On Isolated Umbilic Points
Abstract
Counterexamples to the famous conjecture of Caratheodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean 3space. In particular, Riemannian metrics with a smooth strictly convex 2sphere containing a single umbilic point are constructed explicitly, in contradiction with any direct extension of Caratheodory's conjecture. Additionally, a Riemannian metric with an embedded surface containing an isolated umbilic point of any index is presented, violating Hamburger's umbilic index bound. In both cases, it is shown that the metric can be made arbitrarily close to the flat metric.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.03562
 Bibcode:
 2018arXiv181203562G
 Keywords:

 Mathematics  Differential Geometry;
 53A05;
 53A35
 EPrint:
 14 pages LaTeX. Includes two Appendices with REDUCE computer algebra code to check calculations