Orthogonality preserving bijective maps on real and complex projective spaces

Let F be the field of real numbers or the field of complex numbers and let D, E is an element of F-nxn be invertible matrices, n >= 3. The matrices D and E induce indefinite inner products on F-n. We study maps on the projective space P(F-n) that send D-orthogonal one-dimensional subspaces (elements of the projective space) to E-orthogonal one-dimensional subspaces. We prove that under the assumption of bijectivity such a map T preserves (D, E)-orthogonality if and only if it preserves (D, E)-orthogonality in both directions. In this case it is induced by a linear or conjugate-linear transformation on F-n that is (D, E)-unitary up to a multiplicative constant. The existence of (D, E)-unitary and (D, E)-antiunitary maps is discussed. We also give examples showing the indispensability of the dimension and the bijectivity assumption.