  We discuss the construction of higher-dimensional surfaces based on the
harmonic maps of $S^2$ into $CP^{N-1}$ and other grassmannians. We show that
there are two ways of implementing this procedure - both based on the use of
the relevant projectors. We study various properties of such projectors and
show that the Gaussian curvature of these surfaces, in general, is not
constant. We look in detail at the surfaces corresponding to the Veronese
sequence of such maps and show that for all of them this curvature is constant
but its value depends on which mapping is used in the construction of the
surface.
