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Proceedings of the Steklov Institute of Mathematics

, Volume 258, Issue 1, pp 178–193 | Cite as

Hyperbolic Carathéodory conjecture

  • V. Ovsienko
  • S. Tabachnikov
Article

Abstract

A quadratic point on a surface in ℝP3 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.

Keywords

Normal Form STEKLOV Institute Quadratic Point Hyperbolic Surface Umbilic Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne cedexFrance
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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