Local orthogonal maps and rigidity of holomorphic mappings between real hyperquadrics

We propose a coordinate-free approach to study the holomorphic maps between the real hyperquadrics in complex projective spaces. It is based on a notion of orthogonality on the projective spaces induced by the Hermitian structures that define the hyperquadrics. There are various kinds of special linear subspaces associated to this orthogonality which are well respected by the relevant holomorphic maps and we obtain rigidity theorems by analyzing the images of these linear subspaces, together with techniques in projective geometry. Our method allows us to recover and generalize a number of well-known results in the field with simpler arguments.