On the Combinatorics of Inflexion Arches of Saddle Spheres

G. Panina

Journal for geometry and graphics, Tome 13 (2009) no. 1, pp. 59-73.

Résumé

Each saddle sphere $\Gamma \subset S^3$ is known to generate a spanning arrangement of at least four non-crossing oriented great semicircles on $S^2$. Each semicircle arises as the projection of an inflexion arch of the surface $\Gamma$. In the paper we prove the converse: each spanning arrangement of non-crossing oriented great semicircles is generated by some smooth saddle sphere. In particular, this means the diversity of saddle spheres on $S^3$. Recall that each $C^2$-smooth saddle sphere leads directly to a counterexample to the following conjecture of A. D. Alexandrov: \par {\it Let $K \subset \mathbb{R}^3$ be a smooth convex body. If, for a constant $C$, at every point of $\partial K$, we have $R_1 \leq C \leq R_2$, then $K$ is a ball ($R_1$ and $R_2$ stand for the principal curvature radii of $\partial K$).} \par In the framework of the conjecture, the main result of the paper means that all counterexamples can be classified by non-crossing arrangements of oriented great semicircles.

Détail

Classification : 53C45, 53A10
Mots-clés : Alexandrov's conjecture, inflexion point, inflexion arch, saddle surface, hyperbolic virtual polytope