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http://dx.doi.org/10.5666/KMJ.2017.57.4.683

Real Hypersurfaces in the Complex Hyperbolic Quadric with Killing Shape Operator  

Jeong, Imsoon (Division of Future Capability Education, Pai Chai University)
Suh, Young Jin (Department of Mathematics & Research Institute of Real and Complex Manifolds, Kyungpook National University)
Publication Information
Kyungpook Mathematical Journal / v.57, no.4, 2017 , pp. 683-699 More about this Journal
Abstract
We introduce the notion of Killing shape operator for real hypersurfaces in the complex hyperbolic quadric $Q^{m*}=SO_{m,2}/SO_mSO_2$. The Killing shape operator implies that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^{m*}=SO_{m,2}/SO_mSO_2$ with Killing shape operator.
Keywords
Killing shape operator; A-isotropic; A-principal; $K{\ddot{a}}hler$ structure; complex conjugation; complex quadric;
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1 S. Tachibana, On Killing tensors in a Riemannian space, Tohoku Math. J., 20(1968), 257-264.   DOI
2 J. Berndt, S. Console and C. Olmos, Submanifolds and holonomy, Chapman & Hall/CRC, Boca Raton, FL, 2003.
3 J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatsh. Math., 137(2002), 87-98.   DOI
4 J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math., 23(2012), 1250103, 35 pp.
5 J. Berndt and Y. J. Suh, Contact hypersurfaces in Kahler manifold, Proc. Amer. Math. Soc., 143(2015), 2637-2649.   DOI
6 D. E. Blair, Almost contact manifolds with Killing structure tensors, Pacific J. Math., 39(1971), 285-292.   DOI
7 I. Jeong, H. J. Kim and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator, Publ. Math. Debrecen, 76(2010), 203-218.
8 I. Jeong, C. J. G. Machado, J. D. Perez and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with ${\mathscr{D}}^{\bot}$-parallel structure Jacobi operator, Internat. J. Math., 22(2011), 655-673.   DOI
9 I. Jeong, C. J. G. Machado, J. D. Perez and Y. J. Suh, ${\mathcal{D}}$-parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians, Ann. Mat. Pura Appl., 193(2014), 591-608.   DOI
10 S. Klein, Totally geodesic submanifolds in the complex quadric, Differential Geom. Appl., 26(2008), 79-96.   DOI
11 S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, A Wiley-Interscience Publ., Wiley Classics Library Ed., 1996.
12 M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc., 212(1975), 355-364.   DOI
13 J. D. Perez, Real hypersurfaces of quaternionic projective space satisfying ${\nabla}_U_iA$ = 0, J. Geom., 49(1994), 166-177.   DOI
14 J. D. Perez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying ${\nabla}_U_iR$ = 0, Differential Geom. Appl., 7(1997), 211-217.   DOI
15 H. Reckziegel, On the geometry of the complex quadric, Geometry and Topology of Submanifolds VIII (Brussels/Nordfjordeid 1995), World Sci. Publ., River Edge, NJ, 1995, pp. 302-315.
16 B. Smyth, Differential geometry of complex hypersurfaces, Ann. Math., 85(1967), 246-266.   DOI
17 Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proc. Roy. Soc. Edinburgh Sect. A, 142(2012), 1309-1324.   DOI
18 Y. J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. Appl. Math., 50(2013), 645-659.   DOI
19 Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Internat. J. Math., 25(2014), 1450059, 17 pp.
20 Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Adv. Math., 281(2015), 886-905.   DOI
21 Y. J. Suh, Real hypersurfaces in the complex quadric with harmonic curvature, J. Math. Pures Appl., 106(2016), 393-410.   DOI
22 Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadric with isometric Reeb flow, to appear in Comm. Contemp. Math., (2017).
23 Y. J. Suh and D. H. Hwang, Real hypersurfaces in the complex quadric with commuting Ricci tensor, Sci. China Math., 59(2016), 2185-2198.   DOI
24 Y. J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor, Math. Nachr., 287(2014), 1524-1529.   DOI