A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

被引：7

作者：

Aydin, Muhittin Evren ^{[1
]}

Mihai, Adela ^{[2
]}

机构：

[1] Firat Univ, Dept Math, TR-23000 Elazig, Turkey

[2] Tech Univ Civil Engn, Dept Math & Comp Sci, Bucharest 020396, Romania

来源：

MATHEMATICS | 2020年 / 8卷 / 03期

关键词：

Pythagorean formula; Golden ratio; Gauss curvature; space form; Primary: 11C20; Secondary: 11E25; 53C24; 53C42;

D O I：

10.3390/math8030444

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In the present note we introduce a Pythagorean-like formula for surfaces immersed into 3-dimensional space forms mml:semantics M3(c)mml:semantics of constant sectional curvature mml:semantics c=-1,0,1.mml:semantics> More precisely, we consider a surface immersed into <mml:semantics M3 mml:mfenced open="(" close=")">c mml:mfenced /mml:semantics satisfying mml:semantics 2+II2=III2,mml:semantics where mml:semanticsImml:semantics, mml:semanticsIImml:semantics and mml:semantics>IIImml:semantics> are the matrices corresponding to the first, second and third fundamental forms of the surface, respectively. We prove that such a surface is a totally umbilical round sphere with Gauss curvature mml:semantics phi +c,mml:semantics where mml:semantics phi mml:semantics is the Golden ratio.

引用

页数：5

共 13 条

[1] A new proof of Liebmann classical rigidity theorem for surfaces in space forms [J].