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A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

by Muhittin Evren Aydin 1,† and Adela Mihai 2,*,†
1
Department of Mathematics, Firat University, 23000 Elazig, Turkey
2
Department of Mathematics and Computer Science, Technical University of Civil Engineering, Bucharest, 020396 Bucharest, Romania
*
Author to whom correspondence should be addressed.
The authors contributed equally to this work.
Mathematics 2020, 8(3), 444; https://doi.org/10.3390/math8030444
Received: 18 February 2020 / Revised: 12 March 2020 / Accepted: 13 March 2020 / Published: 19 March 2020
(This article belongs to the Section Mathematics and Computer Science)
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. View Full-Text
Keywords: Pythagorean formula; Golden ratio; Gauss curvature; space form Pythagorean formula; Golden ratio; Gauss curvature; space form
MDPI and ACS Style

Aydin, M.E.; Mihai, A. A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms. Mathematics 2020, 8, 444.

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