In Fedosov's version of quantisation of functions on a symplectic manifold, the product is given in terms of a symplectic connection. I have looked through Fedosov's book in deformation quantisation, and can't find the second order term for the product of two functions (which will involve the curvature), though I can find other expressions to second order. Am I being very unobservant (very likely), or if there really is no expression given there, is there another place where I can find it?
Application: The commutation relations of quantum mechanics arise in constructing quantum theory in classical geometry. Now suppose that we construct quantum mechanics in a geometry that is already noncommutative? In particular, if we have a first-order in some parameter noncommutative algebra of functions, can writing quantum mechanics give information on a second order deformation?