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 ${\displaystyle M\,}$.