Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold $(M,\omega )$ is the canonical principal $\mathrm{S}\mathrm{p}(n,\mathbb{ℝ})$-subbundle ${\pi }_{\mathbf{𝐑}}:\mathbf{𝐑}\to M$ 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 $\mathrm{S}\mathrm{p}(n,\mathbb{ℝ})$-bundle ${\pi }_{\mathbf{𝐑}}:\mathbf{𝐑}\to M$ is the set of all symplectic bases of $T_p(M)$.

The symplectic frame bundle ${\pi }_{\mathbf{𝐑}}:\mathbf{𝐑}\to M$, 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.