A geometric approach to Wigner-type theorems

Let H be a complex Hilbert space and let P(H) be the associated projective space (the set of rank-one projections). Suppose dimH > 3. We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P(H) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P(H) to itself is induced by a linear or conjugate-linear isometry (H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.