Classical solutions of (P’~ non linear a -models; an algebraic geometrical description* R. CATENACCI Dipartimento di Matematica dell’Università Via Strada Nuova 65, Pavia (Italy) INFN, Section of Pavia M. CORNALBA Dipartimento di Matematica dell’Universiti Via Strada Nuova 65, Pavia (Italy) C. REINA t)jpaitimento di Fisica dell’UniversitI Via Celoria 16, Milano (Italy) Abstract. The classical SO (3)-invariant u-modeland its suitably generalized versions are studied from the geometrical point of view. Known mathematical results con- cerning harmonic and holomorphic maps of a Riemann surface into the n-dimensio- nal complex pro jective space are briefly reviewed. These are used to give a classifi- cation of classical solutions (both instantons and a certain subclass of unstable solu- tions) of 62~fl models and to study the properties of their energy spectrum. 1. INTRODUCTION 1.1. In recent years, several theories and models have been proposed in physics, which were realized to be rich in geometrical meaning. Among these, some of the most important to physics seem to be at present the Yang-Mills gauge theories, for which the dynamical field can be regarded as a connection on a principal fibre bundle [1]. For Yang-Mills fields, this geometrical interpretation has been far from a mere (*) Work partially supported by Gruppo Nazionale di Fisica Matematica, C.N.R. and Grup- p0 Nazionale per le Strutture Algebriche e Geometriche e Applicazioni, C.N.R.