Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold formula_1 is the canonical principal formula_2-subbundle formula_3 of the tangent frame bundle formula_4 consisting of linear frames which are symplectic with respect to formula_5. In other words, an element of the symplectic frame bundle is a linear frame formula_6 at point formula_7 i.e. an ordered basis formula_8 of tangent vectors at formula_9 of the tangent vector space formula_10, satisfying
for formula_13. For formula_14, each fiber formula_15 of the principal formula_2-bundle formula_3 is the set of all symplectic bases of formula_10.
The symplectic frame bundle formula_3, a subbundle of the tangent frame bundle formula_4, is an example of reductive G-structure on the manifold formula_21. 