Surfaces in projective differential geometry and integrable systems

E. V. Ferapontov


Abstract. Our aim here is to give a brief review of surfaces in projective 3-space with the emphasize on the nonlinear equations underlying them. As examples we have choosen isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas and projectively minimal surfaces. It is demonstrated, that the corresponding projective `Gauss-Codazzi' equations reduce to integrable systems which are familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in the equations of the Wilczynski moving frame.

AMSclassification. 53A40, 53A20, 35Q58, 35L60

Keywords. Projective differential geometry, integrable systems