We show that the theory of isothermic surfaces in E 3 -one of the oldest branches of differential geometry -can be reformulated within the modern theory of completely integrable (soliton) systems. This enables one to study the geometry of isothermic surfaces in E 3 by means of powerful spectral methods available in the soliton theory. Also the associated non-linear system is interesting in itself since it displays some unconventional soliton features and, physically, could be applied in the theory of infinitesimal deformations of membranes. * The work supported in part by the grants 566/2/91 GR 10 (KBN 2 0168 91 01) and PB 1274/P3/92/02 (KBN 2 2303 91 02).

In this paper, we present invariant recurrence relations for the completely integrable CP N −1 Euclidean sigma model in two dimensions defined on the Riemann sphere S 2 when its action functional is finite. We determine the links between successive projection operators, wave functions of the linear spectral problem, and immersion functions of surfaces in the su(N ) algebra together with outlines of the proofs. Our formulation preserves the conformal and scaling invariance of these quantities. Certain geometrical aspects of these relations are described. We also discuss the singularities of meromorphic solutions of the CP N −1 model and show that they do not affect the invariant quantities. We illustrate the construction procedure through the examples of the CP 2 and CP 3 models.

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In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the CP N −1 sigma model with finite action, defined in the Riemann sphere, are themselves solutions of the Euler-Lagrange equations for sigma models. On the other hand, we show that the Euler-Lagrange equations for surfaces immersed in the Lie algebra su(n), with conformal coordinates, that are extremals of the area functional subject to a fixed polynomial identity are exactly the Euler-Lagrange equations for sigma models. In addition to these differential constraints, the algebraic constraints, in the form of eigenvalues of the immersion functions, are treated systematically. The spectrum of the immersion functions, for different dimensions of the model, as well as its symmetry properties and its transformation under the action of the ladder operators are discussed. Another approach to the dynamics is given, i.e. description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model.