Invariant description of ℂℙ N−1 sigma models

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We propose an invariant formulation of completely integrable ℂℙN−1 Euclidean sigma models in two dimensions defined on the Riemann sphere S2. We explicitly take the scaling invariance into account by expressing all the equations in terms of projection operators, discussing properties of the operators projecting onto one-dimensional subspaces in detail. We consider surfaces connected with the ℂℙN−1 models and determine invariant recurrence relations, linking the successive projection operators, and also immersion functions of the surfaces.