具有简单割迹结构的广义vonMangoldt旋转曲面和非对称双旋转球
Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure
摘要：
众所周知，如果高斯曲率函数沿着旋转表面$(R^2, dr^2+m(r)^2dθ^2)$上的每个子午线减小，则$θ^{-1}(0)$的每个点的割迹是空的或者是相对子午线$θ^{-1}(π).$的子弧。这样的表面被称为冯曼戈尔德旋转表面。旋转曲面$(R^2, dr^2+m(r)^2dθ^2)$称为广义冯曼戈尔特旋转曲面，如果$θ^{-1}(0)$的每个点的割迹是空的或相反子午线$θ^{-1}(π).$的子弧。例如，旋转曲面$(R^2, dr^2+m_0(r)^2dθ^2),$，其中$m_0(x):=x/(1+x^2),$具有与上述相同的割迹结构，并且$r^{-1}( (0, \infty ) )$中每个点的割迹是非空的。请注意，高斯曲率函数不会沿此曲面的子午线减小。本文给出了旋转曲面$(R^2, dr^2+m(r)^2dθ^2)$是广义vonMangoldt旋转曲面的充分条件。证明了对任意有限总曲率旋转曲面$c,$，存在一个具有相同总曲率$c$的广义von Mangoldt旋转曲面，使得沿子午线的高斯曲率函数在$[a,\infty)$上不单调，对任意$a>0.$
英文摘要：
It was known that if the Gaussian curvature function along each meridian on a surface of revolution $(R^2, dr^2+m(r)^2dθ^2)$ 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 $(R^2, dr^2+m(r)^2dθ^2)$ 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 $(R^2, dr^2+m_0(r)^2dθ^2),$ where $m_0(x):=x/(1+x^2),$ has the same cut locus structure as above and the cut locus of each point in $r^{-1}( (0, \infty ) )$ 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 $(R^2, dr^2+m(r)^2dθ^2)$ 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 $[a,\infty)$ for any $a>0.$
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