Hopf Hypersurfaces Research Articles

Ricci operator in a Hopf real Hypersurfaces of complex space form

We initiate the study of real hypersurface of a complex space form and it is proved that Lie parallelism of the Ricci operator in the direction of ξ, locally symmetric Ricci operator in a real hypersurface of a complex space form reduce the hypersurface to Hopf hypersurface.

Real Hypersurfaces in $$Q^m$$ with Commuting Structure Jacobi Operator

In this paper, we study real hypersurfaces in the complex quadric space $$Q^m$$ whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is $$\mathfrak {A}$$ -principal we show that the Reeb curvature $$\alpha $$ is non-vanishing and determine principal curvatures of the hypersurface. In the case of $$\mathfrak {A}$$ -isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.

Biharmonic Hopf Hypersurfaces of Complex Euclidean Space and Odd Dimensional Sphere

Biharmonic Hopf Hypersurfaces of Complex Euclidean Space and Odd Dimensional Sphere

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.

On Hopf hypersurfaces of the homogeneous nearly Kähler $${\mathbf {S}}^3\times {\mathbf {S}}^3$$

In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly K\"ahler) manifold $\mathbf{S}^3\times\mathbf{S}^3$. First, we show that any Hopf hypersurface of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$ does not admit two distinct principal curvatures. Then, for the important class of Hopf hypersurfaces with three distinct principal curvatures, we establish a complete classification under the additional condition that their holomorphic distributions $\{U\}^\perp$ are preserved by the almost product structure $P$ of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$.

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.

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

Abstract We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G 2(ℂ m+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G 2(ℂ m+2) and prove that a real hypersurface in G 2(ℂ m+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 G 2(ℂ m+2), where m = 2n.

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

Commuting structure Jacobi operators for real hypersurfaces in complex space forms II

Let $M$ be a real hypersurface in a complex space form $M_n(c)$, $c \not= 0$. In this paper, we prove that if the structure Jacobi operator $R_\xi$ is $\phi\nabla_{\xi}\xi$-parallel and $R_\xi$ commute with the Ricci tensor, then $M$ is a Hopf hypersurface provided that the mean curvature of $M$ is constant with respect to the structure vector field.

Real Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensor

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.

Ricci tensor of Hopf hypersurfaces in a complex space form

Ricci tensor of Hopf hypersurfaces in a complex space form

Real hypersurfaces with Killing type structure Jacobi operators in $\mathbb{C}P^2$ and $\mathbb{C}H^2$

In this paper, we prove that if the structure Jacobi operator of a $3$-dimen\-sional real hypersurface in a nonflat complex plane is of Killing type, then the hypersurface is either a tube of radius $\frac{\pi}{4}$ over a holomorphic curve in $\mathbb{C}P^2$ or a Hopf hypersurface with vanishing Hopf principal curvature in $\mathbb{C}H^2$. This extends the corresponding results in [6].

Real hypersurfaces with recurrent normal Jacobi operator in the complex quadric

Abstract In this paper, first we give a non-existence theorem for a real hypersurface M with anti-commuting shape operator S in the complex quadric Q m , that is, S ϕ + ϕ S = 0 , where ϕ denotes the structure tensor of M in Q m . By using such a non-existence theorem, we give a classification of Hopf hypersurfaces with recurrent normal Jacobi operator in the complex quadric Q 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.

Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannians II

We show that Hopf hypersurfaces in complex projective space are constructed from half-dimensional totally complex submanifolds in complex 2-plane Grassmannian and Legendrian submanifolds in the twistor space.

Real Hypersurfaces in Complex Two-plane Grassmannians with Reeb Parallel Ricci Tensor in the GTW Connection

AbstractThere are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2). Among them, Suh classified Hopf hypersurfaces in G2(ℂm+2) with Reeb parallel Ricci tensor in Levi–Civita connection. In this paper, we introduce the notion of generalized Tanaka–Webster (GTW) Reeb parallel Ricci tensor for Hopf hypersurfaces in G2(ℂm+2). Next, we give a complete classification of Hopf hypersurfaces in G2(ℂm+2) with GTW Reeb parallel Ricci tensor.

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster D-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.

Hopf Hypersurfaces in Complex Grassmannians of Rank Two

In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of q-umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.