Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 837-845

EQUATORS HAVE AT MOST COUNTABLE MANY SINGULARITIES WITH BOUNDED TOTAL ANGLE

Pilar Herreros, Mario Ponce and J. J. P. Veerman

Pontificia Universidad Católica de Chile, Facultad de Matemáticas
Avda. Vicuña Mackenna 4860, Santiago, Chile; pherrero 'at' mat.puc.cl

Pontificia Universidad Católica de Chile, Facultad de Matemáticas
Avda. Vicuña Mackenna 4860, Santiago, Chile; mponcea 'at' mat.puc.cl

Portland State University, Department of Mathematics and Statistics, and
CCQCN, Dept. of Physics, University of Crete, 71003 Heraklion, Greece; eerman@pdx.edu

Abstract. For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix L_pq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. In the case of a topological sphere, mediatrices are called equators and it can be noticed that there are no branching points, thus an equator is a topological circle with possibly many Lipschitz singularities. This paper establishes that mediatrices have the radial linearizability property. This is a regularity property that implies that at each singular or branching point mediatrices have a geometrically defined derivative in each direction. In the case of equators we show that there are at most countably many singular points and the sum of the angles over all singularities is always finite.

2010 Mathematics Subject Classification: Primary 53C22; Secondary 53A35.

Key words: Mediatrix, equator, bisectors.

Reference to this article: Pilar Herreros, Mario Ponce and J. J. P. Veerman: Equators have at most countable many singularities with bounded total angle. Ann. Acad. Sci. Fenn. Math. 42 (2017), 837-845.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4251