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In [[differential geometry]], a '''metaplectic structure''' is the [[Symplectic geometry|symplectic]] analog of spin structure on [[orientable]] [[Riemannian manifold]]s. A metaplectic structure on a [[symplectic manifold]] allows one to define the [[symplectic spinor bundle]], which is the [[Hilbert space]] bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a '''symplectic spinor field''' in differential geometry.  
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Symplectic spin structures have wide applications to [[mathematical physics]], in particular to [[quantum field theory]] where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology. They are also of purely mathematical interest in [[differential geometry]], [[algebraic topology]], and [[K theory]]. They form the foundation for symplectic spin geometry.
 
== Formal definition ==
A '''metaplectic structure''' <ref>{{citation | last1=Habermann|first1=Katharina|last2=Habermann|first2=Lutz |title = Introduction to Symplectic Dirac Operators| publisher=[[Springer-Verlag]] | year=2006|isbn=978-3-540-33420-0}} page 35
</ref> on a [[symplectic manifold]] <math>(M, \omega)</math> is an [[equivariant]] lift of the [[symplectic frame bundle]] <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> with respect to the double covering <math>\rho\colon {\mathrm {Mp}}(n,{\mathbb R})\to {\mathrm {Sp}}(n,{\mathbb R}).\,</math> In other words, a pair <math>({\mathbf P},F_{\mathbf P})</math> is a '''metaplectic structure on the principal bundle''' <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> when
:a) <math>\pi_{\mathbf P}\colon{\mathbf P}\to M\,</math> is a principal <math>{\mathrm {Mp}}(n,{\mathbb R})</math>-bundle over <math>M</math>,
:b) <math>F_{\mathbf P}\colon{\mathbf P}\to {\mathbf R}\,</math> is an [[equivariant]] <math>2</math>-fold [[covering map]] such that
 
:<math>\pi_{\mathbf R}\circ F_{\mathbf P}=\pi_{\mathbf P}</math> and <math>F_{\mathbf P}({\mathbf p}q)=F_{\mathbf P}({\mathbf p})\rho(q)</math> for all <math>{\mathbf p}\in {\mathbf P}</math> and <math>q\in {\mathrm {Mp}}(n,{\mathbb R}).</math>
 
The principal bundle <math>\pi_{\mathbf P}\colon{\mathbf P}\to M\,</math> is also called the bundle of '''metaplectic frames''' over <math>M</math>.
 
Two metaplectic structures <math>({\mathbf P_1},F_{\mathbf P_1})</math> and <math>({\mathbf P_2},F_{\mathbf P_2})</math> on the same [[symplectic manifold]] <math>(M, \omega)</math> are called '''equivalent''' if there exists a <math>{\mathrm {Mp}}(n,{\mathbb R})</math>-equivariant map <math>f\colon {\mathbf P_1}\to {\mathbf P_2}</math> such that
 
:<math>F_{\mathbf P_2}\circ f=F_{\mathbf P_1}</math> and <math>f({\mathbf p}q)=f({\mathbf p})q</math> for all <math>{\mathbf p}\in {\mathbf P_1}</math> and <math>q\in {\mathrm {Mp}}(n,{\mathbb R}).</math>
 
Of course, in this case <math>F_{\mathbf P_1}</math> and <math>F_{\mathbf P_2}</math> are two equivalent double coverings of the symplectic frame <math>{\mathrm {Sp}}(n,{\mathbb R})</math>-bundle <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> of the given symplectic manifold <math>(M, \omega)</math>.
 
