Hypersurfaces In Non-flat Complex Space Research Articles

Parallelism of Structure Lie Operators on Real Hypersurfaces in Nonflat Complex Space Forms

Parallelism of Structure Lie Operators on Real Hypersurfaces in Nonflat Complex Space Forms

Real Hypersurfaces in Nonflat Complex Space Forms with Reeb Recurrence Condition

In this paper we prove that the structure Lie operator of a real hypersurface in a nonflat complex space form is Reeb recurrent if and only if it is Lie Reeb recurrent, and this is equivalent to that the hypersurface is of type (A).

Some Problems on Ruled Hypersurfaces in Nonflat Complex Space Forms

We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.

REAL HYPERSURFACES IN COMPLEX SPACE FORMS ATTAINING EQUALITY IN AN INEQUALITY INVOLVING A CONTACT δ-INVARIANT

AbstractWe investigate real hypersurfaces in nonflat complex space forms attaining equality in an inequality involving the contact δ-invariant δc(2) introduced by Chen and Mihai in [3].

𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms

Abstract Kaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H 2(c) with tanh ⁡ ( | c | 2 r ) = 1 2 $\begin{array}{} \displaystyle \tanh(\frac{\sqrt{|c|}}{2}r) = \frac{1}{2} \end{array}$ or a ruled real hypersurface.

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.

Ricci Curvature of Real Hypersurfaces in Non-flat Complex Space Forms

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which admit a unit vector field satisfying identically the equality case of the inequality.

Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = ϕ A - A ϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified.

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.

Conditions of Parallelism of $^{*}$-Ricci Tensor of Three Dimensional Real Hypersurfaces in Non-flat Complex Space Forms

This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in complex hyperbolic space are provided. Furthermore, new results concerning $\xi$-parallelism of $^{*}$-Ricci tensor of real hypersurfaces in non-flat complex space forms are presented.

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.

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.

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

PARALLEL*-RICCI TENSOR OF REAL HYPERSURFACES IN $\mathbb{C}P^{2}$ AND $\mathbb{C}H^{2}$

In this paper the idea of studying real hypersurfaces in non-flat complex space forms, whose *-Ricci tensor satisfies geometric conditions is presented. More precisely, three dimensional real hypersurfaces in non-flat complex space forms with parallel *-Ricci tensor are studied. At the end of the paper ideas for further research on $^{*}$-Ricci tensor are given.

*-Ricci solitons of real hypersurfaces in non-flat complex space forms

In this paper the notion of *-Ricci soliton is introduced and real hypersurfaces in non-flat complex space forms admitting a *-Ricci soliton with potential vector field being the structure vector field ξ are studied. At the end of the paper discussion on this new notion and ideas for further work are presented.

A New Condition on the Structure Jacobi Operator of Real Hypersurfaces in Non-flat Complex Space Forms

In this paper a new condition for the Lie derivative of the structure Jacobi operator of real hypersurfaces in non-flat complex space forms is introduced. A classification theorem for real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator satisfies this condition and at the same time the commuting condition of the shape operator with the structure Jacobi operator, is presented. Furthermore, three-dimensional real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator satisfies the new condition, are classified.

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.

Axioms of planes on real hypersurfaces in complex space forms

A Riemannian manifold M satisfies the axiom of 2-planes if at each point $ p\in{M} $ and each tangent 2-plane V at the point, there exists a totally geodesic submanifold L with $ p\in{L} $ and $ T_pL=V $ . On the other hand, real hypersurfaces in complex space forms have nice families of tangent 2-planes. If we restrict the above definition to those planes, several definitions of axioms of 2-planes arise naturally. We classify real hypersurfaces in non-flat complex space forms satisfying these axioms.