Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure

Abstract

It is known that if the Gaussian curvature function along each meridian on a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is decreasing, then the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . Such a surface is called a von Mangoldt’s surface of revolution . A surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) is called a generalized von Mangoldt surface of revolution if the cut locus of each point of θ − 1 (0) is empty or a subarc of the opposite meridian θ − 1 ( π ) . For example, the surface of revolution ( R 2 , dr 2 + m 0 ( r ) 2 dθ 2 ) , where m 0 ( x ) = x/ (1 + x 2 ) , has the same cut locus structure as above and the cut locus of each point in r − 1 ((0 , ∞ )) is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution ( R 2 , dr 2 + m ( r ) 2 dθ 2 ) to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature c, there exists a generalized von Mangoldt surface of revolution with the same total curvature c such that the Gaussian curvature function along a meridian is not monotone on [ a, ∞ ) for any a > 0 .