Fedosov manifold

In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that is, $\backslash omega$ is a symplectic form, a non-degenerate closed exterior 2-form, on a $C^\{\backslash infty\}$-manifold M), and ∇ is a symplectic torsion-free connection on $M.${{cite journal |last1=Gelfand |first1=I. |last2=Retakh |first2=V. |last3=Shubin |first3=M. |arxiv=dg-ga/9707024 |title=Fedosov Manifolds |url=https://archive.org/details/arxiv-dg-ga9707024 |journal=Preprint |year=1997 |bibcode=1997dg.ga.....7024G }} (A connection ∇ is called compatible or symplectic if X ⋅ ω(Y,Z) = ω(∇_XY,Z) + ω(Y,∇_XZ) for all vector fields X,Y,Z ∈ Γ(TM). In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol $\backslash Gamma^i\_\{jk\}=0$. Then choose a partition of unity (subordinate to the cover) and glue the local connections together to a global connection which still preserves the symplectic form. The famous result of Boris Vasilievich Fedosov gives a canonical deformation quantization of a Fedosov manifold.{{cite journal|last=Fedosov|first=B. V.|title=A simple geometrical construction of deformation quantization|journal=Journal of Differential Geometry|volume=40|year=1994|issue=2|pages=213–238|doi=10.4310/jdg/1214455536|mr=1293654|doi-access=free}}