  Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the
associated projective space (the set of rank-one projections). Suppose that
$\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is
finite-dimensional, then every orthogonality preserving transformation of
${\mathcal P}(H)$ 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 ${\mathcal P}(H)$ 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.
