Invariant description of $\mathbb{CP}^{N-1}$ sigma models

We propose an invariant formulation of completely integrable $\mathbb P^{N-1}$ Euclidean sigma models in two dimensions defined on the Riemann sphere $S^2$. 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 $\mathbb P^{N-1}$ models and determine invariant recurrence relations, linking the successive projection operators, and also immersion functions of the surfaces.