Normal Jacobi Operator Research Articles

The preservability of the curvature-adaptedness along the mean curvature flow

In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the shape operator and the normal Jacobi operator of the hypersurface commute.

Killing normal Jacobi operator on real hypersurfaces in the complex hyperbolic quadric

Killing normal Jacobi operator on real hypersurfaces in the complex hyperbolic quadric

Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type

In this paper, we introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex hyperbolic quadric [Formula: see text]. The normal Jacobi operator of Codazzi type implies that the unit normal vector field [Formula: see text] becomes [Formula: see text]-principal or [Formula: see text]-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in [Formula: see text] with normal Jacobi operator of Codazzi type. The result of the classification shows that no such hypersurfaces exist.

Quadratic Killing normal Jacobi operator for real hypersurfaces in the complex quadric

We introduce the notion of quadratic Killing normal Jacobi operator (simply, Killing normal Jacobi operator) and its geometric meaning for real hypersurfaces in the complex quadric. In addition, we give a nonexistence theorem for Hopf real hypersurfaces with quadratic Killing normal Jacobi operator in the complex quadric.

Real Hypersurfaces with Quadratic Killing Normal Jacobi Operator in the Real Grassmannians of Rank Two

In this paper, first we introduce a new notion of (quadratic) Killing normal Jacobi operator (or cyclic parallel normal Jacobi operator) and its geometric meaning for real hypersurfaces in the real Grassmannians of rank two $${\mathbb {Q}}^{m}(\varepsilon )$$ , $$\varepsilon =\pm 1$$ , where $${\mathbb {Q}}^{m}(\varepsilon )$$ denotes the complex quadric $$\mathbb Q^{m}(\varepsilon )=Q^{m}=SO_{m+2}/SO_{m}SO_{2}$$ for $$\varepsilon =1$$ and $${\mathbb {Q}}^{m}(\varepsilon )= Q^{m*}=SO_{m,2}^{0}/SO_{m}SO_{2}$$ for $$\varepsilon =-1$$ , respectively. Next, we give a non-existence theorem for Hopf real hypersurfaces satisfying quadratic Killing normal Jacobi operator in the real Grassmannians of rank two $${\mathbb {Q}}^{m}(\varepsilon )$$ .

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.

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.

New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric

On a real hypersurface M of a complex quadric we have an almost contact metric structure induced by the Kahlerian structure of the ambient space. Therefore, on M we have the Levi-Civita connection $$\nabla $$ and, for any non-null real number k, the so called kth generalized Tanaka Webster connection $${\hat{\nabla }}^{(k)}$$ . We introduce the notions of $$({\hat{\nabla }}^{(k)},\nabla )$$ -Codazzi and $$({\hat{\nabla }}^{(k)},\nabla )$$ -Killing normal Jacobi operator on such a real hypersurface and classify Hopf real hypersurface in a complex quadric whose normal Jacobi operators satisfy any of both conditions.

Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric

In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal C$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal C$-parallel normal Jacobi operator in the complex quadric $Q^{m}$, $m \geq 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $\mathfrak A$-isotropic singular normal vector field.

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

CR -hypersurfaces of conformal Kenmotsu manifolds with ξ-parallel normal Jacobi operator

In this paper, we study CR-hypersurfaces of a conformal Kenmotsu manifold with a ξ-parallel or Lie ξ-parallel normal Jacobi operator.

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

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

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

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.

Commutativity of torsion and normal Jacobi operators on real hypersurfaces in the complex quadric

Commutativity of torsion and normal Jacobi operators on real hypersurfaces in the complex quadric

Commutativity of Cho and normal Jacobi operators on real hypersurfaces in the complex quadric

Commutativity of Cho and normal Jacobi operators on real hypersurfaces in the complex quadric

Real hypersurfaces in the complex quadric with Killing normal Jacobi operator

AbstractWe introduce the notion of Killing normal Jacobi operator for real hypersurfaces in the complex quadricQm=SOm+2/SOmSO2. The Killing normal Jacobi operator implies that the unit normal vector fieldNbecomes 𝔄-principal or 𝔄-isotropic. Then according to each case, we give a complete classification of real hypersurfaces inQm=SOm+2/SOmSO2with Killing normal Jacobi operator.

Real hypersurfaces in the complex quadric with Lie invariant normal Jacobi operator

We introduce the notion of Lie invariant normal Jacobi operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2. The existence of an invariant normal Jacobi operator implies that the unit normal vector field N becomes A-principal or A-isotropic. Using an analysis of cases, we give a complete classification of real hypersurfaces in Qm=SOm+2/SOmSO2 with Lie invariant normal Jacobi operator.

Totally geodesic real hypersurfaces in complex quadric

The Complex quadric is a complex hypersurface in complex projective space. It also can be regarded as a kind of real Grassmann manifold of compact type with rank 2. On the other hand Jacobi fields along geodesics of a given Riemannian manifold satisfy a well known differential equation, called Jacobi equation. This equation naturally inspires the so-called Jacobi operator. The purpose of this article is to give a non-existence result for any totally geodesic real hypersurfaces with recurrent normal Jacobi operator RN and also with recurrent structure Jacobi operator Rξ in Qm. In the continuation, we obtain classifications of totally geodesic real hypersurfaces in Qm with recurrent Ricci tensor Ric.