Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold $(M,\omega )$ is the canonical principal $\mathrm{S}\mathrm{p}(n,ℝ)$-subbundle ${\pi }_{𝐑}:𝐑\to M$ 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 $({𝐞}_1,\dots ,{𝐞}_n,{𝐟}_1,\dots ,{𝐟}_n)$ of tangent vectors at $p$ of the tangent vector space $T_p(M)$, satisfying

${\omega }_p({𝐞}_j,{𝐞}_k)={\omega }_p({𝐟}_j,{𝐟}_k)=0$ and ${\omega }_p({𝐞}_j,{𝐟}_k)={\delta }_{jk}$

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

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

• da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). .
• Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel .