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On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric

  • Juan De Dios Pérez EMAIL logo , David Pérez-López and Young Jin Suh
Published/Copyright: December 18, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 6

ABSTRACT

The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SOm+2/SOmSO2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇^(k) . Associated to this connection, we have Cho and torsion operators FX(k) and TX(k) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by BF(k) and BT(k) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M.

2020 Mathematics Subject Classification: Primary 53C15; 53B25
Keywords: Complex quadric; real hypersurface; structure Lie operator; kth generalized Tanaka-Webster connection; Lie derivative

(Communicated by Tibor Macko)

Funding statement: First author is supported by Projects PID 2020-11 6126GB-I00 from MICINN and PY20-01391 from Junta de Andalucía. Third author by grant Proj. No. NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea. The authors thank the referee for valuable suggestions to improve the paper.

REFERENCES

[1] Berndt, J.—Suh, Y. J.: On the geometry of homogeneous real hypersurfaces in the complex quadric, Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop 16 (2012), 1–9.Search in Google Scholar

[2] Berndt, J.—Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex quadric, Internat. J. Math. 24 (2013), Art. ID 1350050.10.1142/S0129167X1350050XSearch in Google Scholar

[3] Berndt, J.—Suh, Y. J.: Real Hypersurfaces in Hermitian Symmetric Spaces. Advances in Analysis and Geometry, vol. 5, Walter de Gruyter GmbH, Berlin-Boston, 2022.10.1515/9783110689839Search in Google Scholar

[4] Blair, D. E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Math., vol. 203, Birkhauser Boston Inc., Boston, 2002.10.1007/978-1-4757-3604-5Search in Google Scholar

[5] Cho, J. T.: CR-structures on real hypersurfaces of a complex space form, Publ. Math. Debr. 54 (1999), 473–487.10.5486/PMD.1999.2081Search in Google Scholar

[6] Cho, J. T.: Pseudo-Einstein CR-structures on real hypersurfaces in complex space forms, Hokkaido Math. J. 37 (2008), 1–17.10.14492/hokmj/1253539581Search in Google Scholar

[7] Kimura, M.—Lee, H.—Pérez, J. D.—Suh, Y.J: Ruled real hypersurfaces in the complex quadric, J. Geom. Anal. 31 (2021), 7989–8012.10.1007/s12220-020-00564-2Search in Google Scholar

[8] Klein, S.: Totally geodesic submanifolds of the complex quadric and the quaternionic 2-Grassmannians, Trans. Amer. Math. Soc. 361 (2009), 4927–4967.10.1090/S0002-9947-09-04699-6Search in Google Scholar

[9] Kobayashi, S.—Nomizu, K.: Foundations of Differential Geometry, vol. II, A Wiley-Interscience Publ., Wiley Classics Library Ed., 1996.Search in Google Scholar

[10] Pérez, J. D.: Commutativity of torsion and normal Jacobi operators on real hypersurfaces of the complex quadric, Publ. Math. Debrecen 95 (2019), 157–168.10.5486/PMD.2019.8424Search in Google Scholar

[11] Pérez, J. D.—Suh, Y. J.: Derivatives of the shape operator of real hypersurfaces in the complex quadric, Results Math. 73 (2018), 126.10.1007/s00025-018-0888-4Search in Google Scholar

[12] Pérez, J. D.—Suh, Y. J.: Commutativity of Cho and normal Jacobi operators on real hypersurfaces in the complex quadric, Publ. Math. Debrecen 94 (2019), 359–367.10.5486/PMD.2019.8315Search in Google Scholar

[13] Reckziegel, H.: On the geometry of the complex quadric. In: Geometry and Topology of Submanifolds VIII, World Scientific Publishing, Brussels/Nordfjordeid, River Edge, 1995, pp. 302–315.Search in Google Scholar

[14] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Jpn. J. Math. 2 (1976), 131–190.10.4099/math1924.2.131Search in Google Scholar

[15] Tanno, S.: Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349–379.10.1090/S0002-9947-1989-1000553-9Search in Google Scholar

[16] Webster, S. M.: PseudoHermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 25–41.10.4310/jdg/1214434345Search in Google Scholar
Received: 2022-09-14
Accepted: 2023-03-24
Published Online: 2023-12-18

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0113
Keywords for this article
Complex quadric; real hypersurface; structure Lie operator; kth generalized Tanaka-Webster connection; Lie derivative

Articles in the same Issue

  1. ℒ-fuzzy Annihilators in Residuated Lattices
  2. Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
  3. Uniform Local Connectedness and Completion of Metric σ-Frames
  4. On Transcendental Regular Continued Fractions
  5. On Repdigits Which are Sums or Differences of Two k-Pell Numbers
  6. Peirce Decompositions for Evolution Algebras
  7. Generalized Anti-Symmetry Laws in Groupoids
  8. New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
  9. Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
  10. On the Attainable Set of Iterative Differential Inclusions
  11. On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
  12. A Critical p(x)-Laplacian Steklov Type Problem with Weights
  13. Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
  14. Some Approximation Properties of Operators Including Degenerate Appell Polynomials
  15. Operators that are Weakly Coercive and a Compact Perturbation
  16. On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
  17. Paracompactness and Open Relations
  18. On (Strongly) (Θ-)Discrete Homogeneous Spaces
  19. The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
  20. Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
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