A new family of latitudinally corrugated two-spheres of revolution with simple cut locus

TL;DR

This paper introduces a new family of latitudinally corrugated 2-spheres of revolution with simple cut locus structures, expanding the known examples beyond classical surfaces like ellipsoids and tori.

Contribution

The paper presents a novel family of 2-spheres of revolution with simple cut locus structures, where the number of extremal points of Gaussian curvature increases with n.

Findings

New family of 2-spheres with simple cut locus structures

Number of extremal points of Gaussian curvature increases with n

Expands known examples of surfaces with determined cut locus structures

Abstract

There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids, and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family {M_n}_n of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or maximum of the Gaussian curvature function on the meridian goes to infinity as n goes to infinity.