New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric

Abstract

On a real hypersurface M of a complex quadric we have an almost contact metric structure induced by the Kählerian structure of the ambient space. Therefore, on M we have the Levi-Civita connection \(\nabla \) and, for any non-null real number k, the so called kth generalized Tanaka Webster connection \({\hat{\nabla }}^{(k)}\). We introduce the notions of \(({\hat{\nabla }}^{(k)},\nabla )\)-Codazzi and \(({\hat{\nabla }}^{(k)},\nabla )\)-Killing normal Jacobi operator on such a real hypersurface and classify Hopf real hypersurface in a complex quadric whose normal Jacobi operators satisfy any of both conditions.

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Acknowledgements

First author is supported by MINECO-FEDER Project MTM 2016-78807-C2-1-P and second author by grant Proj. No. NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea. The authors thank the referees for valuable comments that have improved the paper.

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Correspondence to Juan de Dios Pérez.

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Communicated by Rosihan M. Ali.

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Pérez, J.d.D., Suh, Y.J. New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric. Bull. Malays. Math. Sci. Soc. 44, 891–903 (2021). https://doi.org/10.1007/s40840-020-00988-7

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Keywords

  • Complex quadric
  • Real hypersurface
  • Normal Jacobi operator
  • kth generalized Tanaka Webster connection
  • \(({\hat{\nabla }}^{(k)}, \nabla )\)-Codazzi normal Jacobi operator
  • \(({\hat{\nabla }}^{(k)}, \nabla )\)-Killing normal Jacobi operator

Mathematics Subject Classification

  • 53C15
  • 53B25