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A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms

Author

Listed:
  • Muhittin Evren Aydin

    (Department of Mathematics, Firat University, 23000 Elazig, Turkey
    The authors contributed equally to this work.)

  • Adela Mihai

    (Department of Mathematics and Computer Science, Technical University of Civil Engineering, Bucharest, 020396 Bucharest, Romania
    The authors contributed equally to this work.)

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.

Suggested Citation

  • Muhittin Evren Aydin & Adela Mihai, 2020. "A Note on Surfaces in Space Forms with Pythagorean Fundamental Forms," Mathematics, MDPI, vol. 8(3), pages 1-5, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:444-:d:334206
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