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We introduced a new coordinate-free approach to study the Cauchy-Riemann (CR)maps between the real hyperquadrics in the complex projective space. Thecentral theme is based on a notion of orthogonality on the projective spaceinduced by the Hermitian structure defining the hyperquadrics. There arevarious kinds of special linear subspaces associated to this orthogonalitywhich are well respected by the relevant CR maps and this is where therigidities come from. Our method allows us to generalize a number of well-knownrigidity theorems for the CR mappings between real hyperquadrics with muchsimpler arguments.

Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics