symplectic frame bundle について

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symplectic frame bundle ：ウィキペディア英語版

symplectic frame bundle
In symplectic geometry, the symplectic frame bundle〔
〕 of a given symplectic manifold

(M, \omega)\,

is the canonical principal

)

\pi_\colon\to M\,

\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_(M)\,

at point

p\in M\, ,

i.e. an ordered basis

(_1,\dots,_n,_1,\dots,_n)\,

of tangent vectors at

p\,

of the tangent vector space

T_(M)\,

, satisfying
:

\omega_(_j,_k)=\omega_(_j,_k)=0\,

and

\omega_(_j,_k)=\delta_\,

for

j,k=1,\dots,n\,

. For

p\in M\,

, each fiber

_p\,

of the principal

)

-bundle

\pi_\colon\to M\,

is the set of all symplectic bases of

T_(M)\,

.
The symplectic frame bundle

\pi_\colon\to M\,

, a subbundle of the tangent frame bundle

\mathrm FM\,

, is an example of reductive G-structure on the manifold

M\,

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