Ovaloids of \(\mathbb{R}^3\) and their umbilics: A differential equation approach
The authors consider the following extension of the Carathéodory conjecture on umbilics: Let \(N:\mathbb{M} \to\mathbb{S}^2\) be the Gauss map of a \(C^3\)-ovaloid \(\mathbb{M}\subset \mathbb{R}^3\). Let \({\mathcal A}\) be the set of umbilics \(q\in \mathbb{M}\) such that \(N(q)=q\). If \(A\) is a connected component of \({\mathcal A}\), then each connected component of \(\mathbb{M}/A\) contains at least an umbilic of \(\mathbb{M}\). This is solved in two very special cases: 1) When the ovaloid is parametrized by a Bonnet chart such that the associated Bonnet function \(\beta: \mathbb{R}^2\to \mathbb{R}\) is the product \(\beta(x,y)= \varphi(x) \cdot\psi (y)\) of two functions \(\varphi,\psi: \mathbb{R}\to \mathbb{R}\) which have compact support. 2) In the case that a connected component of \({\mathcal A}\) is diffeomorphic to a disk, having regular boundary, with an additional generic condition which determines the behavior of the curvature lines near this boundary.
scientific article; zbMATH DE number 1556509
scientific article