Weyl manifold and quantized connection

This paper is two folds. First we give a brief review of Weyl manifold and Deformation quantization. Poincaré–Cartan class is introduced and complete classification of Weyl manifolds is given. Second, Quantized connection or twisted exterior derivative is discussed. Degree operator field is introduced and plays an important role. Classical coordinate is constructed by means of the degree operator field. Quantized connection is then defined on Weyl manifold. In terms of classical coordinates the quantized connection is shown to be the same as Fedosov connection. Finally, we show the Poincaré–Cartan class is equal to the deRham cohomology class of the curvature of Fedosov connection.