Structure Jacobi Operator Research Articles

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

Derivatives of structure Jacobi operator on real hypersurfaces in complex Grassmannians of rank two

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

GTW parallel structure Jacobi operator of real hypersurfaces in nonflat complex space forms

We classify real hypersurfaces in nonflat complex space forms of complex dimension two whose structure Jacobi operator is parallel with respect to the generalized Tanaka-Webster connection. We classify these hypersurfaces when the dimension is more than two under the Hopf assumption.

Real hypersurfaces in S6(1) equipped with structure Jacobi operator satisfying Lxl = ∇xl

The study of hypersurfaces of almost Hermitian manifolds by means of their Jacobi operators has been highly active in recent years. Specially, many recent results answer the question of the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. We investigate real hypersurfaces in nearly K?hler sphere S6(1) whose Lie derivative of structure Jacobi operator coincides with the covariant derivative of it and show that such submanifolds do not exist.

On two tensors associated to the structure Jacobi operator of a real hypersurface in complex projective space

A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space. This almost contact metric structure allows us to define, for any nonzero real number k, the so-called k-th generalized Tanaka–Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R_{xi } of M. We classify real hypersurfaces in complex projective space for which such tensors satisfy a cyclic property.

Notes on parallelism of the structure Jacobi operators on real hypersurfaces

Notes on parallelism of the structure Jacobi operators on real hypersurfaces

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in S6(1)

It is well known that the sphere S6(1) admits an almost complex structure J which is nearly Kähler. If M is a hypersurface of an almost Hermitian manifold with a unit normal vector field N, the tangent vector field ξ=−JN is said to be characteristic or the Reeb vector field. The Jacobi operator with respect to ξ is called the structure Jacobi operator, and is denoted by l=R(·,ξ)ξ, where R is the curvature tensor on M. The study of Riemannian submanifolds in different ambient spaces by means of their Jacobi operators has been highly active in recent years. In particular, many recent results deal with questions around the existence of hypersurfaces with a structure Jacobi operator that satisfies conditions related to their parallelism. In the present paper, we study the parallelism of the structure Jacobi operator of real hypersurfaces in the nearly Kähler sphere S6(1). More precisely, we prove that such real hypersurfaces do not exist.

Transversal Killing operators on real hypersurfaces in nonflat complex space forms

In this paper, we prove that the structure Lie operator on a real hypersurface in a nonflat complex space form is transversally Killing if and only if so is the structure Jacobi operator. One of these two conditions holds if and only if the hypersurface is of type (A).

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.

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.

Real hypersurfaces with Killing structure Jacobi operator in the complex hyperbolic quadric

First we introduce the notion of Killing structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ . Next we give a complete classification of real hypersurfaces in ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Killing structure Jacobi operator.

Killing structure Jacobi operator of a real hypersurface in a complex projective space

Killing structure Jacobi operator of a real hypersurface in a complex projective space

Real hypersurfaces with Reeb invariant structure Jacobi operator in the complex quadric

We introduce a new notion of Reeb invariant structure Jacobi operator and two kinds of singular normal vector field N for a real hypersurface M in the complex quadric Qm, m ? 3. If the unit normal N is U-isotropic, we give a classification of Hopf real hypersurfaces with Reeb invariant structure Jacobi operator in the complex quadric Qm, for m ? 3.

Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi operators.

Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces II

Let M be a real hypersurface in complex projective space. The almost contact metric structure on M allows us to consider, for any nonnull real number k, the corresponding k-th generalized Tanaka-Webster connection on M and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator Rξ on M a tensor field of type (1,2), RξT(k). We obtain some classifications of real hypersurfaces for which RξT(k) is either symmetric or skew symmetric.

Real hypersurfaces in [formula omitted] with constant Reeb sectional curvature

We prove that the structure Jacobi operator Rξ on a real hypersurface M in the complex projective space CP2 satisfies Rξ+κϕ2=0 with κ∈R⁎ (or equivalently, constant Reeb sectional curvature) if and only if either M is of type (A) or M is locally congruent to a non-homogeneous Hopf hypersurface with vanishing Hopf principal curvature.

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.

Real Hypersurfaces in the Complex Hyperbolic Quadric with Reeb Parallel Structure Jacobi Operator

We introduce the notion of Reeb parallel structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^*}^m=SO^0_{2,m}/SO_2 SO_m$, $m \geq 3$, and give a classification theory for real hypersurfaces in ${{Q^*}^m}$, $m \geq 3$, with Reeb parallel structure Jacobi operator.

Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operator

We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the non-existence of real hypersurfaces in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.

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.