Normal Jacobi Operator Research Articles

Real Hypersurfaces in the Complex Hyperbolic Quadric with Parallel Normal Jacobi Operator

First, we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex hyperbolic quadric $${Q^m}^* = SO_{m,2}/SO_mSO_2$$ . Next we give a complete proof of non-existence of real hypersurfaces in $${Q^m}^* = SO_{m,2}/SO_mSO_2$$ with parallel normal Jacobi operator.

On the Normal Jacobi Operator of CR-Hypersurfaces in Conformal Kenmotsu Space Forms

We study the CR-hypersurfaces of a conformal Kenmotsu space form with a ξ-parallel normal Jacobi operator. We also present an illustrative example of a three-dimensional conformal Kenmotsu manifold that is not Kenmotsu.

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 .

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).

Real Hypersurfaces in the Complex Quadric with Normal Jacobi Operator of Codazzi Type

We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) . The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes \({\mathfrak {A}}\)-principal or \({\mathfrak {A}}\)-isotropic. Then, according to each case, we give a complete classification of Hopf real hypersurfaces in \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) with normal Jacobi operator of Codazzi type. The result of the classification is that no such hypersurfaces exist.

Real hypersurfaces with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians

The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassmannians. This answers a question considered in [Monatsh Math, 172 (2013), 167-178] in negative.

Characterizations for Real Hypersurfaces in Complex and Quaternionic Space Forms Related to the Normal Jacobi Operator

Characterizations for Real Hypersurfaces in Complex and Quaternionic Space Forms Related to the Normal Jacobi Operator

Real hypersurfaces in the complex quadric with parallel normal Jacobi operator

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.

Parallelism of normal Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

Parallelism of normal Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

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.

Generalized Tanaka-Webster and Levi-Civita connections for normal Jacobi operator in complex two-plane Grassmannians

We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in G2(ℂm+2) satisfying such conditions.

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator

We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian G 2(ℂm+2) which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in G 2(ℂm+2) and prove non-existence of real hypersurfaces in G 2(ℂm+2) with generalized Tanaka-Webster parallel normal Jacobi operator.

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

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.

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.

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.

$$\mathfrak D $$ -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 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 with F-parallel Normal Jacobi Operator

In this paper we give a non-existence theorem for Hopf hypersurfaces M in complex two-plane Grassmannians whose normal Jacobi operator is parallel on the distribution F defined by , where [] = Span{}, = Span {, , } and , .

HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH LIE PARALLEL NORMAL JACOBI OPERATOR

In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian <TEX>$G_2(\mathbb{C}^{m+2})$</TEX> with Lie parallel normal Jacobi operator <TEX>$\bar{R}_N$</TEX> and totally geodesic D and <TEX>$D^{\bot}$</TEX> components of the Reeb flow.

Real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type

Abstract We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.