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. an ordered basis ( e 1 , … , e n , f 1 , … , f n ) {\displaystyle ({\mathbf {e} }_{1},\dots ,{\mathbf {e} }_{n},{\mathbf {f} }_{1},\dots ,{\mathbf {f} }_{n})\,} of tangent vectors at p {\displaystyle p\,} of the tangent vector space T p ( M ) {\displaystyle T_{p}(M)\,}, satisfying

ω p ( e j , e k ) = ω p ( f j , f k ) = 0 {\displaystyle \omega _{p}({\mathbf {e} }_{j},{\mathbf {e} }_{k})=\omega _{p}({\mathbf {f} }_{j},{\mathbf {f} }_{k})=0\,} and ω p ( e j , f k ) = δ j k {\displaystyle \omega _{p}({\mathbf {e} }_{j},{\mathbf {f} }_{k})=\delta _{jk}\,}

for j , k = 1 , … , n {\displaystyle j,k=1,\dots ,n\,}. For p ∈ M {\displaystyle p\in M\,}, each fiber R p {\displaystyle {\mathbf {R} }_{p}\,} of the principal S p ( n , R ) {\displaystyle {\mathrm {Sp} }(n,{\mathbb {R} })}-bundle π R : R → M {\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,} is the set of all symplectic bases of T p ( M ) {\displaystyle T_{p}(M)\,}.

The symplectic frame bundle π R : R → M {\displaystyle \pi _{\mathbf {R} }\colon {\mathbf {R} }\to M\,}, a subbundle of the tangent frame bundle F M {\displaystyle \mathrm {F} M\,}, is an example of reductive G-structure on the manifold M {\displaystyle M\,}.

See also

Notes

Books

  • Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0
  • da Silva, A.C., , Springer (2001). ISBN 3-540-42195-5. doi:
  • Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.