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Year 2024, Volume: 17 Issue: 1, 221 - 231, 23.04.2024
https://doi.org/10.36890/iejg.1466325

Abstract

References

  • [1] Berndt, J. and Suh, Y. J.: On the geometry of homogeneous real hypersurfaces in the complex quadric. In: Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop 16, 1-9 (2012).
  • [2] Berndt, J. and Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex quadric. Internat. J. Math. 24, 1350050 (18pp) (2013).
  • [3] Cho, J.T.: CR-structures on real hypersurfaces of a complex space form. Publ. Math. Debr. 54, 473-487(1999).
  • [4] Kimura, M.,Lee, H., Pérez, J.D. and Suh, Y.J.: Ruled real hypersurfaces in the complex quadric., J. Geom. Anal., 31 (2021), 7989-8012.
  • [5] Klein, S. : Totally geodesic submanifolds of the complex quadric and the quaternionic 2-Grassmannians. Trans. Amer. Math. Soc. 361, 4927- 4967(2009).
  • [6] Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. II, AWiley-Interscience Publ.,Wiley Classics Library Ed., 1996.
  • [7] Lee, H. and Suh, Y.J.: Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric. Math. Phys. Analysis and Geom. 23, 44 (2020).
  • [8] Pérez, J.D. and Suh, Y.J.: Derivatives of the shape operator of real hypersurfaces in the complex quadric. Results Math. 73, 126(2018).
  • [9] Reckziegel, H.: On the geometry of the complex quadric, in Geometry and Topology of Submanifolds VIII. World Scientific Publishing, Brussels/Nordfjordeid, River Edge, 302-315(1995).
  • [10] Smyth, B.: Differential geometry of complex hypersurfaces. Ann. of Math. 85, 246-266(1967).
  • [11] Suh, Y.J.: Real hypersurfaces in the complex quadric with Reeb parallel shape operator. International J. Math. 25, 1450059 17pp (2014).
  • [12] Suh, Y.J.:Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. in Math. 281, 886-905(2015) .
  • [13] Suh, Y.J.: Real hypersurfaces in complex quadric with harmonic curvature. J. Math. Pures Appl. 106, 393-410(2016).

New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

Year 2024, Volume: 17 Issue: 1, 221 - 231, 23.04.2024
https://doi.org/10.36890/iejg.1466325

Abstract

A real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ inherits an almost contact metric structure . This structure allows to define, for any nonnull real number $k$, the so called $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. If $\nabla$ denotes the Levi-Civita connection on $M$, we introduce the concepts of $(\hat{\nabla}^{(k)},\nabla)$-Codazzi and $(\hat{\nabla}^{(k)},\nabla)$-Killing shape operator $S$ of the real hypersurface and classify real hypersurfaces in $Q$ satisfying any of these conditions.

References

  • [1] Berndt, J. and Suh, Y. J.: On the geometry of homogeneous real hypersurfaces in the complex quadric. In: Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop 16, 1-9 (2012).
  • [2] Berndt, J. and Suh, Y. J.: Real hypersurfaces with isometric Reeb flow in complex quadric. Internat. J. Math. 24, 1350050 (18pp) (2013).
  • [3] Cho, J.T.: CR-structures on real hypersurfaces of a complex space form. Publ. Math. Debr. 54, 473-487(1999).
  • [4] Kimura, M.,Lee, H., Pérez, J.D. and Suh, Y.J.: Ruled real hypersurfaces in the complex quadric., J. Geom. Anal., 31 (2021), 7989-8012.
  • [5] Klein, S. : Totally geodesic submanifolds of the complex quadric and the quaternionic 2-Grassmannians. Trans. Amer. Math. Soc. 361, 4927- 4967(2009).
  • [6] Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. II, AWiley-Interscience Publ.,Wiley Classics Library Ed., 1996.
  • [7] Lee, H. and Suh, Y.J.: Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric. Math. Phys. Analysis and Geom. 23, 44 (2020).
  • [8] Pérez, J.D. and Suh, Y.J.: Derivatives of the shape operator of real hypersurfaces in the complex quadric. Results Math. 73, 126(2018).
  • [9] Reckziegel, H.: On the geometry of the complex quadric, in Geometry and Topology of Submanifolds VIII. World Scientific Publishing, Brussels/Nordfjordeid, River Edge, 302-315(1995).
  • [10] Smyth, B.: Differential geometry of complex hypersurfaces. Ann. of Math. 85, 246-266(1967).
  • [11] Suh, Y.J.: Real hypersurfaces in the complex quadric with Reeb parallel shape operator. International J. Math. 25, 1450059 17pp (2014).
  • [12] Suh, Y.J.:Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. in Math. 281, 886-905(2015) .
  • [13] Suh, Y.J.: Real hypersurfaces in complex quadric with harmonic curvature. J. Math. Pures Appl. 106, 393-410(2016).
There are 13 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Juan De Dios Perez

David Pérez-lópez This is me

Early Pub Date April 7, 2024
Publication Date April 23, 2024
Submission Date January 12, 2024
Acceptance Date February 15, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Perez, J. D. D., & Pérez-lópez, D. (2024). New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric. International Electronic Journal of Geometry, 17(1), 221-231. https://doi.org/10.36890/iejg.1466325
AMA Perez JDD, Pérez-lópez D. New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric. Int. Electron. J. Geom. April 2024;17(1):221-231. doi:10.36890/iejg.1466325
Chicago Perez, Juan De Dios, and David Pérez-lópez. “New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 221-31. https://doi.org/10.36890/iejg.1466325.
EndNote Perez JDD, Pérez-lópez D (April 1, 2024) New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric. International Electronic Journal of Geometry 17 1 221–231.
IEEE J. D. D. Perez and D. Pérez-lópez, “New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 221–231, 2024, doi: 10.36890/iejg.1466325.
ISNAD Perez, Juan De Dios - Pérez-lópez, David. “New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric”. International Electronic Journal of Geometry 17/1 (April 2024), 221-231. https://doi.org/10.36890/iejg.1466325.
JAMA Perez JDD, Pérez-lópez D. New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric. Int. Electron. J. Geom. 2024;17:221–231.
MLA Perez, Juan De Dios and David Pérez-lópez. “New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 221-3, doi:10.36890/iejg.1466325.
Vancouver Perez JDD, Pérez-lópez D. New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric. Int. Electron. J. Geom. 2024;17(1):221-3.