Parallel Structure Jacobi Operator Research Articles

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.

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.

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 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.

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.

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

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

Real hypersurfaces in the complex quadric with parallel structure Jacobi operator

First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2. Next we give a complete classification of real hypersurfaces in Qm=SOm+2/SOmSO2 with parallel structure Jacobi operator.

Real Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Structure Jacobi Operator

AbstractIn this paper we give a characterization of a real hypersurface of Type (A) in complex two-plane GrassmanniansG2(ℂm+2), which means a tube over a totally geodesicG2(ℂm+1) inG2(ℂm+2), by means of the Reeb parallel structure Jacobi operator ∇εRε= 0.

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator

AbstractRegarding 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 G 2(ℂm+2) and proved that a real hypersurface in G 2(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 ℍP n in G 2(ℂm+2), where m = 2n.

$$\mathfrak D $$ -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 IN COMPLEX TWO-PLANE GRASSMANNIANS WITH 𝔇⊥-PARALLEL STRUCTURE JACOBI OPERATOR

In this paper we give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with 𝔇⊥-parallel structure Jacobi operator, where 𝔇⊥= Span {ξ1, ξ2, ξ3}.

ON THE STRUCTURE JACOBI OPERATOR AND RICCI TENSOR OF REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS

It is known that there are no real hypersurfaces with parallel structure Jacobi operator <TEX>$R_{\xi}$</TEX> (cf.[16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator <TEX>$R_{\xi}$</TEX> which are weaker than <TEX>${\nabla}R_{\xi}$</TEX> = 0. Under further condition <TEX>$S\phi={\phi}S$</TEX> for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

Real hypersurfaces in complex two-plane grassmannians with parallel structure Jacobi operator

We give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with parallel structure Jacobi operator Rξ.

REAL HYPERSURFACES IN COMPLEX SPACE FORMS WITH ε-PARALLEL RICCI TENSOR AND STRUCTURE JACOBI OPERATOR

We know that there are no real hypersurfaces with parallel Ricci tensor or parallel structure Jacobi operator in a nonflat complex space form (See [4], [6], [10] and [11]). In this paper we investigate real hypersurfaces M in a nonflat complex space form Mn(c) under the condition that ∇ξS = 0 and ∇ξRξ = 0, where S and Rξ respectively denote the Ricci tensor and the structure Jacobi operator of M in Mn(c). 0. Introduction A Kaehler manifold of constant holomorphic sectional curvature c is called a complex space form, which is denoted by Mn(c). It is well known that complete and simply connected complex space forms are isometric to a complex projective space PnC, a complex Euclidean space C or a complex hyperbolic space HnC according as c > 0, c = 0 and c < 0. In this paper we consider a real hypersurface M in a complex space form Mn(c), c6=0. Then M has an almost contact metric structure (φ, ξ, η, g) induced from the complex structure J and the Kaehler metric of Mn(c). The structure vector field ξ is said to be principal if Aξ = αξ is satisfied, where A denotes the shape operator of M and α = η(Aξ). A real hypersurface is said to be a Hopf hypersurface if the structure vector field ξ of M is principal. In the study of real hypersurfaces in PnC, Takagi [12] classified all homogeneous real hypersurfaces and Cecil and Ryan [2] showed that they can be regarded as the tubes of constant radius over Kaehler submanifolds when the structure vector field ξ is principal. Such tubes can be divided into six kinds of type A1, A2, B, C, D and E. Received August 8, 2005; Revised May 30, 2006. 2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms

We prove the nonexistence of real hypersurfaces in nonflat complex space forms whose Jacobi operator associated to the structure vector field is parallel. In order to prove this result we also obtain the nonexistence of several classes of non homogeneous real hypersurfaces in complex projective space.