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Open AccessOpen Access JUSTC Mathematics 11 May 2022

Inequalities of warped product submanifolds in a Riemannian manifold of quasi-constant curvature

  • School of Mathematics and Physics, Anhui University of Technology, Maanshan 243002, China

Funds:  Supported by NSFC (No.12026262)
Cite this:
https://doi.org/10.52396/JUSTC-2021-0217
More Information
  • Author Bio:

    Jiahui Wang is currently a graduate student at the Anhui University of Technology. Her research interests mainly focus on warped product submanifolds and isoparametric hypersurfaces

    Yecheng Zhu received his PhD from the University of Science and Technology of China. He is currently an associate professor at the Anhui University of Technology. He is mainly engaged in differential geometry

  • Corresponding author: E-mail: zhuyc929@mail.ustc.edu.cn
  • Received Date: 09 October 2021
  • Accepted Date: 21 January 2022
    • Available Online: 11 May 2022
    Abstract

  • Abstract

    By optimization methods on Riemannian submanifolds, we establish two inequalities between the intrinsic and extrinsic invariants, for generalized normalized δ-Casorati curvatures of warped product submanifolds in a Riemannian manifold of quasi-constant curvature. We generalize the conclusions of the optimal inequalities of submanifolds in real space forms.

    Graphical abstract

      The process of establishing the generalized normalized δ-Casorati curvatures inequality.

    Abstract

    By optimization methods on Riemannian submanifolds, we establish two inequalities between the intrinsic and extrinsic invariants, for generalized normalized δ-Casorati curvatures of warped product submanifolds in a Riemannian manifold of quasi-constant curvature. We generalize the conclusions of the optimal inequalities of submanifolds in real space forms.

    • Public Summary

    • We establish Chen-like inequalities for generalized normalized δ-Casorati curvatures of warped product submanifolds in a Riemannian manifold of quasi-constant curvature.
    • Our inequalities extend the optimal inequalities involving the scalar curvature and the Casorati curvature of a Riemannian submanifold in a real space form.

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    • References(11)
  • [1]
    Chen B Y. Some pinching and classification theorems for minimal submanifolds. Arch. Math., 1993, 60: 568–578. doi: 10.1007/BF01236084
    [2]
    Chen B Y. A Riemannian invariant and its applications to submanifold theory. Results in Mathematics, 1995, 27: 17–26. doi: 10.1007/BF03322265
    [3]
    Casorati F. Mesure de la courbure des surfaces suivant l'idée commune.: Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta. Math., 1890, 14: 95–110. doi: 10.1007/BF02413317
    [4]
    Decu S, Haesen S, Verstraelen L. Optimal inequalities involving Casorati curvatures. Bull.Transilv. Univ. Brasov Ser. B, 2007, 14: 85–93.
    [5]
    Decu S, Haesen S, Verstraelen L. Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequal. Pure and Appl. Math., 2008, 9: 79.
    [6]
    Park K S. Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians. Taiwanese J. Math., 2018, 22: 63–77. doi: 10.11650/tjm/8124
    [7]
    Choudhary M A, Blaga A M. Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in metallic Riemannian space forms. J. Geom., 2020, 111: 39. doi: 10.1007/s00022-020-00552-5
    [8]
    Chen B Y, Yano K. Hypersurfaces of a conformally flat space. Tensor, N. S., 1972, 26: 318–322.
    [9]
    Chen B Y. Another general inequality for CR-warped products in complex space forms. Hokkaido Math. J., 2003, 32: 415–444.
    [10]
    Oprea T. Chen's inequality in the Lagrangian case. Colloq. Math., 2007, 108: 163–169. doi: 10.4064/cm108-1-15
    [11]
    Vîlcu G E. An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature. J. Math. Anal. Appl., 2018, 465: 1209–1222. doi: 10.1016/j.jmaa.2018.05.060
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    [1]
    Chen B Y. Some pinching and classification theorems for minimal submanifolds. Arch. Math., 1993, 60: 568–578. doi: 10.1007/BF01236084
    [2]
    Chen B Y. A Riemannian invariant and its applications to submanifold theory. Results in Mathematics, 1995, 27: 17–26. doi: 10.1007/BF03322265
    [3]
    Casorati F. Mesure de la courbure des surfaces suivant l'idée commune.: Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta. Math., 1890, 14: 95–110. doi: 10.1007/BF02413317
    [4]
    Decu S, Haesen S, Verstraelen L. Optimal inequalities involving Casorati curvatures. Bull.Transilv. Univ. Brasov Ser. B, 2007, 14: 85–93.
    [5]
    Decu S, Haesen S, Verstraelen L. Optimal inequalities characterising quasi-umbilical submanifolds. J. Inequal. Pure and Appl. Math., 2008, 9: 79.
    [6]
    Park K S. Inequalities for the Casorati curvatures of real hypersurfaces in some Grassmannians. Taiwanese J. Math., 2018, 22: 63–77. doi: 10.11650/tjm/8124
    [7]
    Choudhary M A, Blaga A M. Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in metallic Riemannian space forms. J. Geom., 2020, 111: 39. doi: 10.1007/s00022-020-00552-5
    [8]
    Chen B Y, Yano K. Hypersurfaces of a conformally flat space. Tensor, N. S., 1972, 26: 318–322.
    [9]
    Chen B Y. Another general inequality for CR-warped products in complex space forms. Hokkaido Math. J., 2003, 32: 415–444.
    [10]
    Oprea T. Chen's inequality in the Lagrangian case. Colloq. Math., 2007, 108: 163–169. doi: 10.4064/cm108-1-15
    [11]
    Vîlcu G E. An optimal inequality for Lagrangian submanifolds in complex space forms involving Casorati curvature. J. Math. Anal. Appl., 2018, 465: 1209–1222. doi: 10.1016/j.jmaa.2018.05.060
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