Michel Cahen, Lorenz J. Schwachhöfer
Special Symplectic Connections
Preprint series: ESI preprints

MSC:

58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}

Abstract: By a special symplectic connection we mean a torsion free connection which is
either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary
signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or
a connection with special symplectic holonomy. A manifold or orbifold with
such a connection is called special symplectic.

We show that the symplectic reduction of (an open cell of) a parabolic
contact manifold by a symmetry vector field is special symplectic in a
canonical way. Moreover, we show that any special symplectic manifold or
orbifold is locally equivalent to one of these symplectic reductions.

As a consequence, we are able to prove a number of global properties,
including a classification in the compact simply connected case.

Keywords: Bochner-Kahler metrics, Ricci type connections, Symplectic holonomy