QA-deformation of surface of negative Gaussian curvature

Abstract

An infinitesimal deformation with the given law of changing the element of area of a surface in Euclidean three-space was considered in this article. Such deformation in the article was called the quasiareal infinitesimal deformation or, briefly, the QA-deformation. The problem of finding the QA-deformation, under which the unit normal vector to the surface is preserved, was reduced to the study of one nonhomogeneous partial differential equation of the second order with respect to one unknown function. The initial conditions, under which the only one QA-deformation with the stationary unit normal vector exists, were defined for the surfaces of a negative Gaussian curvature. In this case, for the above equation, the Cauchy and Goursat problems were applied. The initial conditions of these tasks were expressed through the deforming vector.