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, [math]\displaystyle{ \omega }[/math] is a symplectic form, a non-degenerate closed exterior 2-form, on a [math]\displaystyle{ C^{\infty} }[/math]-manifold M), and ∇ is a symplectic torsion-free connection on [math]\displaystyle{ M. }[/math] (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 [math]\displaystyle{ \Gamma^i_{jk}=0 }[/math]. 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, [math]\displaystyle{ \R^{2n} }[/math] with the standard symplectic form [math]\displaystyle{ dx_i \wedge dy_i }[/math] has the symplectic connection given by the exterior derivative [math]\displaystyle{ d. }[/math] Hence, [math]\displaystyle{ \left(\R^{2n}, \omega, d\right) }[/math] is a Fedosov manifold.

References

• Esrafilian, Ebrahim; Hamid Reza Salimi Moghaddam (2013). "Symplectic Connections Induced by the Chern Connection". arXiv:1305.2852 [math.DG].