Generalized Tanaka-Webster Connection Research Articles

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.

On the conharmonic curvature tensor of Kenmotsu manifolds with generalized Tanaka-Webster connection

In this paper, we study a generalized Tanaka-Webster connection on a Kenmotsu manifold. We study the conharmonic curvature tensor with respect to the generalized Tanaka-Webster connection er and also characterize conharmonically flat and locally �-conharmonically symmetric Kenmotsu manifold with respect to the connectioner. Besides these we also classify Kenmotsu manifolds which satisfyeK �eR D 0 andeP �eK D 0, whereeK andeP are the conharmonic curvature tensor, the projective curvature tensor and Riemannian curvature tensor, respectively with respect to the connection

Derivatives on Real Hypersurfaces of Two-Dimensional Non-flat Complex Space Forms

The aim of this paper is to answer the question if there are three-dimensional real hypersurfaces in non-flat complex space forms whose covariant derivative with respect to the kth generalized Tanaka–Webster connection of a tensor field of type (1, 1) coincides with the Lie derivative of it either in direction of any vector field orthogonal to \(\xi \) or in direction of \(\xi \). The answer is given, in case the tensor field is the shape operator of a real hypersurface or the structure Jacobi operator of it.

Kenmotsu manifolds with generalized Tanaka-Webster connection

We study the g-Tanaka-Webster connection associated to a Kenmotsu structure. With the help of g-Tanaka-Webster connection we characterize Kenmotsu manifolds and find certain curvature properties of this connection on Kenmotsu manifolds. Finally an illustrative example is given to verify some results.

A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of their Generalized Tanaka–Webster Lie Derivative

AbstractOn a real hypersurface M in a non-flat complex space form there exist the Levi–Civita and the k-th generalized Tanaka–Webster connections. The aim of this paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operatorwith respect to the Levi–Civita connection coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in direction of any vector field orthogonal to the structure vector field.

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.

The structure Jacobi operator of three-dimensional real hypersurfaces in non-flat complex space forms

In this paper new results concerning three dimensional real hypersurfaces in non-flat complex space forms in terms of their stucture Jacobi operator are presented. More precisely, the conditions of 1) the structure Jacobi operator being of Codazzi type with respect to the generalized Tanaka-Webster connection and commuting with the shape operator and 2) η-invariance of the structure Jacobi operator and commutativity of it with the shape operator are studied. Furthermore, results concerning Hopf hypersurfaces and ruled hypersurfaces of dimension greater than three satisfying the previous conditions are also included.

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.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP2AND ℂH2WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Abstract. In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

Generalized Tanaka-Webster and Levi-Civita connections for normal Jacobi operator in complex two-plane Grassmannians

We study classifying problems of real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka-Webster connection, we consider that the generalized Tanaka-Webster derivative of the normal Jacobi operator coincides with the covariant derivative. In this case, we prove complete classifications for real hypersurfaces in G2(ℂm+2) satisfying such conditions.

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 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>.

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.

On the Ricci tensor and the generalized Tanaka--Webster connection of real hypersurfaces in a complex space form

We prove that the Ricci tensor Ŝ with respect to the generalized Tanaka-Webster connection of a real hypersurface with the almost contact structure (η, φ, ξ, g) in a complex space form of complex dimension n ≥ 3 satisfies Ŝ(X,φY ) = λg(X,φY ) for any vector field X and Y , λ being a function, if and only if the real hypersurface is locally congruent to some type (A) hypersurface.

Adapted connections on metric contact manifolds

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka–Webster connection and to describe adapted connections whose Dirac operators are essentially self-adjoint. Finally, Schrödinger–Lichnerowicz-type formulae are provided for these operators.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH GENERALIZED TANAKA–WEBSTER 𝔇⊥-PARALLEL SHAPE OPERATOR

In a paper due to [I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster parallel shape operator, Kodai Math. J.34 (2011) 352–366] we have shown that there does not exist a hypersurface in G2(ℂm+2) with parallel shape operator in the generalized Tanaka–Webster connection (see [N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math.20 (1976) 131–190; S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc.314(1) (1989) 349–379]). In this paper, we introduce a new notion of generalized Tanaka–Webster 𝔇⊥-parallel for a hypersurface M in G2(ℂm+2), and give a characterization for a tube around a totally geodesic ℍ Pnin G2(ℂm+2) where m = 2n.

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

An $f$-structure on a manifold $M$ is an endomorphism field $\phi\in\Gamma(M,\End(TM))$ such that $\phi^3+\phi=0$. Any $f$-structure $\phi$ determines an almost CR structure $E_{1,0}\subset T_\C M$ given by the $+i$-eigenbundle of $\phi$. Using a compatible metric $g$ and connection $\nabla$ on $M$, we construct an odd first-order differential operator $D$, acting on sections of $\S=\Lambda E_{0,1}^*$, whose principal symbol is of the type considered in arXiv:0810.0338. In the special case of a CR-integrable almost $\S$-structure, we show that when $\nabla$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator $D$ is given by $D = \sqrt{2}(\dbbar+\dbbar^*)$, where $\dbbar$ is the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with $f$-structure that reduce to familiar methods in symplectic geometry in the case that $\phi$ is a compatible almost complex structure, and to the contact quantization defined in \cite{F4} when $\phi$ comes from a contact metric structure. The first is an index-theoretic approach involving the operator $D$; for certain group actions $D$ will be transversally elliptic, and using the results in arXiv:0810.0338, we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization.

The Tanaka-Webster connection and real hypersurfaces in a complex space form

We classify parallel real hypersurfaces in a complex space form for the generalized Tanaka-Webster connection.

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel shape operator

We introduce the notion of generalized Tanaka-Webster connection for hypersurfaces in complex two-plane Grassmannians G2(Cm+2) and give a non-existence theorem for Hopf hypersurfaces in G2(Cm+2) with parallel shape operator in this connection.