Minoru Tanaka, Toyohiro Akamatsu, Robert Sinclair, Masaru Yamaguchi

Tokyo J. Math. Advance Publication, (2022) DOI: 10.3836/tjm/1502179366
KEYWORDS: 53C22, cut point, half period function, simple cut locus structure, surface of revolution

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 ${\left\{{M}_{n}\right\}}_{n}$ 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 $n$ tends to infinity.

Thus, our family includes surfaces which have arbitrarily many bands of alternately increasing or decreasing Gaussian curvature, although each member of this family has a simple cut locus structure.