A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms - Aydin, Muhittin Evren; Mihai, Adela (Mdpi, 2020)
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 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.
Keywords: Pythagorean formula; Golden ratio; Gauss curvature; space form; Primary: 11C20; Secondary: 11E25; 53C24; 53C42