Hypersurfaces In Complex Two-plane Grassmannians Research Articles

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

Hopf Hypersurfaces in Complex Two-Plane Grassmannians with GTW Killing Shape Operator

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.

Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians

In this paper, first we introduce a new notion of generalized Killing Ricci tensor for a real hypersurface M in complex two-plane Grassmannians G2(ℂm+2). Next we give a complete classification for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with generalized Killing Ricci tensor.

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.

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.

Lie Derivatives and Ricci Tensor on Real Hypersurfaces in Complex Two-plane Grassmannians

AbstractOn a real hypersurface M in a complex two-plane Grassmannian G2() we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces M on G2() satisfying , where ξ is the Reeb vector field on M and S the Ricci tensor of M.

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.

Ricci Semi-symmetric Hypersurfaces in Complex Two-Plane Grassmannians

In this paper, we considered Ricci semi-symmetric real hypersurface in complex two-plane Grassmannians. Then we prove the non-existence of Ricci semi-symmetric Hopf hypersurfaces in complex two-plane Grassmannians by using the method of simultaneous diagonalization for pairwise commutative matrices.

Hopf Hypersurfaces in Complex Two-Plane Grassmannians With $\mathfrak{D}$-Parallel Shape Operator

In this paper we consider a generalized condition for shape operator of a real hypersurface $M$ in complex two-plane Grassmannian $G_2(\mathsf{C}^{m+2})$, namely, $\mathfrak{D}$-parallel shape operator of $M$. Using such a notion, we prove that there does not exist a real hypersurface in complex two-plane Grassmannian $G_2(\mathsf{C}^{m+2})$ with $\mathfrak{D}$-parallel shape operator.

Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Harmonic Curvature

AbstractWe prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians with harmonic curvature with respect to the generalized Tanaka–Webster connection if they satisfy some further conditions.

Real hypersurfaces in complex two-plane Grassmannians with recurrent Ricci tensor

In this paper, we have introduced a new notion of generalized Tanaka–Webster Reeb recurrent Ricci tensor of real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2). Next, we show a non-existence property for real hypersurfaces M in G2(ℂm+2) with such a curvature condition.

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

Semi-parallel symmetric operators for Hopf hypersurfaces in complex two-plane Grassmannians

In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians \(G_2({\mathbb C}^{m+2})\). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in \(G_2({\mathbb C}^{m+2})\).

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.

Real Hypersurfaces in Complex Two-Plane Grassmannians with GTW Reeb Lie Derivative Structure Jacobi Operator

Using GTW connection, we considered a real hypersurface M in a complex two-plane Grassmannian \({G_{2}({\mathbb{C}}^{m+2})}\) when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a complete classification for a real hypersurface in \({G_{2}({\mathbb{C}}^{m+2})}\) satisfying such a condition. In this case, we have proved that M is an open part of a tube around a totally geodesic \({G_{2}({\mathbb{C}}^{m+1})}\) in \({G_{2}({\mathbb{C}}^{m+2})}\).

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION

In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians <TEX>$G_2(\mathbb{C}^{m+2})$</TEX>, <TEX>$m{\geq}3$</TEX>, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection <TEX>$\hat{\nabla}^{(k)}$</TEX>.

Real hypersurfaces in complex two-plane Grassmannians whose shape operator is recurrent for the generalized Tanaka--Webster connection

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose shape operator $A$ is generalized Tanaka-Webster recurrent if the principal curvature of the structure vector field is not equal to trace(A).