Pullback coherent and squeezed states and quantization

In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states and show that they satisfy some properties on an integral K$ddot{rm{a}}$hler manifold which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold and show that they satisfy similar properties. Finally we show a Berezin -type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$ where $U$ is an open set in ${mathbb C}P^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization holds for totally real submanifolds of real dimension $n$ of a general homogeneous K$ddot{rm{a}}$hler manifold of real dimension $2n$. In the appendix we review the Rawnsley and generalized Perelomov coherent states on ${mathbb C}P^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.