Infinitesimal, first order bendings of smooth, convex surfaces of revolution subject to conic, sleeve-like constraints along the boundary

Abstract

We prove that on a closed, smooth, convex surface of revolution Φ, whose poles are not flattening points, there exists only a countable set of parallels γn. Each of these parallels cuts surface Φ into two parts so that one of the parts, $$\Phi _{\gamma _n }$$ , admits nontrivial, infinitesimal bendings in the process of which all the points of its boundary γn are displaced on a preassigned, conic sleeve Kα that is coaxial with the surface. The sequence of such parallels γn converges to parallel γ*, which has the following properties: 1) the tangent cone to surface Φ along γ* is orthogonal to sleeve Kα; 2) surface $$\Phi _{\gamma ^* }$$ , cut off from surface Φ by parallel γ*, has rigidity of first order in the indicated class of bendings.