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Biconservative Submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\)

The Journal of Geometric Analysis volume 29, pages 283–298 (2019)Cite this article

Abstract

In this paper we study biconservative submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\) with parallel mean curvature vector field and codimesion 2. We obtain some sufficient and necessary conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of 3-dimensional biconservative submanifolds in \(\mathbb {S}^4\times \mathbb {R}\) and \(\mathbb {H}^4\times \mathbb {R}\) with nonzero parallel mean curvature vector field. We also get some results for biharmonic submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\).

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Acknowledgements

The third author gratefully thanks for the support from the National Post-doctoral Fellowship of Science and Engineering Research Board (SERB), Government of India.

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Authors and Affiliations

  1. Universidade de São Paulo, Av. Trabalhador São-carlense, 400, São Carlos, SP, 13566-590, Brazil

    F. Manfio

  2. Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University, 34469, Maslak, Istanbul, Turkey

    N. C. Turgay

  3. Department of Mathematics, Indian Institute of Science, CV Raman Road, Bengaluru, 560012, Karnataka, India

    A. Upadhyay

Authors
  1. F. Manfio
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  2. N. C. Turgay
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  3. A. Upadhyay
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Correspondence to A. Upadhyay.

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Manfio, F., Turgay, N.C. & Upadhyay, A. Biconservative Submanifolds in \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\). J Geom Anal 29, 283–298 (2019). https://doi.org/10.1007/s12220-018-9990-9

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  • DOI: https://doi.org/10.1007/s12220-018-9990-9

Keywords

  • Biconservative submanifolds
  • Biharmonic submanifolds
  • Product spaces \(\mathbb {S}^n\times \mathbb {R}\) and \(\mathbb {H}^n\times \mathbb {R}\)

Mathematics Subject Classification

  • Primary 53A10
  • Secondary 53C40, 53C42