RKHS, Odzijewicz, Berezin and Fedosov-type quantizations on smooth compact manifolds

In this article we define Odzijewicz, Berezin and Fedosov-type quantization on compact smooth manifolds. The method is as follows. We embed the smooth manifold of real dimension n into CPⁿ (and in the Fedosov quantization case embed into any real 2n dimensional symplectic manifold). The pullback coherent states are defined in the usual way. In the Odzijewicz-type, Berezin-type quantization the Hilbert space of geometric quantization is the pullback by the embedding of the Hilbert space of geometric quantization of CPⁿ. In the Berezin case, the operators that are quantized are those induced from the ambient space. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Kohinoor Ghosh (where we had needed totally real embedding). The Fedosov-type quantization is carried out by restriction to the submanifold given by the embedding.