|
|
| Line 1: |
Line 1: |
| In [[symplectic geometry]], the '''symplectic frame bundle'''<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 23
| | Binder and Finisher Gritz from Holland Landing, has hobbies including house brewing, ganhando dinheiro na internet and texting. Discovers the charm in visiting spots throughout the planet, of late only returning from Vredefort Dome.<br><br>Stop by my webpage; [http://ganhardinheiropelainternet.comoganhardinheiro101.com ganhe dinheiro] |
| </ref> of a given [[symplectic manifold]] <math>(M, \omega)\,</math> is the canonical principal <math>{\mathrm {Sp}}(n,{\mathbb R})</math>-[[subbundle]] <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> of the [[tangent frame bundle]] <math>\mathrm FM\,</math> consisting of linear frames which are symplectic with respect to <math>\omega\,</math>. In other words, an element of the '''symplectic frame bundle''' is a linear frame <math>u\in\mathrm{F}_{p}(M)\,</math> at point <math>p\in M\, ,</math> i.e. an ordered basis <math>({\mathbf e}_1,\dots,{\mathbf e}_n,{\mathbf f}_1,\dots,{\mathbf f}_n)\,</math> of tangent vectors at <math>p\,</math> of the tangent vector space <math>T_{p}(M)\,</math>, satisfying
| |
| :<math>\omega_{p}({\mathbf e}_j,{\mathbf e}_k)=\omega_{p}({\mathbf f}_j,{\mathbf f}_k)=0\,</math> and <math>\omega_{p}({\mathbf e}_j,{\mathbf f}_k)=\delta_{jk}\,</math>
| |
| for <math>j,k=1,\dots,n\,</math>. For <math>p\in M\,</math>, each fiber <math>{\mathbf R}_p\,</math> of the principal <math>{\mathrm {Sp}}(n,{\mathbb R})</math>-bundle <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math> is the set of all symplectic bases of <math>T_{p}(M)\,</math>.
| |
| | |
| The '''symplectic frame bundle''' <math>\pi_{\mathbf R}\colon{\mathbf R}\to M\,</math>, a subbundle of the tangent frame bundle <math>\mathrm FM\,</math>, is an example of '''reductive''' [[G-structure]] on the manifold <math>M\,</math>.
| |
| | |
| ==See also==
| |
| * [[Metaplectic group]]
| |
| * [[Metaplectic structure]]
| |
| * [[Symplectic structure]]
| |
| * [[Symplectic geometry]]
| |
| * [[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}}
| |
| *da Silva, A.C., ''[http://www.springerlink.com/content/hq3au3baggr3/ Lectures on Symplectic Geometry]'', Springer (2001). ISBN 3-540-42195-5.
| |
| * Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.
| |
| | |
| | |
| [[Category:Symplectic geometry]]
| |
| [[Category:Structures on manifolds]]
| |
| [[Category:Algebraic topology]]
| |
| | |
| {{differential-geometry-stub}}
| |
| {{topology-stub}}
| |
Binder and Finisher Gritz from Holland Landing, has hobbies including house brewing, ganhando dinheiro na internet and texting. Discovers the charm in visiting spots throughout the planet, of late only returning from Vredefort Dome.
Stop by my webpage; ganhe dinheiro