In symplectic geometry, the symplectic frame bundle[1] of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic with respect to . In other words, an element of the symplectic frame bundle is a linear frame at point i.e. an ordered basis of tangent vectors at of the tangent vector space , satisfying
- and
for . For , each fiber of the principal -bundle is the set of all symplectic bases of .
The symplectic frame bundle , a subbundle of the tangent frame bundle , is an example of reductive G-structure on the manifold .
See also
editNotes
edit- ^ Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, p. 23, ISBN 978-3-540-33420-0
Books
edit- Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic Dirac Operators, Springer-Verlag, ISBN 978-3-540-33420-0
- da Silva, A.C., Lectures on Symplectic Geometry, Springer (2001). ISBN 3-540-42195-5. doi:10.1007/978-3-540-45330-7
- Maurice de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 3-7643-7574-4.