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http://dx.doi.org/10.5666/KMJ.2017.57.3.525

Real Hypersurfaces with k-th Generalized Tanaka-Webster Connection in Complex Grassmannians of Rank Two  

Jeong, Imsoon (Division of Future Capability Education, Pai Chai University)
Lee, Hyunjin (The Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University)
Publication Information
Kyungpook Mathematical Journal / v.57, no.3, 2017 , pp. 525-535 More about this Journal
Abstract
In this paper, we consider two kinds of derivatives for the shape operator of a real hypersurface in a $K{\ddot{a}}hler$ manifold which are named the Lie derivative and the covariant derivative with respect to the k-th generalized Tanaka-Webster connection ${\hat{\nabla}}^{(k)}$. The purpose of this paper is to study Hopf hypersurfaces in complex Grassmannians of rank two, whose Lie derivative of the shape operator coincides with the covariant derivative of it with respect to ${\hat{\nabla}}^{(k)}$ either in direction of any vector field or in direction of Reeb vector field.
Keywords
real hypersurface; complex two-plane Grassmannian; complex hyperbolic two-plane Grassmannian; Hopf hypersurface; Levi-Civita connection; Lie derivative; k-th generalized Tanaka-Webster connection; shape operator;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 J. Berndt, Riemannian geometry of complex two-plane Grassmannians, Rend. Sem. Mat. Univ. Politec. Torino, 55(1997), 19-83.
2 J. Berndt, H. Lee and Y. J. Suh, Contact hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math., 24(11)(2013), 1350089, 11 pp.   DOI
3 J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math., 127(1999), 1-14.   DOI
4 J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb ows in complex two-plane Grassmannians, Monatsh. Math., 137(2002), 87-98.   DOI
5 J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math., 23(10)(2012), 1250103, 35 pp.   DOI
6 D. E. Blair, Riemannian geometry of contact and symplectic manifolds, Birkhauser, Boston, 2002.
7 I. Jeong, H. Lee and Y. J. Suh, Levi-Civita and generalized Tanaka-Webster covariant derivatives for real hypersurfaces in complex two-plane Grassmannians, Ann. Mat. Pura Appl., 194(3)(2015), 919-930.   DOI
8 I. Jeong, C. Machado, J.D. Perez and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with $D^{\bot}$-parallel structure Jacobi operator, Internat. J. Math., 22(5)(2011), 655-673.   DOI
9 H. Lee and Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bull. Korean Math. Soc., 47(3)(2010), 551-561.   DOI
10 H. Lee, Y. J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators, Meditterr. J. Math., 13(2016), 3389-3407.   DOI
11 J. D. Perez and Y. J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. Korean Math. Soc., 44(2007), 211-235.   DOI
12 Y. J. Suh, Hypersurfaces with isometric Reeb ow in complex hyperbolic two-plane Grassmannians, Adv. in Appl. Math., 50(2013), 645-659.   DOI
13 Y. J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field, Adv. in Appl. Math., 55(2014), 131-145.   DOI
14 N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math. (N.S.), 2(1)(1976), 131-190.   DOI
15 S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 314(1)(1989), 349-379.   DOI
16 S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., 13(1978), 25-41.   DOI
17 S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.