中心仿射度量的两类具有常截面曲率的闭凸超曲面  

Two Types of Closed Convex Hypersurfaces with Constant Cross-sectional Curvature for the Central Affine Metric

作　　者：李明[1] 左莹 LI Ming;ZUO Ying(Mathematical Sciences Research Center,Chongqing University of Technology,Chongqing 400054,China)

机构地区：[1]重庆理工大学数学科学研究中心,重庆400054

出　　处：《西南师范大学学报（自然科学版）》2023年第3期31-38,共8页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11871126)。

摘　　要：本文主要研究Minkowski空间的Laugwitz猜测.首先给出了参数化超曲面的中心仿射几何的直接描述,在此基础上刻画了Minkowski空间黎曼几何与其单位超球面的中心仿射几何的关系,将Laugwitz猜测等价描述为:闭凸超曲面的中心仿射几何截面曲率为常数必为椭球面的刚性问题.然后建立了超曲面中心仿射几何量与欧氏几何量的联系,由此说明欧氏空间凸超曲面n+1-仿射表面积正是该超曲面的中心仿射体积,进而运用关于仿射表面积的等周不等式及其取得等号的几何条件给出了Schneider定理的新证明.最后研究了Simon 3-形式模长的Laplace在常截面曲率条件下的表达式,应用极大值原理证明了具有常截面曲率且具有平行无迹Tchebychev算子的闭凸超曲面具有消失的Simon 3-形式,再根据结构方程证明了该超曲面为中心在原点的椭球面.This paper focuses on the Laugwitz conjecture for Minkowski spaces.Firstly,a direct description of the central affine geometry of a parametric hypersurface is given,based on which the relationship between the Riemannian geometry of Minkowski space and its central affine geometry of a unit hypersphere is inscribed,and the Laugwitz conjecture is equivalently described as a rigid problem where the curvature of the central affine geometry section of a closed convex hypersurface is constant and must be ellipsoidal.The connection between the central affine geometry of a hypersurface and the Euclidean geometry is then established,and it is shown that the n+1-affine surface area of a convex hypersurface in Euclidean space is the central affine volume of the hypersurface.Furthermore,a new proof of Schneider’s theorem is given by using the isoperimetric inequality about the affine surface area and the geometric conditions for obtaining the equal sign.At last,the expression of Laplace with Simon 3-form module length under the condition of constant section curvature is studied.By using the maximum principle,it is proved that a closed convex hypersurface with constant section curvature and parallel traceless Tchebychev operator has a vanishing Simon 3-form.Then,according to the structural equation,it is proved that the hypersurface is an ellipsoid whose center is at the origin.

关 键 词：Laugwitz猜测 中心仿射几何等周不等式 Tchebychev算子 椭球面 截面曲率 

分 类 号：O186.1[理学—数学]