the following conditions is called a '''{{math|(''G'', ''X'')}}-manifold''':<ref name="Thurston notes p27">{{harvnb|Thurston|2002|p=27}}</ref> 
# There exists an [[open cover]] <math>\{U_\alpha\}</math> of {{math|''M''}} and a family <math>\{\varphi_\alpha\colon U_\alpha \to V_\alpha\}</math> of diffeomorphisms taking each {{math|''U''<sub>''&alpha;''</sub>}} to an open subset {{math|''V''<sub>''&alpha;''</sub>}} of {{math|''X''}}.
# For each {{math|''U''<sub>''&alpha;''</sub>, ''U''<sub>''&beta;''</sub>}} with nonempty intersection, there exists a <math>g\in G</math> such that <math>gx=\phi_\alpha\circ\phi_\beta^{-1}(x)</math> for all <math>x\in V_\alpha\cap V_\beta</math>. In other words, viewing the elements of {{math|''G''}} as diffeomorphisms, each transition map <math>\phi_\alpha\circ\phi_\beta^{-1}\colon V_\alpha\cap V_\beta\to V_\alpha\cap V_\beta</math> is the restriction of an element of {{math|''G''}} to <math>V_\alpha\cap V_\beta</math>.

==Citations==
{{reflist}}

==References==
* {{cite web
 |url          =http://library.msri.org/books/gt3m/PDF/3.pdf
 |title        =The Geometry and Topology of 3-Manifold, Chapter 3
 |last         =Thurston
 |first        =William P.
 |authorlink   =William Thurston
 |date         =March 2002
 |format       =PDF
 |archiveurl   =http://web.archive.org/web/20120319160503/http://library.msri.org/books/gt3m/PDF/3.pdf
 |archivedate  =22 July 2012
 |ref          =harv
}}
* {{cite web
 |last        =Bonahon
 |first       =Francis
 |title       =Geometric structures on 3–manifolds
 |url         =http://www-bcf.usc.edu/~fbonahon/Research/Preprints/Handbook.pdf
 |date        =7 July 2000
 |format      =PDF
 |archiveurl  =
 |archivedate =
 |ref         =harv
}}

[[Category:Structures on manifolds]]


{{Differential-geometry-stub}}