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

Juan De Dios Perez , David Pérez-lópez

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.

Keywords

Complex quadric , real hypersurface , shape operator , $k$-th generalized Tanaka-Webster connection , Cho operators

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).