  Let M be a manifold endowed with a symmetric affine connection $\Gamma.$ The
aim of this paper is to describe a quantization map between the space of
second-order polynomials on the cotangent bundle T^{*} M and the space of
second-order linear differential operators, both viewed as modules over the
group of diffeomorphisms and the Lie algebra of vector fields on M. This map is
an isomorphism, for almost all values of certain constants, and it depend only
on the projective class of the affine connection $\Gamma.$
