Journal of Mathematical Sciences, Vol. 74, No. 3, 1995 BENDING OF SURFACES. PART II I. Ivanova-Karatopraklieva and I. Kh. Sabitov UDC 514.772.35 The first part of this work was published in 1991 [45]. According to the general scheme of the whole survey, outlined in the introduction (see [45]), the previous part was devoted to studies on bendings and infinitesimal bendings of surfaces in R a, with the surface classes set off with respect to the sign of the curvature. According to the same scheme, the part presented here is a review of studies on two special classes of surfaces: surfaces of revolution and polyhedrons. Section 7 is the result of the authors' joint work, while Section 8 was written by I. Kh. Sabitov alone. The references to the items of the first part of the survey are marked, for example, in the following way: i.l.2,I. 7. BENDINGS AND INFINITESIMAL BENDINGS OF SURFACES OF REVOLUTION The finding of infinitesimal bendings of surfaces of revolution is reduced to the solution of a certain system of ordinary differential equations. This leads to the fact that the surfaces of revolution occupy a remarkable position in the theory .of infinitesimal bendings because, in the first place, they represent a class of surfaces with a wide range of concrete properties of their infinitesimal bendings, in the second place, they are model surfaces which are used for verification of various hypotheses, and in the third place, they serve as a theoretical source of new hypotheses. 7.1. Equation of Infinitesimal Bendings of the Surfaces of Revolution The main method of investigating infinitesimal bendings of the surfaces of revolution was introduced by S. Cohn-Vossen in 1929 (see [154]). This method implies that the position vector r of the surface of revolution S as well as the fields zj of the infinitesimal deformation of order n, 1 < j < n, are refered to the movable frame {k, e(0), e'(O)}, where k is a unit vector directed along the rotational axis Oz, e(6) = cos 0i+ sin 0j is a unit vector located perpendicularly to the axis Oz, which describes a circle under alteration of the angle Of revolution 0 from 0 to 27r; here i, j, k are the standard notations of basis vectors along the axes Ox, Oy, Oz. In the general case, when the equation of meridian L is thought to have a parametric representation z = ~(r), p = r (1) (where p is the distance from the rotational axis), the position vector r is represented as r(r,e) = ~(r)k + r (2) and the fields zj are sought in the following form: zS = as( ~, e)k + ~j(r, e)e(e) + ~(r, e)d(e), 1 < j _ n. (3) Since the equations for determining the field Zl of the infinitesimal deformation of order n are linear ones, by expansion of the components zx(r, 0) in the Fourier series = dk)Cr) 'kt = = (4) JR JR R Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 8, Geometriya-1, 1993. 1072-3374/95/7403-0997512.50 9 Plenum Publishing Corporation 997