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

It is known that if the Gaussian curvature function along each meridian on a surface of revolution (R2, dr2 + m(r) 2 do2 ) is decreasing, then the cut locus of each point of 0-1(0) is empty or a subarc of the opposite meridian 0-1(4 Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution (]I ?2, dr2 + m(r)2 is called a generalized von Mangoldt surface of revolution if the cut locus of each point of 0-1(0) is empty or a subarc of the opposite meridian 0-1(7r).For example, the surface of revolution (R2, dr2 + mo(r)2d & theta;2), where mo(x) = x/(1 + x2), has the same cut locus structure as above and the cut locus of each point in r-1((0, oo)) 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 (R2, dr2 + m(7.)202) 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, oo) for any a > 0.