# Fedosov Manifolds

### Mikhail Shubin

### Northeastern University

### Thursday, March 5, 1998

4:00 PM

### Room 102, Bradley Hall

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.