REAL HYPERSURFACES WITH Φ-Invariant SHAPE OPERATOR IN a COMPLEX PROJECTIVE SPACE

We characterize geodesic spheres with su‰ciently small radii in a complex hyperbolic space of constant holomorphic sectional curvature cð< 0Þ by using their geometric three properties. These properties are based on their contact forms, geodesics and shape operators. These geodesic spheres are the only examples of hypersurfaces of type (A) which are of nonnegative sectional curvature in this ambient space. Moreover, in particular, when À1 e c < 0, the class of these geodesic spheres has just one example of Sasakian space forms.

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“…M is locally congruent to a hypersurface of type ðAÞ with radius r ¼ p=ð2 ffiffi ffi c p Þ (see [11]).…”

Section: Sectional Curvatures Of Geodesic Spheresmentioning

confidence: 99%

Sectional curvatures of geodesic spheres in a complex hyperbolic space

Kajiwara

Maeda

2015

Kodai Math. J.

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“…(cf. [14,21]). It is clear that A is φ-invariant if and only if it satisfies (1) and hence Theorem 1 completely classifies real hypersurfaces with φ-invariant shape operator.…”

Section: Introductionmentioning

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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Hopf Hypersurfaces

Cecil¹,

Ryan²

2015

Springer Monographs in Mathematics

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REAL HYPERSURFACES WITH Φ-Invariant SHAPE OPERATOR IN a COMPLEX PROJECTIVE SPACE

Cited by 4 publications

References 8 publications

Sectional curvatures of geodesic spheres in a complex hyperbolic space

Sectional curvatures of geodesic spheres in a complex hyperbolic space

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

Hopf Hypersurfaces

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