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2010-06-30T09:25:36Z

Correct cap in header and/or general fixes.

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In mathematics, a '''Fedosov manifold''' is a [[symplectic manifold]] with a compatible torsionfree [[Connection (mathematics)|connection]], that is, a triple (''M'', ω, ∇), where (''M'', ω) is a [[symplectic manifold]] (i.e., ω is a symplectic form, a non-degenerate closed exterior 2-form, on a ''C''∞-manifold ''M''), and ∇ is a symplectic torsionfree connection on ''M''.<ref name="Fedosov">{{cite paper |last=Gelfand |first=I. |last2=Retakh |first2=V. |last3=Shubin |first3=M. |id={{ArXiv|dg-ga|9707024}} |title=Fedosov Manifolds |work=Preprint |year=1997 |accessdate=2009-10-27 }}</ref> (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 torsionfree connection. Cover the manifold with [[Darboux chart]]s and on each chart define a connection ∇ with Christoffel symbol <math>\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 Fedosov|Boris Vasilievich Fedosov]] gives a canonical [[deformation quantization]] of a Fedosov manifold.<ref>{{cite journal |first=B. V. |last=Fedosov |title=A simple geometrical construction of deformation quantization |journal=Journal of Differential Geometry |volume=40 |year=1994 |issue=2 |pages=213–238 |doi= |mr=1293654 }}</ref>

==References==
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[[Category:Mathematical physics]]

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en>SmackBot: Correct cap in header and/or general fixes.