Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle 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

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