Complex Hyperbolic Two-plane Grassmannians Research Articles

Real hypersurfaces in the complex hyperbolic two-plane Grassmannians satisfying the Ricci-Bourguignon soliton

In this paper we studied real hypersurfaces in the complex hyperbolic two-plane Grassmannian G2⁎(Cm+2) and proved that a Hopf real hypersurface in G2⁎(Cm+2) does not admit Ricci-Bourguignon soliton if we use the notion of pseudo-anti commuting Ricci tensor. In addition to this one, we have proved that a non-trivial gradient Ricci-Bourguignon soliton (M,Df,ν,ρ,γ,g) on real hypersurfaces with isometric Reeb flow in the complex hyperbolic two-plane Grassmannian G2⁎(Cm+2) does not exist. In the class of contact hypersurface in G2⁎(Cm+2), it has been also proved that there does not exist a non-trivial gradient Ricci-Bourguignon soliton in G2⁎(Cm+2).

Reeb lie derivatives on real hypersurfaces in complex hyperbolic two-plane Grassmannians

In complex two-plane Grassmannians G2(Cm+2) = SU2+m/S(U2?Um), it is known that a real hypersurface satisfying the condition (L?(k)?R?)Y = (L?R?)Y is locally congruent to an open part of a tube around a totally geodesic G2(Cm+1) in G2(Cm+2). In this paper, as an abient space, we consider a complex hyperbolic two-plane Grassmannian SU2,m/S(U2?Um) and give a complete classification of Hopf real hypersurfaces in SU2,m/S(U2?Um) with the above condition.

Fischer–Marsden conjecture on real hypersurfaces in the complex hyperbolic two-plane Grassmannians

Fischer–Marsden conjecture on real hypersurfaces in the complex hyperbolic two-plane Grassmannians

Yamabe and gradient Yamabe solitons in the complex hyperbolic two-plane Grassmannians

First, we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons for real hypersurfaces in the complex hyperbolic two-plane Grassmannians [Formula: see text]. Next, as an application we also give a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on real hypersurfaces in the complex hyperbolic two-plane Grassmannians [Formula: see text].

Semisymmetric hypersurfaces in complex hyperbolic two-plane Grassmannians

In this paper, we introduce new notions of symmetric operators such as semisymmetric shape operator and structure Jacobi operator in complex hyperbolic two-plane Grassmannians. Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians $S U_{2, m} / S\left(U_{2} \cdot U_{m}\right)$ with such notions.

Generalized Killing–Ricci Tensor for Real Hypersurfaces in Complex Hyperbolic Two-Plane Grassmannians

In this paper, first we introduce a new notion of generalized Killing–Ricci tensor for a real hypersurface M in complex hyperbolic two-plane Grassmannians $$SU_{2,m}/S(U_{2}{\cdot }U_{m})$$ . Next we give a complete classification for Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians $$SU_{2,m}/S(U_{2}{\cdot }U_{m})$$ with generalized Killing–Ricci tensor.

Harmonic curvature for real hypersurfaces in complex hyperbolic two plane Grassmannians

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) from the equation of Gauss and derive a new formula for the Ricci tensor of M in SU2,m∕S(U2⋅Um). Finally we give a complete classification for Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m∕S(U2⋅Um) with harmonic curvature or harmonic Weyl tensor.

Cyclic parallel hypersurfaces in complex Grassmannians of rank 2

The object of the paper is to study cyclic parallel hypersurfaces in complex (hyperbolic) two-plane Grassmannians which have a remarkable geometric structure as Hermitian symmetric spaces of rank 2. First, we prove that if the Reeb vector field belongs to the orthogonal complement of the maximal quaternionic subbundle, then the shape operator of a cyclic parallel hypersurface in complex hyperbolic two-plane Grassmannians is Reeb parallel. By using this fact, we classify all cyclic parallel hypersurfaces in complex hyperbolic two-plane Grassmannians with non-vanishing geodesic Reeb flow. Next, we give a non-existence theorem for cyclic Hopf hypersurfaces in complex two-plane Grassmannians.

English

We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2·U m ) with commuting conditions between the restricted normal Jacobi operator $${\bar R_{N\varphi }}$$ and the shape operator A (or the Ricci tensor S).

Inequalities for the Casorati Curvatures of Real Hypersurfaces in Some Grassmannians

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.

Addendum to “Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians” [Adv. in Appl. Math. 50 (2013) 645–659

We classify all of real hypersurfaces M with Reeb invariant shape operator in the complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m≥2. These are either a tube over a totally geodesic SU2,m−1/S(U2⋅Um−1) in SU2,m/S(U2⋅Um) or a horosphere whose center is at infinity and is singular and of type JN∈JN for a unit normal vector field N of M.

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb invariant 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 S 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 Reeb invariant Ricci tensor, that is, LξS=0. 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 Hyperbolic Two-Plane Grassmannians with Commuting Structure Jacobi Operators

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, \({R_{\xi} \phi T = TR_{\xi} \phi}\), where T = A or T = S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. Using simultaneous diagonalization for commuting symmetric operators, we give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition, respectively.

Pseudo-Einstein real hypersurfaces in complex hyperbolic two-plane Grassmannians

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 equations of Gauss. Next we derive a new formula for the Ricci tensor of M in SU2,m/S(U2⋅Um). As an application of this result we prove that there do not exist Einstein Hopf or Q⊥-invariant Einstein real hypersurfaces in SU2,m/S(U2⋅Um).

The normal Jacobi operator of real hypersurfaces in complex hyperbolic two-plane Grassmannians

The aim of this paper is to introduce the notion of normal Jacobi operator of real hypersurfaces in complex hyperbolic two-plane Grassmannians. Furthermore, results concerning real hypersurfaces in complex hyperbolic two-plane Grassmannians whose normal Jacobi operator satisfies conditions of parallelism will be given.

Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator

Abstract In this paper we prove a non-existence of real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2.m/S(U2·Um), m≥3, whose structure tensors {ɸi}i=1,2,3 commute with the shape operator.

Real hypersurfaces in complex hyperbolic two-plane Grassmannians with 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 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.

Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians

We classify the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians SU2,m/S(U2⋅Um), m⩾2. 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.