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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 differential form, closed, nondegenerate form, nondegenerate different ...
, 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 sympl ...
(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 subbundle ...
\pi_\colon\to M\, of the
tangent frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts natu ...
\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 var ...
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 differential form, closed, nondegenerate form, nondegenerate different ...
*
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 grou ...
* 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