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.

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 CYCLIC-PARALLEL STRUCTURE JACOBI OPERATORS IN A NONFLAT COMPLEX SPACE FORM

2009 ◽  
Vol 81 (2) ◽  
pp. 260-273 ◽  
Author(s):  
U-HANG KI ◽  
HIROYUKI KURIHARA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Jacobi Operator ◽  
Complex Space Form ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Jacobi Operators

AbstractIt is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.

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.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH 𝔇⊥-PARALLEL STRUCTURE JACOBI OPERATOR

2011 ◽  
Vol 22 (05) ◽  
pp. 655-673 ◽  
Author(s):  
IMSOON JEONG ◽  
CARLOS J. G. MACHADO ◽  
JUAN DE DIOS PÉREZ ◽  
YOUNG JIN SUH
Keyword(s):  
Existence Theorems ◽  
Jacobi Operator ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator

In this paper we give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with 𝔇⊥-parallel structure Jacobi operator, where 𝔇⊥= Span {ξ1, ξ2, ξ3}.

Cyclic parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

2021 ◽  
pp. 1-26
Author(s):  
Hyunjin Lee ◽  
Young Jin Suh ◽  
Changhwa Woo
Keyword(s):  
Jacobi Operator ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Equivalent Relation

In this paper, from the property of Killing for structure Jacobi tensor $\mathbb {R}_{\xi }$ , we introduce a new notion of cyclic parallelism of structure Jacobi operator $R_{\xi }$ on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of $R_{\xi }$ and Killing property of $\mathbb {R}_{\xi }$ . Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.

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