RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold

In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension $n$ into ${\mathbb C}P^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of ${\mathbb C}P^n$. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space ${\mathbb C}P^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$.