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In this paper, we provide an invariant formulation of completely integrable$CP^{N-1}$ Euclidean sigma models in two dimensions defined on the Riemannsphere $S^2$. The scaling invariance is explicitly taken into account byexpressing all the equations in terms of projection operators. Properties ofthe 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.

Invariant formulation of surfaces associated with $CP^{N-1}$ models