Orthogonality preserving transformations on complex projective spaces

It is well known that transformations of ${\mathbb{C}}^n$ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $\mathbb{C}{P}^n$ preserving $H$-orthogonality in both directions, called $H$-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $H$-orthogonal transformations on some other sets of $\mathbb{C}{P}^n$ were also determined.