  In this paper, we provide an invariant formulation of completely integrable
$CP^{N-1}$ Euclidean sigma models in two dimensions defined on the Riemann
sphere $S^2$. The scaling invariance is explicitly taken into account by
expressing all the equations in terms of projection operators. Properties of
the projectors mapping onto one-dimensional subspaces are discussed in detail.
The paper includes a discussion of surfaces connected with the $CP^{N-1}$
models and the wave functions of their linear spectral problem.
