On curvature of surfaces immersed in normed spaces

被引:2
作者
Balestro, Vitor [1 ]
Martini, Horst [2 ]
Teixeira, Ralph [1 ]
机构
[1] Univ Fed Fluminense, Inst Matemat & Estat, BR-24210201 Niteroi, RJ, Brazil
[2] Tech Univ Chemnitz, Fak Math, D-09107 Chemnitz, Germany
来源
MONATSHEFTE FUR MATHEMATIK | 2020年 / 192卷 / 02期
关键词
Alexandrov's theorem; Birkhoff-Gauss map; Finsler manifold; Minkowski curvature; Normed space; Relative differential geometry; Weyl's tube formula;
D O I
10.1007/s00605-020-01394-8
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. The relations of this setting to the field of relative differential geometry are clarified. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand-Diguet-Puiseux are derived. A generalization of Weyl's formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtained, and in addition we discuss also two-dimensional subcases of the results in more detail.
引用
收藏
页码:291 / 309
页数:19
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