𝔇⊥-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection

2020 ◽  
Vol 20 (2) ◽  
pp. 163-168
Author(s):  
Eunmi Pak ◽  
Young Jin Suh
Keyword(s):  
Projective Space ◽  
Real Hypersurface ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Quaternionic Projective Space ◽  
Totally Geodesic ◽  
Open Part ◽  
Jacobi Operators ◽  
Hopf Hypersurfaces ◽  
Normal Jacobi Operator

AbstractWe study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove that a real hypersurface in G2(ℂm+2) with generalized Tanaka–Webster 𝔇⊥-parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n.

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Eunmi Pak ◽  
Young Suh
Keyword(s):  
Projective Space ◽  
Real Hypersurface ◽  
Jacobi Operator ◽  
Parallel Structure ◽  
Quaternionic Projective Space ◽  
Structure Jacobi Operator ◽  
Totally Geodesic ◽  
Open Part ◽  
Hopf Hypersurfaces

AbstractRegarding the generalized Tanaka-Webster connection, we considered a new notion of $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

scholarly journals REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350050 ◽  
Author(s):  
JURGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Kähler Manifolds ◽  
Real Hypersurfaces ◽  
Kahler Manifolds ◽  
Totally Geodesic ◽  
Open Part ◽  
Complex Submanifold ◽  
Complex Projective

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator

2014 ◽  
Vol 57 (4) ◽  
pp. 821-833 ◽  
Author(s):  
Imsoon Jeong ◽  
Seonhui Kim ◽  
Young Jin Suh
Keyword(s):  
Real Hypersurface ◽  
Jacobi Operator ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Totally Geodesic

AbstractIn this paper we give a characterization of a real hypersurface of Type (A) in complex two-plane GrassmanniansG2(ℂm+2), which means a tube over a totally geodesicG2(ℂm+1) inG2(ℂm+2), by means of the Reeb parallel structure Jacobi operator ∇εRε= 0.

scholarly journals Real hypersurfaces with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3917-3923
Author(s):  
Avik De ◽  
Tee-How Loo
Keyword(s):  
Real Hypersurface ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Normal Jacobi Operator

The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassmannians. This answers a question considered in [Monatsh Math, 172 (2013), 167-178] in negative.

scholarly journals Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting Ricci tensor

2015 ◽  
Vol 26 (01) ◽  
pp. 1550008 ◽  
Author(s):  
Young Jin Suh
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Real Hypersurface ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Totally Geodesic ◽  
Hopf Hypersurfaces ◽  
New Formula ◽  
Commuting Ricci Tensor

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um), m ≥ 2 from the equation of Gauss. Next we derive a new formula for the Ricci tensor of M in SU2,m/S(U2 ⋅ Um). Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um) with commuting Ricci tensor. Each can be described as a tube over a totally geodesic SU2,m-1/S(U2 ⋅ Um-1) in SU2,m/S(U2 ⋅ Um) or a horosphere whose center at infinity is singular.

Real Hypersurfaces in Complex Projective Space Whose Structure Jacobi Operator is Lie 𝔻-parallel

2013 ◽  
Vol 56 (2) ◽  
pp. 306-316 ◽  
Author(s):  
Juan de Dios Pérez ◽  
Young Jin Suh
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Complex Projective

AbstractWe prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie 𝔻-parallel and satisfies a further condition.

scholarly journals On real hypersurfaces in quaternionic projective space with𝒟⊥-recurrent second fundamental tensor

1999 ◽  
Vol 22 (1) ◽  
pp. 109-117
Author(s):  
Young Jin Suh ◽  
Juan De Dios Pérez
Keyword(s):  
Projective Space ◽  
Complete Classification ◽  
Real Hypersurfaces ◽  
Quaternionic Projective Space ◽  
Fundamental Tensor ◽  
Orthogonal Distribution

In this paper, we give a complete classification of real hypersurfaces in a quaternionic projective spaceQPmwith𝒟⊥-recurrent second fundamental tensor under certain condition on the orthogonal distribution𝒟.

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