Complex Two-plane Grassmannians Research Articles

Addendum to the paper “Real hypersurfaces in complex two-plane Grassmannians with ξ-invariant Ricci tensor”

In this paper, we introduce some notions of invariancy for the Ricci tensor on real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) , namely, F -invariant and invariant Ricci tensor. Using these notions, we give non-existence theorem and characterization for the special case among the real hypersurfaces of Type ( A ) in G 2 ( C m + 2 ) , respectively. Here the distribution F is defined by F = [ ξ ] ∪ D ⊥ where D ⊥ = Span { ξ 1 , ξ 2 , ξ 3 } and [ ξ ] = Span { ξ } .

Semi-parallel real hypersurfaces in complex two-plane Grassmannians

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces M of Type (A) in complex two plane Grassmannians G 2(ℂm+2) with a commuting condition between the shape operator A and the structure tensors φ and φ 1 for M in G 2(ℂm+2). Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator A and a new operator φφ 1 induced by two structure tensors φ and φ 1. That is, this commuting shape operator is given by φφ 1 A = A φφ 1. Using this condition, we prove that M is locally congruent to a tube of radius r over a totally geodesic G 2(ℂm+1) in G 2(ℂm+2).

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field

In this paper we give a characterization of real hypersurfaces in the noncompact complex two-plane Grassmannian SU(2,m)/S(U(2)⋅U(m)), m⩾2, with Reeb vector field ξ belonging to the maximal quaternionic subbundle Q. Then we show that such a hypersurface must be a tube over a totally real totally geodesic HHn, m=2n, in the noncompact complex two-plane Grassmannian SU(2,m)/S(U(2)⋅U(m)), a horosphere whose center at the infinity is singular or an exceptional case.

Levi-Civita and generalized Tanaka–Webster covariant derivatives for real hypersurfaces in complex two-plane Grassmannians

It is known that submanifolds in Kaehler manifolds have many kinds of connections. Among them, we consider two connections, that is, Levi-Civita and Tanaka–Webster connections for real hypersurfaces in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\). When they are equal to each other, we give some characterizations in \(G_2({\mathbb C}^{m+2})\).

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

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.

Semi-parallelism of normal Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians

It is proved the non-existence of Hopf hypersurfaces in \(G_{2}({\mathbb{C }}^{m+2}), m\ge 3\), whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle \(\mathfrak{D }\) or its orthogonal complement \(\mathfrak{D } ^{\bot }\) is invariant by the shape operator.

Real Hypersurfaces in Complex Two-plane Grassmannians with $${\mathfrak{D}^{\bot}}$$ -Parallel Shape Operator

In this paper we consider a new notion of \({\mathfrak{D}^{\bot}}\)-parallel shape operator for real hypersurfaces in complex two-plane Grassmannians \({G_2(\mathbb{C}^{m+2})}\) and give a non-existence theorem for a Hopf hypersurface in \({G_2(\mathbb{C}^{m+2})}\) with \({\mathfrak{D}^{\bot}}\)-parallel shape operator.

RECURRENT JACOBI OPERATOR OF REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

In this paper we give a non-existence theorem for Hopf hy- persurfaces in the complex two-plane Grassmannian G2(C m+2 ) with re- current normal Jacobi operator ¯ RN.

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator

In this paper, we consider a new notion of generalized Tanaka–Webster D-parallel shape operator for a real hypersurface in a complex two-plane Grassmannian and prove a non-existence theorem of a real hypersurface.

Real hypersurfaces of type (A) in complex two-plane Grassmannians related to the commuting shape operator

We give some characterizations of real hypersurfaces of type (A) in complex two plane Grassmannians , that is, a tube over a totally geodesic in with the commuting condition for the shape operator A, the structure tensors and , together with additional geometric conditions.

Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor

We introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2) from the Gauss equation. We then derive a new formula for the Ricci tensor of M in G2(ℂm+2). Finally, we prove that there does not exist any Hopf real hypersurface in complex two-plane Grassmannians G2(ℂm+2) with parallel Ricci tensor.

Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor

In this paper we introduce a new notion of Reeb parallel Ricci tensor for real hypersurfaces in complex two-plane Grassmannians G2(Cm+2). Next we give a complete classification of Hopf real hypersurfaces in complex two-plane Grassmannians G2(Cm+2) with Reeb parallel Ricci tensor.

Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G2(Cm+2) from the equation of Gauss and derive a new formula for the Ricci tensor of M in G2(Cm+2). Finally we give a complete classification for Hopf real hypersurfaces in complex two-plane Grassmannians G2(Cm+2) with harmonic curvature or harmonic Weyl tensor.

$$\mathfrak D $$ -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $$G_2(\mathbb{C }^{m+2})$$ with $$\mathfrak D $$ -parallel normal Jacobi operator $${\bar{R}}_N$$ and $$\mathfrak D $$ -parallel structure Jacobi operator $$R_{\xi }$$ if the distribution $$\mathfrak D $$ or $$\mathfrak D ^{\bot }$$ component of the Reeb vector field is invariant by the shape operator, respectively.

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ1A = Aφ1φ between the shape operator A and the structure tensors φ and φ1 for real hypersurfaces in G2(ℂm+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G2(ℂm+1) in complex two plane Grassmannians G2(ℂm+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G2(ℂm+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ1A = Aφ1φ.

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

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 Pnin G2(ℂm+2) where m = 2n.

REAL HYPERSURFACES OF TYPE A IN COMPLEX TWO-PLANE GRASSMANNIANS RELATED TO THE NORMAL JACOBI OPERATOR

In this paper we give a characterization of real hypersurfaces of type (A) in a complex two-plane Grassmannian <TEX>$G_2(\mathbb{C}^{m+2})$</TEX> which is a tube over a totally geodesic <TEX>$G_2(\mathbb{C}^{m+1})$</TEX> in <TEX>$G_2(\mathbb{C}^{m+2})$</TEX>, in terms of two commuting conditions related to the normal Jacobi operator and the shape operator.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS SOME OF WHOSE JACOBI OPERATORS ARE ξ-INVARIANT

We classify Hopf real hypersurfaces in complex two-plane Grassmannians for which either the Jacobi structure operator or the Jacobi operators corresponding to the directions in the distribution [Formula: see text] are Lie ξ-parallel if they satisfy a further condition.

On the structure vector field of a real hypersurface in complex two-plane Grassmannians

Abstract Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.