Nonflat Complex Space Form Research Articles

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

AbstractLet M be a Hopf hypersurface in a nonflat complex space form , , of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].

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.

3-Dimensional real hypersurfaces with transversal $\eta$-Killing Ricci tensor field of a complex space form

We classify real hypersurfaces of a non-flat complex space form of complex dimension 2 satisfying the condition that the Ricci tensor S is the transversal Killing tensor field on the direction of the structure vector field $\xi$ and $S\xi = \beta\xi$ where $\beta$ is a function.

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.

English

In a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space), real hypersurfaces admit an almost contact metric structure (φ, ξ, η, g) induced from the ambient space. As a matter of course, many geometers have investigated real hypersurfaces in a nonflat complex space form from the viewpoint of almost contact metric geometry. On the other hand, it is known that the tensor field $$h( = \frac{1}{2}{\cal L}\xi \varphi )$$ plays an important role in contact Riemannian geometry. In this paper, we investigate real hypersurfaces in a nonflat complex space form from the viewpoint of the parallelism of the tensor field h.

Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

Let [Formula: see text] be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that [Formula: see text] is [Formula: see text]-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in [Formula: see text] or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in [Formula: see text]. Moreover, [Formula: see text]-parallel real hypersurfaces in [Formula: see text] and [Formula: see text] under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with [Formula: see text]-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

Cyclic η-parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms

Cyclic η-parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms

Hypersurfaces of the homogeneous nearly Kähler $\mathbb{S}^6$ and $\mathbb{S}^3\times\mathbb{S}^3$ with anticommutative structure tensors

Each hypersurface of a nearly K\"ahler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $\phi$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $A\phi+\phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly K\"ahler manifold $\mathbb{S}^3\times\mathbb{S}^3$ does not admit a hypersurface that satisfies the condition $A\phi+\phi A=0$.

Integral curves of the characteristic vector field of minimal ruled real hypersurfaces in non-flat complex space forms

The purpose of this paper is to give geometric characterizations of minimal ruled real hypersurfaces $M$ in a non-flat complex space form by paying particular attention to integral curves of the characteristic vector field on $M$.

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.

On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

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.

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form

AbstractWe first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.

Real Hypersurfaces with $^{*}$-Ricci Solitons of Non-flat Complex Space Forms

Kaimakamis and Panagiotidou in \cite{KP} introduced the notion of $^*$-Ricci soliton and studied the real hypersurfaces of a non-flat complex space form admitting a $^*$-Ricci soliton whose potential vector field is the structure vector field. In this article, we consider that a real hypersurface of a non-flat complex space form admits a $^*$-Ricci soliton whose potential vector field belongs to the principal curvature space and the holomorphic distribution.

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.

Curvature properties of homogeneous real hypersurfaces in nonflat complex space forms

In this paper, we study curvature properties of all homogeneous real hypersurfaces in nonflat complex space forms, and determine their minimalities and the signs of their sectional curvatures completely. These properties reflect the sign of the constant holomorphic sectional curvature $c$ of the ambient space. Among others, for the case of $c$ < 0 there exist homogeneous real hypersurfaces with positive sectional curvature and also ones with negative sectional curvature, whereas for the case of $c$ > 0 there do not exist any homogeneous real hypersurfaces with nonpositive sectional curvature.

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.

Conformally flat real hypersurfaces in nonflat complex planes

In this paper, we prove that there are no conformally flat real hypersurfaces in nonflat complex space forms of complex dimension two provided that the structure vector field is an eigenvector field of the Ricci operator. This extends some recent results by Cho (Conformally flat normal almost contact 3-manifolds, Honam Math. J. 38 (2016) 59–69) and Kon (3-dimensional real hypersurfaces with η-harmonic curvature, in: Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155–164).

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.

Certain types of real hypersurfaces in complex space forms

A nice characterization of real hypersurfaces of a nonflat complex space form is very much familiar when its structure vector field $$\xi $$ is Killing. This motivates us to classify a real hypersurface of a nonflat complex space form when the structure vector field $$\xi $$ is 2-Killing (i.e., $$\pounds _{\xi }\pounds _{\xi }g = 0)$$ , where $$\pounds _{\xi }$$ denotes the Lie derivative along the vector field $$\xi $$ . Further, we classify real hypersurfaces of a nonflat complex space form when the tensor field $$T (= \pounds _{\xi }g)$$ is (i) weakly Lie $$\xi $$ -parallel, and (ii) weakly parallel.