Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator

2016 ◽  
Vol 196 (4) ◽  
pp. 1307-1326 ◽  
Author(s):  
Young Jin Suh ◽  
Doo Hyun Hwang
Keyword(s):  
Shape Operator ◽  
Real Hypersurfaces ◽  
Hyperbolic Quadric ◽  
Parallel Shape Operator ◽  
Complex Hyperbolic Quadric ◽  
Reeb Parallel Shape Operator

Real hypersurfaces in the complex quadric with Reeb parallel shape operator

2014 ◽  
Vol 25 (06) ◽  
pp. 1450059 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Shape Operator ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Complete Proof ◽  
Complex Quadric ◽  
Parallel Shape Operator ◽  
Codazzi Type ◽  
Reeb Parallel Shape Operator

First, we introduce the notion of shape operator of Codazzi type for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2. Next, we give a complete proof of non-existence of real hypersurfaces in Qm = SOm+2/SOmSO2 with shape operator of Codazzi type. Motivated by this result we have given a complete classification of real hypersurfaces in Qm = SOm+2/SOmSO2 with Reeb parallel shape operator.

Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type

2021 ◽  
Author(s):  
Imsoon Jeong ◽  
Eunmi Pak ◽  
Young Jin Suh
Keyword(s):  
Jacobi Operator ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Hyperbolic Quadric ◽  
Complex Hyperbolic Quadric ◽  
Unit Normal Vector Field ◽  
Codazzi Type ◽  
Normal Jacobi Operator

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

scholarly journals Real hypersurfaces in complex two-plane Grassmannians with parallel shape operator

2003 ◽  
Vol 67 (3) ◽  
pp. 493-502 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Shape Operator ◽  
Real Hypersurfaces ◽  
Two Dimensional ◽  
Parallel Shape Operator

In this paper we show that there do not exist any real hypersurfaces in a complex two dimensional Grassmannians G2(ℂn+2) with parallel shape operator.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR

2012 ◽  
Vol 09 (04) ◽  
pp. 1250032 ◽  
Author(s):  
IMSOON JEONG ◽  
HYUNJIN LEE ◽  
YOUNG JIN SUH
Keyword(s):  
Lie Algebras ◽  
Riemannian Manifolds ◽  
Shape Operator ◽  
Variational Problems ◽  
Real Hypersurfaces ◽  
Graded Lie Algebras ◽  
Totally Geodesic ◽  
Cartan Connections ◽  
Parallel Shape Operator ◽  
Kodai Math

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G2(ℂm+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇⊥-parallel for a hypersurface M in G2(ℂm+2), and give a characterization for a tube around a totally geodesic ℍ Pn in G2(ℂm+2) where m = 2n.

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