Infinitesimal and global rigidity and inflexibility of surfaces of revolution with flattening at the poles

Abstract

The subject of this article is one of the most important questions of classical geometry: the theory of bendings and infinitesimal bendings of surfaces. These questions are studied for surfaces of revolution and, unlike previous well-known works, we make only minimal smoothness assumptions (the class ) in the initial part of our study. In this class we prove local existence and uniqueness theorems for infinitesimal bendings. We then consider the analytic class and establish simple criteria for rigidity and inflexibility of compact surfaces. These criteria depend on the values of certain integer characteristics related to the order of flattening of the surface at its poles. We also show that in the nonanalytic situation there exist nonrigid surfaces with any given order of flattening at the poles.Bibliography: 22 titles.