Abstract:
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.
Citation:
Z. D. Usmanov, “On infinitesimal deformations of surfaces of positive curvature with an isolated flat point”, Math. USSR-Sb., 12:4 (1970), 595–614
\Bibitem{Usm70}
\by Z.~D.~Usmanov
\paper On~infinitesimal deformations of surfaces of positive curvature with an isolated flat point
\jour Math. USSR-Sb.
\yr 1970
\vol 12
\issue 4
\pages 595--614
\mathnet{http://mi.mathnet.ru/eng/sm3531}
\crossref{https://doi.org/10.1070/SM1970v012n04ABEH000940}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=273550}
\zmath{https://zbmath.org/?q=an:0219.53049|0219.53050}
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https://www.mathnet.ru/eng/sm3531
https://doi.org/10.1070/SM1970v012n04ABEH000940
https://www.mathnet.ru/eng/sm/v125/i4/p596
This publication is cited in the following 7 articles:
B. de Lessa Victor, Abdelhamid Meziani, “Infinitesimal bendings for classes of two-dimensional surfaces”, Complex Variables and Elliptic Equations, 69:1 (2024), 122
S. A. Abdymanapov, “Initial-boundary value problems for a class of homogeneous second-order elliptic systems on a plane with a singular point”, Journal of Mathematical Sciences, 164:4 (2010), 471–477
Z. D. Usmanov, “On Efimov surfaces that are rigid 'in the small'”, Sb. Math., 187:6 (1996), 903–915
A. Tungatarov, “Continuous solutions of a generalized Cauchy–Riemann system with a finite number of singular points”, Math. Notes, 56:1 (1994), 722–729
Z.D. Usmanov, “Generalized cauchy—riemann systems with a singular point”, Complex Variables, Theory and Application: An International Journal, 26:1-2 (1994), 41
A. Tungatarov, “On the theory of the Carleman–Vekua equation with a singular point”, Russian Acad. Sci. Sb. Math., 78:2 (1994), 357–365
Z. D. Usmanov, “On a problem concerning the deformation of a surface with a flat point”, Math. USSR-Sb., 18:1 (1972), 61–81
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