Real Hypersurfaces With Cyclic-Parallel Structure Jacobi Operators in a Nonflat Complex Space Form

Ki, U-Hang; Kurihara, Hiroyuki

doi:10.1017/s0004972709000860

Cited by 10 publications

(8 citation statements)

References 15 publications

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“…In the past two decades, a number of papers dealt with the problem of characterizing real hypersurfaces in a non-flat complex space form under certain additional properties on which the real hypersurfaces being classified consist of subclasses of the Takagi's list, Montiel's list and ruled real hypersurfaces (see [6,8,9,11,[15][16][17][18]20], etc, for some recent papers and also the papers cited in [24]). …”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

On a Class of Real Hypersurfaces in a Complex Space Form with Weakly $$\phi $$ ϕ -Invariant Shape Operator

Loo

2015

Bull. Malays. Math. Sci. Soc.

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In this paper, we study real hypersurfaces in a complex space form with weakly φ-invariant shape operator, where φ is the almost contact structure on the real hypersurfaces induced by the complex structure on its ambient space. We first construct a class of real hypersurfaces with weakly φ-invariant shape operator in complex Euclidean spaces and complex projective spaces and then give a characterization of such a class of real hypersurfaces. With this results, we classify minimal real hypersurfaces with weakly φ-invariant shape operator in complex Euclidean spaces and in complex projective spaces.Keywords Complex space forms · Hopf hypersurfaces · Ruled real hypersurfaces · Weakly φ-invariant shape operator Mathematics Subject ClassificationPrimary 53B25 · Secondary 53C15 T.-H. Loo N a unit vector normal to M (cf. [3]). For a vector bundle V over M, we denote by (V) the module of all differentiable sections on V.Hopf hypersurfaces with constant principal curvatures, appearing in the Takagi's list (for c > 0) and Montiel's list (for c < 0), have been the main focus of the theory (cf. [22,27]). A typical example of non-Hopf real hypersurfaces in M n (c), for c = 0, is the class of ruled real hypersurfaces. Ruled real hypersurfaces in M n (c) are characterized by having a one-codimensional foliation whose leaves are totally geodesic complex hypersurfaces in M n (c) (cf. [19]).These real hypersurfaces mentioned above appeared to be standard spaces and play a central role in the study of real hypersurfaces in a non-flat complex space form. In the past two decades, a number of papers dealt with the problem of characterizing real hypersurfaces in a non-flat complex space form under certain additional properties on which the real hypersurfaces being classified consist of subclasses of the Takagi's list, Montiel's list and ruled real hypersurfaces (see [6,8,9,11,[15][16][17][18]20], etc, for some recent papers and also the papers cited in [24]).Real hypersurfaces, other than these standard spaces, have little been investigated, partly because the properties studied are too restrictive to be used for characterizing other classes of real hypersurfaces. It is natural to investigate certain conditions weak enough to include other classes of real hypersurfaces into the classification. This paper is a contribution along this line. We shall now briefly discuss the motivation for the paper as well as the results obtained.Besides the submanifold structure, represented by the shape operator A, on M, there is an almost contact metric structure (φ, ξ, η, , ) on M induced by the complex structure J of the ambient space. In [23,25] and [26], having considered the condition Theorem 2 Let M be a real hypersurface in M n (c), n ≥ 3. Suppose that M satisfies the condition dα(ξ ) is nowhere zero in an open dense subset of M,where α = Aξ, ξ . If the shape operator is weakly φ-invariant then M is locally congruent to a ruled real hypersurface.It is worthwhile to remark that there exist many examples of ruled real hypersurfaces M in M n (c) o...

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Section: Mathematics Subject Classificationmentioning

confidence: 99%

On a Class of Real Hypersurfaces in a Complex Space Form with Weakly $$\phi $$ ϕ -Invariant Shape Operator

Loo

2015

Bull. Malays. Math. Sci. Soc.

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“…Theorem 1.2 (Ki and Kurihara [7]). Let M be a real hypersurface in a nonflat complex space form which satisfies ∇ φ∇ ξ ξ R ξ = 0.…”

Section: Introductionmentioning

confidence: 99%

On the Structure Jacobi Operator and Ricci Tensor of Real Hypersurfaces in Nonflat Complex Space Forms

Kim¹

2010

Honam Mathematical Journal

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Abstract. It is known that there are no real hypersurfaces with parallel structure Jacobi operator R ξ (cf. [16], [17]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the structure Jacobi operator R ξ which are weaker than ∇R ξ = 0. Under further condition Sφ = φS for the Ricci tensor S we characterize Hopf hypersurfaces in a complex space form.

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“…Recently we have some classification theorems for real hypersurfaces in a nonflat complex space form with respect to the parallelism of R ξ (cf. [3], [4], [5]). …”

Section: Introductionmentioning

confidence: 99%

Jacobi Operators Along the Structure Flow on Real Hypersurfaces in a Nonflat Complex Space Form Ii

Ki¹,

Kurihara²

2011

Bulletin of the Korean Mathematical Society

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Abstract. Let M be a real hypersurface of a complex space form with almost contact metric structure (ϕ, ξ, η, g). In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator R ξ = R(·, ξ)ξ is ξ-parallel. In particular, we prove that the condition ∇ ξ R ξ = 0 characterizes the homogeneous real hypersurfaces of type A in a complex projective space or a complex hyperbolic space when R ξ ϕS = R ξ Sϕ holds on M , where S denotes the Ricci tensor of type (1,1) on M .

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Real Hypersurfaces With Cyclic-Parallel Structure Jacobi Operators in a Nonflat Complex Space Form

Cited by 10 publications

References 15 publications

On a Class of Real Hypersurfaces in a Complex Space Form with Weakly $$\phi $$ ϕ -Invariant Shape Operator

On a Class of Real Hypersurfaces in a Complex Space Form with Weakly $$\phi $$ ϕ -Invariant Shape Operator

On the Structure Jacobi Operator and Ricci Tensor of Real Hypersurfaces in Nonflat Complex Space Forms

Jacobi Operators Along the Structure Flow on Real Hypersurfaces in a Nonflat Complex Space Form Ii

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