2006/11/21 by Valentin Ovsienko, Ovsienko, Valentin, Serge Tabachnikov +1
Mathematics · #53A20 #53C99 #58K50 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #History and Theory of Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53A20 #msc:53C99 #msc:58K50
paper · pdf · doi:10.48550/arxiv.math/0611630
Latex 25 pages, 10 figures
arxiv created 2006/11/21 · openalex publication_date 2006/11/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 non-degenerate 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 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing Tresse-Wilczynski's theorem.