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Pythagorean Submanifolds in Euclidean Spaces

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Abstract

In this paper we introduce the notion of a Pythagorean submanifold isometrically immersed into a Euclidean space. This definiton will be based on the shape operator of the submanifold. We prove that any Pythagorean submanifold in a Euclidean space is isoparametric, the principal curvatures along any parallel normal vector field being given in terms of the golden or conjugate golden ratios. In addition, we obtain that a Pythagorean submanifold of codimension 1 (resp. \(\ge 2\)) is totally umbilical (resp. pseudo-umbilical), classifying those submanifolds.

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Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Firat University, 23200, Elazig, Turkey

    Muhittin Evren Aydin

  2. Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396, Bucharest, Romania

    Adela Mihai

  3. Interdisciplinary Doctoral School, Transilvania University of Braşov, 500036, Brasov, Romania

    Adela Mihai

  4. Department of Mathematics, Izmir Democracy University, 35140, Karabaglar, Izmir, Turkey

    Cihan Ozgur

Authors
  1. Muhittin Evren Aydin
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  2. Adela Mihai
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  3. Cihan Ozgur
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Adela Mihai.

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Aydin, M.E., Mihai, A. & Ozgur, C. Pythagorean Submanifolds in Euclidean Spaces. Results Math 78, 211 (2023). https://doi.org/10.1007/s00025-023-01985-5

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  • DOI: https://doi.org/10.1007/s00025-023-01985-5

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