Infinitesimal bendings of rotational surfaces with changing signs curvature

Abstract

The set of Liebmann's parallels of first order on a non-rigid rotational surfaces $S$ with changing signs curvature $K$ is investigates. $S$ is closed (of genus 0 or 1) or with a boundary. It is proved that there is a countable set of Liebmann's parallels on $S$ outside of its parts which are circular cylinders if $S$ has got an infite number non-trivial fundamental fields of bending. On each belt with $K<0$ these parallels are everywhere densely. On each belt $S_0=S_{L_0{L}_1}$ with $K\ge 0$, bordered by an asymptotic parallel $L_0$, there exist Liebmann's parallels if and only if $S_0$ contains a subbelt $\widehat{S_0}=S_{L\ast L_1}$($L^{\ast }$ is the most right maximal parallel of $S_0$). The Liebmann's parallels a subblet $\widehat{S_0}=S_{L\ast L_1}$($L^{\ast }$ is the most right maximal parallel of $S_0$). The Liebmann's parallels on $S_0$ are a countable set, belong to $\widehat{S_0}$ and are condensed to $L^{\ast }$. Some sufficient conditions for rigidity of $S$ are given.