On~infinitesimal deformations of surfaces of positive curvature with an isolated flat point

Z. D. Usmanov

Sbornik. Mathematics, Tome 12 (1970) no. 4, pp. 595-614

Résumé

In this paper we study infinitesimal deformations of convex pieces of surfaces with boundary. It is assumed that the surface has positive gaussian curvature $K>0$. We investigate infinitesimal deformations, subject on the boundary of the surface to the condition $\lambda\delta k_n+\mu\delta\tau_g=\sigma$, where $\delta k_n$ and $\sigma\tau_g$ are variations of the normal curvature and geodesic torsion of the boundary, $\lambda$ and $\mu$ are fixed known functions, and $\sigma$ an arbitrary given function. We establish necessary and sufficient conditions for the rigidity of the surface under these boundary conditions. Bibliography: 12 titles.