Quantization of forms on the cotangent bundle

We consider the following construction, of quantization. For a Riemannian manifold M the space of forms on T*M is made into a. space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the ("super")commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenbock like identities appear here.) Formulae for the traces corresponding to natural gradings of Omega (T*M) are established. Using these formulae, we give a simple direct proof of the Gauss-Bonnet-Chern Theorem. We discuss these results in connection with a general question of the quantization of forms on a Poisson manifold.