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.
|Original language||English (US)|
|Number of pages||16|
|Journal||Proceedings of the Steklov Institute of Mathematics|
|State||Published - Sep 2007|
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)