Abstract:
We consider rigid (in the class of analytic infinitesimal bendings) analytic surfaces with an isolated point of flattening and positive Gaussian curvature around this point. It is proved that such surfaces are rigid 'in the small' in the class C∞. The proof is based on the study of the asymptotic behaviour of the field of infinitesimal bending in a neighbourhood of the point of flattening and subsequent application of the techniques of the theory of generalized Cauchy–Riemann systems with a singularity in the coefficients.
\Bibitem{Usm96}
\by Z.~D.~Usmanov
\paper On Efimov surfaces that are rigid 'in the~small'
\jour Sb. Math.
\yr 1996
\vol 187
\issue 6
\pages 903--915
\mathnet{http://mi.mathnet.ru/eng/sm140}
\crossref{https://doi.org/10.1070/SM1996v187n06ABEH000140}
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Linking options:
https://www.mathnet.ru/eng/sm140
https://doi.org/10.1070/SM1996v187n06ABEH000140
https://www.mathnet.ru/eng/sm/v187/i6/p119
This publication is cited in the following 4 articles:
Meziani A., “Nonrigidity of a Class of Two Dimensional Surfaces with Positive Curvature and Planar Points”, Proc. Amer. Math. Soc., 141:6 (2013), 2137–2143
Meziani, A, “Infinitesimal bendings of high orders for homogeneous surfaces with positive curvature and a flat point”, Journal of Differential Equations, 239:1 (2007), 16
Meziani A., “Planar complex vector fields and infinitesimal bendings of surfaces with nonnegative curvature”, Recent Progress on Some Problems in Several Complex Variables and Partial Differential Equations, Contemporary Mathematics Series, 400, 2006, 189–201
Meziani, A, “Infinitesimal bendings of homogeneous surfaces with nonnegative curvature”, Communications in Analysis and Geometry, 11:4 (2003), 697
Citation in format AMSBIB
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