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

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.
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