Symplectic frame bundle
In symplectic geometry, the symplectic frame bundle of a given symplectic manifold ( M , ω ) {\displaystyle (M,\omega )\,} is the canonical principal S p ( n , R ) {\displaystyle {\mathrm {Sp} }(n,{\mathbb {R} })} -subbundle π R : R → M {\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,} of the tangent frame bundle F M {\displaystyle \mathrm {F} M\,} consisting of linear frames which are symplectic with respect to ω {\displaystyle \omega \,} . In other words, an element of the symplectic frame bundle is a linear frame u ∈ F p ( M ) {\displaystyle u\in \mathrm {F} _{p}(M)\,} at point p ∈ M , {\displaystyle p\in M\,,} i.e.