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On \(\delta \)-Casorati curvature invariants of Lagrangian submanifolds in quaternionic Kähler manifolds of constant q-sectional curvature

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Lagrangian submanifolds, a class of Riemannian submanifolds that arose naturally in the context of Hamiltonian mechanics, play an important role in some modern theories of physics. In this paper, using an optimization technique on submanifolds immersed in Riemannian manifolds, we first obtain some sharp inequalities for \(\delta \)-Casorati curvature invariants of Lagrangian submanifolds in quaternionic space forms, i.e. quaternionic Kähler manifolds of constant q-sectional curvature. Then we show that in the class of Lagrangian submanifolds in quaternionic space forms, there are only two subclasses of ideal Casorati submanifolds, namely the family of totally geodesic submanifolds and a particular subfamily of H-umbilical submanifolds. Finally, we provide some examples to illustrate the obtained results. In particular, we point out that an entire family of ideal Casorati Lagrangian submanifolds can be constructed using the concept of quaternionic extensor introduced by Oh and Kang in the early 2000s.

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Acknowledgements

G.-E. Vîlcu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI - UEFISCDI, project number PN-III-P4-ID-PCE-2020-0025, within PNCDI III.

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Author notes

  1. Mohd Aquib, Mohamd Saleem Lone, Crina Neacşu and Gabriel-Eduard Vîlcu contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi, 110021, India

    Mohd Aquib

  2. Department of Mathematics, School of Physical and Mathematical Sciences, University of Kashmir, Srinagar, 190006, India

    Mohamd Saleem Lone

  3. Department of Mathematics and Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independenţei 313, Bucharest, 060042, Romania

    Crina Neacşu & Gabriel-Eduard Vîlcu

  4. “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie Nr. 13, Bucharest, 050711, Romania

    Gabriel-Eduard Vîlcu

  5. Research Center in Geometry, Topology and Algebra, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, Bucharest, 010014, Romania

    Gabriel-Eduard Vîlcu

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  1. Mohd Aquib
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Correspondence to Gabriel-Eduard Vîlcu.

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Aquib, M., Lone, M.S., Neacşu, C. et al. On \(\delta \)-Casorati curvature invariants of Lagrangian submanifolds in quaternionic Kähler manifolds of constant q-sectional curvature. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 107 (2023). https://doi.org/10.1007/s13398-023-01438-2

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