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In
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
, the symplectic frame bundle of a given
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...
(M, \omega)\, is the canonical principal (n,)-
subbundle In mathematics, a subbundle U of a vector bundle V on a topological space X is a collection of linear subspaces U_xof the fibers V_x of V at x in X, that make up a vector bundle in their own right. In connection with foliation theory, a subbundl ...
\pi_\colon\to M\, of the tangent frame bundle \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 (n,)-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 In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes vari ...
on the manifold M\,.


See also

*
Metaplectic group In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field ...
* Metaplectic structure *
Symplectic basis In linear algebra, a standard symplectic basis is a basis _i, _i of a symplectic vector space, which is a vector space with a nondegenerate alternating bilinear form \omega, such that \omega(_i, _j) = 0 = \omega(_i, _j), \omega(_i, _j) = \delta_. A ...
*
Symplectic structure Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
*
Symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
*
Symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic g ...
* Symplectic spinor bundle


Notes


Books

* *da Silva, A.C.,
Lectures on Symplectic Geometry
', Springer (2001). . * Maurice de Gosson: ''Symplectic Geometry and Quantum Mechanics'' (2006) Birkhäuser Verlag, Basel . Symplectic geometry Structures on manifolds Algebraic topology {{differential-geometry-stub