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 Tr. Mat. Inst. Steklova, 2007, Volume 258, Pages 185–200 (Mi tm483)

Hyperbolic Carathéodory Conjecture

S. L. Tabachnikova, V. Yu. Ovsienkob

a Department of Mathematics, Pennsylvania State University
b Institut Camille Jordan, Université Claude Bernard Lyon 1

Abstract: A quadratic point on a surface in $\mathbb R\mathrm P^3$ is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the six-vertex and the four-vertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of the 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing the Tresse–Wilczynski theorem.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2007, 258, 178–193

Bibliographic databases:

UDC: 514.7
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Citation: S. L. Tabachnikov, V. Yu. Ovsienko, “Hyperbolic Carathéodory Conjecture”, Analysis and singularities. Part 1, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 258, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 185–200; Proc. Steklov Inst. Math., 258 (2007), 178–193

Citation in format AMSBIB
\Bibitem{TabOvs07} \by S.~L.~Tabachnikov, V.~Yu.~Ovsienko \paper Hyperbolic Carath\'eodory Conjecture \inbook Analysis and singularities. Part~1 \bookinfo Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday \serial Tr. Mat. Inst. Steklova \yr 2007 \vol 258 \pages 185--200 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm483} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2400530} \zmath{https://zbmath.org/?q=an:1163.53004} \elib{http://elibrary.ru/item.asp?id=9549689} \transl \jour Proc. Steklov Inst. Math. \yr 2007 \vol 258 \pages 178--193 \crossref{https://doi.org/10.1134/S0081543807030133} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-35148894665} 

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This publication is cited in the following articles:
1. Freitas B.R., Garcia R.A., “Inflection Points on Hyperbolic Tori of S-3”, Q. J. Math., 69:2 (2018), 709–728
2. Uribe-Vargas R., “On Projective Umbilics: a Geometric Invariant and An Index”, J. Singul., 17 (2018), 81–90
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