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, ${\displaystyle \omega }$ is a symplectic form, a non-degenerate closed exterior 2-form, on a ${\displaystyle C^{\infty }}$-manifold M), and ∇ is a symplectic torsion-free connection on ${\displaystyle M.}$^[1] (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 ${\displaystyle \Gamma _{jk}^{i}=0}$. 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.^[2]

Examples

For example, ${\displaystyle \mathbb {R} ^{2n}}$ with the standard symplectic form ${\displaystyle dx_{i}\wedge dy_{i}}$ has the symplectic connection given by the exterior derivative ${\displaystyle d.}$ Hence, ${\displaystyle \left(\mathbb {R} ^{2n},\omega ,d\right)}$ is a Fedosov manifold.

References

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

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