On the reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of christoffel symbols of the second kind under infinitesimal deformations of surfaces in the euclidean space E 3

Abstract

We investigate the problem of reconstruction of the variation of the metric tensor of a surface on the basis of a given variation of Christoffel symbols of the second kind under infinitesimal deformations of surfaces in the Euclidean space E 3 .In the theory of infinitesimal deformations of surfaces in the three-dimensional Euclidean space, of special interest is the question of conditions under which a tensor field δΓ ij h symmetric with respect to its subscripts, on the one hand, defines the variation of Christoffel symbols of the second kind and, on the other hand, enables one to reconstruct the variation δg ij of the metric tensor of a surface under a certain regular infinitesimal deformation, as well as the question of how arbitrary this procedure may be.To a certain extent, this problem is similar to the problem of reconstruction of the metric tensor of a surface on the basis of given Christoffel functions Γ ij h of the second kind symmetric with respect to subscripts, the original solution of which was given in [1, pp. 18 -20].The more general problem of conditions under which a space of affine connection (A n ) reduces to a Weyl space (W n ) was solved in [2] for a binary domain (n = 2). This problem generalizes the main problem of nonRiemann geometry posed by Eisenhart and Veblen in [3]. The problem considered in the present paper and the problems indicated above can be reduced to a system of partial differential equations with similar left-hand sides and substantially different right-hand sides. The difference lies in the fact that the variation of Christoffel symbols δΓ ij h of the second kind, in contrast to Christoffel symbols Γ ij h of the second kind, is a tensor under infinitesimal deformations of the first kind. Since all arguments presented below deal with the metric of a surface rather than with the surface itself, it is reasonable to speak of a family of surfaces with given regular metric. However, without loss of generality, we consider a certain specific surface of families in the proof of our results.We now present some definitions necessary for what follows and prove several important lemmas. For an analytic representation of the process of deformation of regular surfaces of the Euclidean space E 3 in the class C m (G), m ≥ 1, and the definition of deformation S t continuous in a parameter t , see [4, pp. 54, 55; 5]. Definition 1. Let S t be a regular family of surfaces that depends on a small parameter t (a deformation continuous in the parameter t ) and let r t (x 1 , x 2 , t) = r (x 1 , x 2 ) + t