Manhart相对法向量场的一个特征
A Characterisation of Manhart's Relative Normal Vector Fields
摘要：
本文给出了欧氏空间中曲面的曲率泛函与相对微分几何中面积泛函之间的关系。相对微分几何可以描述为仿射空间中曲面的几何，赋予了一个独特的“相对法向量场”，它推广了欧几里得微分几何中单位法向量场N的概念。一个相对微分几何的简明回顾将被提出。第三部分将给出本文标题所指的主要结果。在这里，我们认为，对于一个功能$f$的两个变量，相对法向量字段的形式$f(H,K)\,N-\grad_{\II}(f(H,K))$的非退化表面在欧几里德三维空间。通过比较曲率泛函$\int f(H,K)\,\ddΩ$的变化与由上述相对法向量场得到的相对面积泛函，得到了F. Manhart，并通过选择$f(H,K)=|K|^α$（其中我们将假设$α\neq 1$）获得。更确切地说，下面将在定理6中示出："{临界点与关于相对法向量场$(\dagger)$的相对极小曲面重合的曲率泛函$(\ast)$，本质上是从Manhart族中得到的那些曲率泛函}。在第四部分中，我们利用支撑函数与曲率之间的关系给出了球面的一个特征。在最后一节中，我们结合前面描述的结果，并在一个变分表征的领域。
英文摘要：
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" which generalises the notion of unit normal vector field N from Euclidean differential geometry. A concise review of relative differential geometry will be presented. The main result, to which the title of this article refers, will be given in the third section. Here we consider, for a function $f$ of two variables, relative normal vector fields of the form $f(H,K)\,N-\grad_{\II}(f(H,K))$ for non-degenerate surfaces in the Euclidean three-dimensional space. A comparison of the variation of the curvature functional $\int f(H,K)\,\ddΩ$ with the relative area functional obtained from the above relative normal vector field, results in a distinguishing property for the one-parameter family of relative normal vector fields which was introduced by F.\ Manhart, and which is obtained by choosing $f(H,K)=|K|^α$ (where we will assume that $α\neq 1$). More precisely, the following will be shown in theorem~6: ``{The curvature functionals $(\ast)$ for which the critical points coincide with the relative-minimal surfaces with respect to the relative normal vector field $(\dagger)$, are essentially those obtained from Manhart's family}." In the fourth section, we give a characterisation of the sphere by means of relations between the support function and the curvatures. In the last section, we combine the previously described results and arrive at a variational characterisation of the sphere.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
