Proceedings of the Steklov Institute of Mathematics

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

Hyperbolic Carathéodory conjecture

  • V. OvsienkoEmail author
  • S. Tabachnikov


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.


Normal Form STEKLOV Institute Quadratic Point Hyperbolic Surface Umbilic Point 
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© 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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