Symplectic frame bundle
In symplectic geometry, the symplectic frame bundle[1] of a given symplectic manifold [math]\displaystyle{ (M, \omega)\, }[/math] is the canonical principal [math]\displaystyle{ {\mathrm {Sp}}(n,{\mathbb R}) }[/math]-subbundle [math]\displaystyle{ \pi_{\mathbf R}\colon{\mathbf R}\to M\, }[/math] of the tangent frame bundle [math]\displaystyle{ \mathrm FM\, }[/math] consisting of linear frames which are symplectic with respect to [math]\displaystyle{ \omega\, }[/math]. In other words, an element of the symplectic frame bundle is a linear frame [math]\displaystyle{ u\in\mathrm{F}_{p}(M)\, }[/math] at point [math]\displaystyle{ p\in M\, , }[/math] i.e. an ordered basis [math]\displaystyle{ ({\mathbf e}_1,\dots,{\mathbf e}_n,{\mathbf f}_1,\dots,{\mathbf f}_n)\, }[/math] of tangent vectors at [math]\displaystyle{ p\, }[/math] of the tangent vector space [math]\displaystyle{ T_{p}(M)\, }[/math], satisfying
- [math]\displaystyle{ \omega_{p}({\mathbf e}_j,{\mathbf e}_k)=\omega_{p}({\mathbf f}_j,{\mathbf f}_k)=0\, }[/math] and [math]\displaystyle{ \omega_{p}({\mathbf e}_j,{\mathbf f}_k)=\delta_{jk}\, }[/math]
for [math]\displaystyle{ j,k=1,\dots,n\, }[/math]. For [math]\displaystyle{ p\in M\, }[/math], each fiber [math]\displaystyle{ {\mathbf R}_p\, }[/math] of the principal [math]\displaystyle{ {\mathrm {Sp}}(n,{\mathbb R}) }[/math]-bundle [math]\displaystyle{ \pi_{\mathbf R}\colon{\mathbf R}\to M\, }[/math] is the set of all symplectic bases of [math]\displaystyle{ T_{p}(M)\, }[/math].
The symplectic frame bundle [math]\displaystyle{ \pi_{\mathbf R}\colon{\mathbf R}\to M\, }[/math], a subbundle of the tangent frame bundle [math]\displaystyle{ \mathrm FM\, }[/math], is an example of reductive G-structure on the manifold [math]\displaystyle{ M\, }[/math].
See also
- Metaplectic group
- Metaplectic structure
- Symplectic basis
- Symplectic structure
- Symplectic geometry
- Symplectic group
- Symplectic spinor bundle
Notes
- ↑ Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, p. 23, ISBN 978-3-540-33420-0
Books
- Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0
- da Silva, A.C., Lectures on Symplectic Geometry[yes|permanent dead link|dead link}}], Springer (2001). ISBN 3-540-42195-5.
- Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.
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