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, $\omega$ is a symplectic form, a non-degenerate closed exterior 2-form, on a $C^$-manifold ''M''), and ∇ is a symplectic torsion-free connection on $M.$ (A connection ∇ is called compatible or symplectic if ''X'' ⋅ ω(''Y,Z'') = ω(∇_''X''''Y'',''Z'') + ω(''Y'',∇_''X''''Z'') for all vector fields ''X,Y,Z'' ∈ Γ(T''M''). 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 $\Gamma^i_=0$. 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.

Examples

For example, $\R^$ with the standard symplectic form $dx_i \wedge dy_i$ has the symplectic connection given by the exterior derivative $d.$ Hence, $\left(\R^, \omega, d\right)$ is a Fedosov manifold.

References

Symplectic Connections Induced by the Chern Connection

Mathematical physics

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