A Fedosov manifold is a symplectic manifold with a symmetric connection which preserves the symplectic form. The Kaehler manifolds are a particular case when the connection is the Levi-Civita connection of a Riemannian metric which is related to the symplectic structure by an almost complex structure. Fedosov manifolds appear naturally as classical objects associated with the deformation quantization in the Fedosov quantization construction.
The talk will be based on joint work by I. M. Gelfand, V. Retakh and the speaker. The geometry of the Fedosov manifolds will be discussed. Unlike the usual symplectic geometry, this geometry has local invariants, in particular, the curvature tensor, Ricci curvature, and sectional curvature, which exhibit behavior different from that of the Riemannian case.
There are many open questions in the geometry of Fedosov manifolds. Some of these will be discussed in the talk.