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, is a symplectic form, a non-degenerate closed exterior 2-form, on a -manifold M), and ∇ is a symplectic torsion-free connection on [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 . 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

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For example,   with the standard symplectic form   has the symplectic connection given by the exterior derivative   Hence,   is a Fedosov manifold.

References

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  1. ^ Gelfand, I.; Retakh, V.; Shubin, M. (1997). "Fedosov Manifolds". Preprint. arXiv:dg-ga/9707024. Bibcode:1997dg.ga.....7024G.
  2. ^ Fedosov, B. V. (1994). "A simple geometrical construction of deformation quantization". Journal of Differential Geometry. 40 (2): 213–238. doi:10.4310/jdg/1214455536. MR 1293654.
  • Esrafilian, Ebrahim; Hamid Reza Salimi Moghaddam (2013). "Symplectic Connections Induced by the Chern Connection". arXiv:1305.2852 [math.DG].