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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2012, Volume 9, Pages 639–652
(Mi semr387)
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Geometry and topology
Around the A. D. Alexandrov's theorem on a characterization of a sphere
V. A. Aleksandrovab a Novosibirsk State University, Physics Department
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Abstract:
This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: Let $S$ be an analytic convex sphere-homeomorphic surface in $\mathbb R^3$ and let $k_1(\boldsymbol{x})\leqslant k_2(\boldsymbol{x})$ be its principal curvatures at the point $\boldsymbol{x}$. If the inequalities $k_1(\boldsymbol{x})\leqslant k\leqslant k_2(\boldsymbol{x})$ thold true with some constant $k$ for all $\boldsymbol{x}\in S$ then $S$ is a sphere. The imphases is on a result of Y. Martinez-Maure who first proved that the above statement is not valid for convex $C^2$-surfaces. For convenience of the reader, in addendum we give a Russian translation of that paper by Y. Martinez-Maure originally published in French in C. R. Acad. Sci., Paris, Sér. I, Math. 332 (2001), 41–44.
Keywords:
normal section, principal curvature, Weingarten surface, convex surface, herisson, virtual polytope.
Received September 10, 2012, published December 11, 2012
Citation:
V. A. Aleksandrov, “Around the A. D. Alexandrov's theorem on a characterization of a sphere”, Sib. Èlektron. Mat. Izv., 9 (2012), 639–652
Linking options:
https://www.mathnet.ru/eng/semr387 https://www.mathnet.ru/eng/semr/v9/p639
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Abstract page: | 218 | Full-text PDF : | 59 | References: | 35 |
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