Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle 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

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.