Generalized Tanaka-Webster Connection Research Articles

On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric

ABSTRACT The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SO m+2/SO m SO 2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇ ^ ( k ) . Associated to this connection, we have Cho and torsion operators F X ( k ) and T X ( k ) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by B F ( k ) and B T ( k ) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M.

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.

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.

A NOTE ON KENMOTSU MANIFOLDS ADMITTING GENERALIZED TANAKA-WEBSTER CONNECTION

In the present paper, we investigate some tensor conditions of Kenmotsu manifolds with the generalized Tanaka-Webster connection. Using
 the Q tensor whose trace is the well-known Z-tensor, we prove the conditions ξ − Q∗ flat, ϕ − Q∗ flat Kenmotsu manifold with respect to the generalized Tanaka-Webster connection.

К геометрии обобщенных неголономных многообразий Кенмоцу

The concept of a generalized nonholonomic Kenmotsu manifold is introduced. In contrast to the previously defined nonholonomic Kenmotsu manifold, the manifold studied in the article is an almost normal almost contact metric manifold of odd rank. The manifold is equipped with a metric connection with torsion, which is called the canonical connection in this work. The main properties of the canonical connection are studied. The canonical connection is an analogue of the generalized Tanaka-Webster connection. In this paper, we prove that the canonical connection is the only metric connection with torsion of a special structure that pre­serves the structural 1-form and the Reeb vector field. We study the in­trinsic geometry of a generalized nonholonomic Kenmotsu manifold equipped with a canonical connection. It is proved that if a generalized nonholonomic Kenmotsu manifold is an Einstein manifold with respect to a canonical connection, then it is Ricci-flat with respect to this connec­tion. An example of a generalized nonholonomic Kenmotsu manifold that is not a nonholonomic Kenmotsu manifold is given.

Lie Derivatives of the Shape Operator of a Real Hypersurface in a Complex Projective Space

We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, B^{(k)}_T, related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor A^{(k)}_T associated with the shape operator A of M.

Symmetries of Sasakian Generalized Sasakian-Space-Form Admitting Generalized Tanaka–Webster Connection

The object of this paper is to study symmetric properties of Sasakian generalized Sasakian-space-form with respect to generalized Tanaka–Webster connection. We studied semisymmetry and Ricci semisymmetry of Sasakian generalized Sasakian-space-form with respect to generalized Tanaka–Webster connection. Further we obtain results for Ricci pseudosymmetric and Ricci-generalized pseudosymmetric Sasakian generalized Sasakian-space-form.

N(k)-contact metric manifolds with generalized Tanaka-Webster connection

In this paper, we characterize N(k)-contact metric manifolds with generalized Tanaka-Webster connection. We obtain some curvature properties. It is proven that if an N(k)-contact metric manifold with generalized Tanaka-Webster connection is K-contact then it is an example of generalized Sasakian space form. Also, we examine some flatness and symmetric conditions of concircular curvature tensor on an N(k)-contact metric manifolds with generalized Tanaka-Webster connection.

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.

A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified.

Slant Curves in Contact Lorentzian Manifolds with CR Structures

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the ∇ ^ -geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when c ≤ 0 , there does not exist a non-geodesic slant Frenet curve satisfying the ∇ ^ -Jacobi equations for the ∇ ^ -geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms M 1 3 ( H ^ ) for H ^ = 2 c > 0 .

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.

Concircular curvature tensor of Kenmotsu manifolds admitting generalized Tanaka-Webster connection

The objective of the present paper is to study concircular curvature tensor of Kenmotsu manifold with respect to generalized Tanaka-Webster connection, whose concircular curvature tensor satisifies certain conditions and it is shown that if the curvature tensor of a Kenmotsu manifold admitting generalized Tanaka-Webster connection $\nabla^{*}$ vanishes, then the Kenmotsu manifold is locally isometric to the hyperbolic space $H^{2n+1}(-1)$. Further we have studied $\xi$-concircularly flat, $\phi$-concircularly flat, pseudo-concircularly flat, $C^{*} . \phi =0$, $C^{*}.S^{*}=0$ and we have shown that $R^{*} . C^{*}=R^{*} . R^{*}$. Finally, an example of a $5$-dimensional Kenmotsu manifold with respect to the generalized Tanaka-Webster connection is given to verify our result.

