Symplectic frame bundle

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

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

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

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

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. doi:10.1007/978-3-540-45330-7
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.