Infinitesimal first- and second-order deformations of ribbed surfaces of revolution, preserving the normal curvature or geodesic torsion of the boundary parallel

Infinitesimal deformations of ribbed surfaces of revolution $S_n$ with preservation of the normal curvature $(A)$ or geodesic torsion $(B)$ of the boundary parallel are investigated. The following are proved: a convex surface $S_n$ is rigid under deformations $(A)$ and $(B)$; there are nonconvex surfaces $S_n$ that are nonrigid under deformations $(A)$ and $(B)$; any surface $S_n$ has second-order rigidity under deformations $(A)$; a surface $S_n$ that is nonrigid under these deformations.