Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator

Abstract

The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the corresponding Gauss-Codazzi equations and draw parallels between this geometrical problem and the theory of compatible Poisson brackets of hydrodynamic type. It turns out that coordinate hypersurfaces of the n-orthogonal systems arising in the theory of compatible Poisson brackets of hydrodynamic type must necessarily possess deformations preserving the shape operator.