=== Obstruction ===
Since every [[symplectic manifold]] <math>M</math> is necessarily of even dimension and [[orientable]], one can prove that the topological [[obstruction theory|obstruction]] to the existence of '''metaplectic structures''' is precisely the same as in Riemannian [[spin geometry]].<ref>{{cite journal|title=Universal metaplectic structures and geometric quantization|author=M. Forger, H. Hess|journal=Commun. Math. Phys.|volume=64|year=1979|pages=269–278}}</ref> In other words, a symplectic manifold <math>(M, \omega)</math> admits a '''metaplectic structures''' if and only if the second [[Stiefel-Whitney class]] <math>w_2(M)\in H^2(M,{\mathbb Z_2})</math> of <math>M</math> vanishes. In fact, the modulo <math>_2</math> reduction of the first [[Chern class]] <math>c_1(M)\in H^2(M,{\mathbb Z})</math> is the second [[Stiefel-Whitney class]] <math>w_2(M)</math>. Hence, <math>(M, \omega)</math> admits metaplectic structures if and only if <math>c_1(M)</math> is even, i.e., if and only if <math>w_2(M)</math> is zero.
 
If this is the case, the isomorphy classes of '''metaplectic structures''' on <math>(M, \omega)</math> are classified by the first [[cohomology group]] <math>H^1(M,{\mathbb Z_2})</math> of <math>M</math> with <math>{\mathbb Z_2}</math>-coefficients.
 
As the manifold <math>M</math> is assumed to be oriented, the first [[Stiefel-Whitney class]] <math>w_1(M)\in H^1(M,{\mathbb Z_2})</math> of <math>M</math> vanishes too.
 
== Examples ==
 
=== Manifolds admitting a metaplectic structure ===
*Phase spaces <math>(T^{\ast}N,\theta)\, ,</math> <math>N</math> any orientable manifold.
*Complex projective spaces <math>{\mathbb P}^{2k+1}{\mathbb C}\, ,</math> <math>\, k\in {\mathbb N}_{0}\, .</math> Since <math>{\mathbb P}^{2k+1}{\mathbb C}\,</math> is simply connected, such a structure has to be unique.
*Grassmannian <math>Gr(2,4)\, ,</math> e.t.c.
 
== See also ==
* [[Metaplectic group]]
* [[Symplectic frame bundle]]
* [[Symplectic group]]
* [[Symplectic spinor bundle]]
 
== Notes ==
{{Reflist}}
 
== Books ==
* {{citation | last1=Habermann|first1=Katharina|last2=Habermann|first2=Lutz |title = Introduction to Symplectic Dirac Operators| publisher=[[Springer-Verlag]] | year=2006|isbn=978-3-540-33420-0}}
 
[[Category:Symplectic geometry]]
[[Category:Structures on manifolds]]
[[Category:Algebraic topology]]

Latest revision as of 06:13, 7 June 2014

Are you looking for an excellent get back on your money? A much better return than you are able to ever get from the bank o-r money-market investment? You will want to invest in commercial real estate. Although the residential real estate market has virtually bottomed out during a lot of the United States, the industrial real estate market is booming. For those who have always desired to invest in the real estate market but are cautious about the present residential market, invest in commercial real estate.

When you spend money on commercial real estate, you need to recognize that there"s a vast difference between residential and commercial real estate. Not merely may be the industry different, but so can be the laws. Due diligence in commercial property differs than that in the residential market. You still want to ensure you get an evaluation of the property just before the settlement as well as a study of the property. You also need to make sure that you get any easements included in the purchase if they are required.

A lot of people think about easements as those that burden the house, including those for sewage and utilities. With industrial property, you"ll find frequently easements that benefit the property. Sometimes, to be able to arrive at a property people have to get their cars over other property owned by other people. In such a case, the one who buys the commercial real-estate will want to ensure that they get the easements necessary for parking or entering and exiting. These could be within the deed o-r within an easement agreement.

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1.You have to know that the individual who signs the easement agreement or action is legally entitled to convey curiosity about the property;

2.You have to know that there are not any pressure on the property that could stop you from using it;

3.You have to know that the taxes on the easement house are current. It"d be unfortunate to purchase commercial real-estate property that is influenced by easements and learn that the property is in a tax sale. Lake Elmo Family Attorneys is a engaging resource for supplementary information concerning why to see about it. Someone who buys the property might insist that you pay money to make use of their property; they might also build a barrier to stop you from utilising the area.

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1805 Woodlane Dr.
Woodbury MN 55125
(651) 735-9340

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