  Being motivated by the orthogonal maps studied in \cite{GN1}, orthogonal
pairs between the projective spaces equipped with possibly degenerate Hermitian
forms were introduced. In addition, orthogonal pairs are generalizations of
holomorphic Segre maps between Segre families of real hyperquadrics. We showed
that non-degenerate holomorphic orthogonal pairs also have certain rigidity
properties under a necessary codimension restriction. As an application, we got
a rigidity theorem for Segre maps related to Heisenberg hypersurfaces obtained
by Zhang in \cite{Zh}.
