论文标题
革命的革命和不对称的两次革命的von Mangoldt表面，简单切割基因座结构
Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure
论文作者
论文摘要
众所周知，如果在革命$（r^2，dr^2+m（r）^2dθ^2）$沿每个子午线上的高斯曲率函数正在减少，那么每个点的切割基因座的缩短基因座是空的，则是空的或相反的Meridian $ thece $ theme $ theme sede sepertion Assepers。革命。革命的表面$（r^2，dr^2+m（r）^2dθ^2）$称为革命的广义von mangoldt革命表面，如果每个点的切割基因座的$θ^{ - 1}（-1}（0）$是空的，或者是相反的子午线$θ^{ - 1}（-1}（ - 1}（π）。 dr^2+m_0（r）^2dθ^2），$其中$ m_0（x）：= x/（1+x^2），$具有与上述相同的切割基因座结构，并且每个点的切割基因座和$ r^{ - 1}（（（0，\ infty））中的每个点的切割基因座是非空的。请注意，对于该表面，高斯曲率函数并没有沿着子午线降低。在本文中，我们为革命的表面提供了足够的条件$（r^2，dr^2+m（r）^2dθ^2）$成为革命的广义von Mangoldt表面。此外，我们证明，对于有限的总曲率$ c的任何革命表面，存在着广义的von mangoldt革命表面，具有相同的总曲率$ c $，因此，沿着子午线的高斯曲率功能并不是$ [a，\ infty）$。
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 $a>0.$
