Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle^[1] of a given symplectic manifold $(M,\omega )\,$ is the canonical principal ${{\mathrm {Sp}}}(n,{{\mathbb R}})$-subbundle $\pi _{{{\mathbf R}}}\colon {{\mathbf R}}\to M\,$ of the tangent frame bundle ${\mathrm F}M\,$ consisting of linear frames which are symplectic with respect to $\omega\,$. In other words, an element of the symplectic frame bundle is a linear frame $u\in {\mathrm {F}}_{{p}}(M)\,$ at point $p\in M\,,$ i.e. an ordered basis $({{\mathbf e}}_{1},\dots ,{{\mathbf e}}_{n},{{\mathbf f}}_{1},\dots ,{{\mathbf f}}_{n})\,$ of tangent vectors at $p\,$ of the tangent vector space $T_{{p}}(M)\,$, satisfying

$\omega _{{p}}({{\mathbf e}}_{j},{{\mathbf e}}_{k})=\omega _{{p}}({{\mathbf f}}_{j},{{\mathbf f}}_{k})=0\,$ and $\omega _{{p}}({{\mathbf e}}_{j},{{\mathbf f}}_{k})=\delta _{{jk}}\,$

for $j,k=1,\dots ,n\,$. For $p\in M\,$, each fiber ${{\mathbf R}}_{p}\,$ of the principal ${{\mathrm {Sp}}}(n,{{\mathbb R}})$-bundle $\pi _{{{\mathbf R}}}\colon {{\mathbf R}}\to M\,$ is the set of all symplectic bases of $T_{{p}}(M)\,$.

The symplectic frame bundle $\pi _{{{\mathbf R}}}\colon {{\mathbf R}}\to M\,$, a subbundle of the tangent frame bundle ${\mathrm F}M\,$, is an example of reductive G-structure on the manifold $M\,$.

See also

• Metaplectic group
• Metaplectic structure
• Symplectic basis
• Symplectic structure
• Symplectic geometry
• Symplectic group
• Symplectic spinor bundle

Notes

1. 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, Springer (2001). ISBN 3-540-42195-5.
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.