Vari\'et\'es CR polaris\'ees et G-polaris\'ees, partie I

Polarized and $G$-polarized CR manifolds are smooth manifolds endowed with a double structure: a real foliation $\text{Cal}F$ (given by the action of a Lie group $G$ in the $G$-polarized case) and a transverse CR distribution $(E,J)$. Polarized means that $(E,J)$ is roughly speaking invariant by $\text{Cal}F$. Both structures are therefore linked up. The interplay between them gives to polarized CR-manifolds a very rich geometry. In this paper, we study the properties of polarized and $G$-polarized manifolds, putting special emphasis on their deformations.