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 $$\mathrm FM\,$$ 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 $$\mathrm FM\,$$, is an example of reductive G-structure on the manifold $$M\,$$.

Books

 * 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.
 * Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.