A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

Muhittin Evren Aydin and Adela Mihai
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Muhittin Evren Aydin: Department of Mathematics, Firat University, 23000 Elazig, Turkey
Adela Mihai: Department of Mathematics and Computer Science, Technical University of Civil Engineering, Bucharest, 020396 Bucharest, Romania

Mathematics, 2020, vol. 8, issue 3, 1-5

Abstract: In the present note we introduce a Pythagorean-like formula for surfaces immersed into 3-dimensional space forms M 3 ( c ) of constant sectional curvature c = − 1 , 0 , 1 . More precisely, we consider a surface immersed into M 3 c satisfying I 2 + II 2 = III 2 , where I , II and III 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 φ + c , where φ is the Golden ratio.

Keywords: Pythagorean formula; Golden ratio; Gauss curvature; space form (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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