Some new results on real hypersurfaces with generalized Tanaka-Webster connection

AbstractLetMbe a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator ofMis transversally Killing with respect to the generalized Tanaka-Webster connection if and only ifMis locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator ofMcommutes with Cho operator on holomorphic distribution if and only ifMis locally congruent to a ruled real hypersurface.

Some Real Hypersurfaces in Complex and Complex Hyperbolic Quadrics

We will consider real hypersurfaces in the complex quadric or the complex hyperbolic quadric. From the almost contact metric structure of such hypersurfaces, we can define on any real hypersurface, for any nonnull real number k, a kth generalized Tanaka–Webster connection and a differential operator of first order of Lie type associated with such a connection. We classify real hypersurfaces in the complex quadric and the complex hyperbolic quadric for which the Lie derivative and the Lie-type differential operator coincide when we apply them to the shape operator of the real hypersurface either in the direction of the structure vector field or in any direction of the maximal holomorphic distribution.

K-th generalized Tanaka–Webster Einstein real hypersurfaces in non-flat complex space forms

In this paper the notion of k-th generalized Tanaka–Webster Einstein real hypersurfaces in non-flat complex space forms is introduced. Furthermore, three dimensional real hypersurfaces in non-flat complex space forms, which are k-th generalized Tanaka–Webster Einstein, are classified. Moreover, the notion of Ricci soliton for the k-th generalized Tanaka–Webster connection of a real hypersurface is also introduced and first results concerning the new notion and real hypersurfaces are provided. At the end of the paper, discussion and ideas of further research on the new notions is included.

Concircular curvature tensor of Kenmotsu manifolds admitting generalized Tanaka-Webster connection

The objective of the present paper is to study concircular curvature tensor of Kenmotsu manifold with respect to generalized Tanaka-Webster connection, whose concircular curvature tensor satisifies certain conditions and it is shown that if the curvature tensor of a Kenmotsu manifold admitting generalized Tanaka-Webster connection $\nabla^{*}$ vanishes, then the Kenmotsu manifold is locally isometric to the hyperbolic space $H^{2n+1}(-1)$. Further we have studied $\xi$-concircularly flat, $\phi$-concircularly flat, pseudo-concircularly flat, $C^{*} . \phi =0$, $C^{*}.S^{*}=0$ and we have shown that $R^{*} . C^{*}=R^{*} . R^{*}$. Finally, an example of a $5$-dimensional Kenmotsu manifold with respect to the generalized Tanaka-Webster connection is given to verify our result.

Derivatives of the Operator $$\phi A-A\phi $$ϕA-Aϕ on a Real Hypersurface in Non-flat Complex Space Forms

On a real hypersurface M in a non-flat complex space form, two types of connections can be defined: the Levi-Civita connection, which is torsion free and, for any nonnull constant k, the kth generalized Tanaka-Webster connection, which is an affine connection with torsion. So the associated covariant derivatives can be considered. Moreover, the Lie derivative and a derivative of Lie type associated with the kth generalized Tanaka-Webster connection can be defined. In this paper, real hypersurfaces for which both covariant derivatives or Lie derivative and the derivative of Lie type associated with the kth generalized Tanaka-Webster connection coincide when they act on the operator $$\phi A-A\phi $$ , where $$\phi $$ denotes the structure operator and A the shape operator of M, either in the direction of the structure vector field $$\xi $$ or in any direction of the maximal holomorphic distribution of M are classified.

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.

Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

On a real hypersurface in the complex quadric we can consider the Levi-Civita connection and, for any nonnull constant k, the k-th generalized Tanaka-Webster connection. We also have a differential operator of first order of Lie type associated to the k-th generalized Tanaka-Webster connection. We prove non-existence of real hypersurfaces in the complex quadric for which the covariant derivatives associated to both connections coincide or Lie derivative and Lie type differential operator coincide when they act on the shape operator of the real hypersurface.