Geometric version of Wigner's theorem for Hilbert Grassmannians

Let H be a complex Hilbert space of dimension not less than 3 and let Gk(H) be the Grassmannian formed by k-dimensional subspaces of H. Suppose that dim⁡H≥2k>2. We show that the transformations of Gk(H) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality, adjacency and ortho-adjacency relations). As a consequence, we get the following: if the dimension of H is finite and greater than 2k, then every transformation of Gk(H) preserving the orthogonality relation in both directions is a bijection induced by a unitary or anti-unitary operator.