In this chapter we discuss the theory of complex projective structures on compact surfaces and its connections with Teichmüller theory, 2-and 3-dimensional hyperbolic geometry, and representations of surface groups into PSL 2 (C). Roughly speaking, a complex projective structure is a type of 2-dimensional geometry in which Möbius transformations play the role of geometric congru-ences (this is ...

We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull number $k$ symmetric tensor field of type (1,1) $L$ on we can define a (1,2) $M$, $L^{(k)}_F$, related to study symmetry skewsymmetry $A^{(k)}_F$ associated shape operator$A$ $M$.