New equations of infinitesimal deformations of surfaces in E 3

Abstract

We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3 .In the classical theory of surfaces in a three-dimensional Euclidean space, there is the well-known Bonnet theorem [1], which states that if the coefficients of two forms one of which is positive definite satisfy the main equations (the Gauss equations and Mainardi -Peterson -Codazzi equations), then, up to motion and mirror reflection, there exists a surface for which these forms are the coefficients of the first and the second quadratic forms.In the present paper, we obtain a new form of integrability conditions for the existence of a displacement vector of an infinitesimal deformation of a surface in E 3 . This form consists of equations analogous to the Gauss equations and Mainardi -Peterson -Codazzi equations in the classical theory of surfaces.In the three-dimensional Euclidean space E 3 , we consider a regular surface S of the class C k , k ≥ 3, that is homeomorphic to a planar, two-dimensional, simply connected domain with vector-parametric equation