Complex Two-plane Grassmannians Research Articles

Real hypersurfaces with Ricci–Bourguignon soliton in the complex two-plane Grassmannians

The study of Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian [Formula: see text] is first investigated. It is proved that there exists a shrinking Ricci-Bourguignon soliton on a Hopf real hypersurface [Formula: see text] in [Formula: see text] by using pseudo-anticommuting Ricci tensor. Moreover, we have proved that there does not exist a nontrivial gradient Ricci-Bourguignon soliton ([Formula: see text]) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian [Formula: see text]. Among the class of contact hypersurface in [Formula: see text], we also prove that there does not exist a nontrivial gradient Ricci-Bourguignon in [Formula: see text] over the totally geodesic and totally real quaternionic projective space [Formula: see text] in [Formula: see text], [Formula: see text].

Real hypersurfaces with pseudo-Ricci-Bourguignon soliton in the complex two-plane Grassmannians

In this paper, we have investigated a pseudo-Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian G2(Cm+2). By using pseudo-anti commuting Ricci tensor, we give a complete classification of Hopf pseudo-Ricci-Bourguignon soliton real hypersurfaces in G2(Cm+2). Moreover, we have proved that there exists a non-trivial classification of gradient pseudo-Ricci-Bourguignon soliton (M,ξ,η,Ω,θ,γ,g) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian G2(Cm+2). In the class of contact hypersurface in G2(Cm+2), we prove that there does not exist a gradient pseudo-Ricci-Bourguignon soliton in G2(Cm+2).

Some New Characterizations of Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane Grassmannians G 2 ℂ m + 2 , involving the shape operator A and the Reeb vector field ξ . Moreover, this integral inequality is optimal in the sense that the real hypersurfaces attaining the equality are completely determined. As direct consequences, some new characterizations of the real hypersurfaces in G 2 ℂ m + 2 with isometric Reeb flow can be presented.

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.

Holomorphic isometric embeddings of the complex two-plane Grassmannian into quadrics

The present article studies holomorphic isometric embeddings of the complex two-plane Grassmannian into quadrics. We discuss the moduli space of these embeddings up to gauge and image equivalence using a generalisation of do Carmo–Wallach theory.

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

Structure Lie operator on real hypersurfaces of complex two-plane Grassmannians

It is proved that on a real hypersurface $M$ in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, the structure Lie operator is parallel if and only if $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb {C}^{m+1})$ in $G_2(\ma

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

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.

Einstein–Weyl structures on real hypersurfaces of complex two-plane Grassmannians

We study real Hopf hypersurfaces with Einstein–Weyl structures in the complex two-plane Grassmannian $G_2(\mathbb {C}^{m+2})$, $m\geq 3$. First we prove that a real Hopf hypersurface with a closed Einstein–Weyl structure $W=(g,\theta )$ is of type (B) if

Quadratic Killing normal Jacobi operator for real hypersurfaces in complex Grassmannians of rank 2

In this paper, we introduce a new notion of quadratic Killing normal Jacobi operator (simply, Killing normal Jacobi operator) and its geometric meaning for real hypersurfaces in the complex Grassmannians of rank two G2m+2(c), c≠0. In addition, we give two classification theorems for Hopf real hypersurfaces with quadratic Killing normal Jacobi operator in complex two-plane Grassmannians of compact type and of non-compact type.

Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians

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

Nonexistence of Hopf hypersurfaces in complex two-plane Grassmannians with GTW parallel normal Jacobi operator

We prove that there are no Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb {C}^{m+2})$ if the normal Jacobi operator is $\mathfrak {D}$-parallel or $\mathfrak {D}^\perp $-parallel with respect to the generalized Tanaka--Webster connection and the Hopf principal curvature is invariant along the Reeb flow.

Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians

Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians

Hopf Hypersurfaces in Complex Two-plane Grassmannians with Generalized Tanaka-Webster Reeb-parallel Structure Jacobi Operator

Regarding 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 G2(ℂm+2) and proved that a real hypersurface in G2(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 ℍPn in G2(ℂm+2), where m = 2n.

Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

AbstractOn a real hypersurface M in a complex two-plane Grassmannian G2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces M on G2() satisfying , where ξ is the Reeb vector field on M and S the Ricci tensor of M.

Real hypersurfaces of complex two-plane Grassmannians with certain parallel conditions

In Chen (Bull Korean Math Soc 54(3):975–992, 2017), we introduce the notion of *-Ricci tensor in a real hypersurface of complex two-plane Grassmannians $$G_2({\mathbb {C}}^{m+2}),m\ge 3$$ , and in the present paper we study the characterizations of the Hopf hypersurfaces in $$G_2({\mathbb {C}}^{m+2})$$ with parallel, Reeb parallel and Lie Reeb parallel *-Ricci tensor, respectively.

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.

REAL HYPERSURFACES WITH ∗-RICCI TENSORS IN COMPLEX TWO-PLANE GRASSMANNIANS

In this article, we consider a real hypersurface of complex two-plane Grassmannians <TEX>$G_2({\mathbb{C}}^{m+2})$</TEX>, <TEX>$m{\geq}3$</TEX>, admitting commuting <TEX>${\ast}$</TEX>-Ricci and pseudo anti-commuting <TEX>${\ast}$</TEX>-Ricci tensor, respectively. As the applications, we prove that there do not exist <TEX>${\ast}$</TEX>-Einstein metrics on Hopf hypersurfaces as well as <TEX>${\ast}$</TEX>-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

On the Riemannian Curvature Tensor of a Real Hypersurface in Complex Two-Plane Grassmannians

We prove the nonexistence of Hopf real hypersurfaces in complex two-plane Grassmannians such that the covariant derivatives with respect to Levi-Civita and kth generalized Tanaka–Webster connections in the direction of the Reeb vector field applied to the Riemannian curvature tensor coincide when the shape operator and the structure operator commute on the \(\mathcal Q\)-component of the Reeb vector field.