Unitarily invariant metrics on the Grassmann space

Let G(m,n) be the Grassmann space of m-dimensional subspaces of F-n. Denote by theta(1)(X, Y),...,.m( X, Y) the canonical angles between subspaces X, Y is an element of G(m,)n. It is shown that Phi(theta(1)(X, Y),...,.theta(m)(X, Y)) defines a unitarily invariant metric on G(m,n) for every symmetric gauge function F. This provides a wide class of new metrics on G(m,n). Some related results on perturbation and approximation of subspaces in G(m,n), as well as the canonical angles between them, are also discussed. Furthermore, the equality cases of the triangle inequalities for several unitarily invariant metrics are analyzed.