The aim of the present paper is the study of some classes of real hypersurfaces equipped with the condition φl = lφ, (l = R(., ξ)ξ).
MSC: 53C40, 53D15Keywords: real hypersurfaces, almost contact manifold, Jacobi structure operator.
Introduction.An n -dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called complex space form, which is denoted by M n (c). The complete and simply connected complex space form is a projective space CP n if c > 0, a hyperbolic space CH n if c < 0, or a Euclidean space C n if c = 0. The induced almost contact metric structure of a real hypersurface M of M n (c) will be denoted by (φ, ξ, η, g).Real hypersurfaces in CP n which are homogeneous, were classified by R. Takagi ([15]). J. Berndt ([1]) classified real hypersurfaces with principal structure vector fields in CH n , which are divided into the model spaces A 0 , A 1 , A 2 and B.Another class of real hypersurfaces were studied by Okumura [13], and Montiel and Romero [12], who proved respectively the following theorems.
The aim of the present paper is to prove the nonexistence of real hypersurfaces with D-recurrent structure Jacobi operator, in non-flat complex planes. Our results complement work of several other authors who worked in CP n and CH n for n ≥ 3.Keywords Real hypersurface · Structure Jacobi operator · Recurrent tensor field Mathematics Subject Classification (2010) 53C40 · 53D15
IntroductionAn n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called complex space form, which is denoted by M n (c). The complete and simply connected complex space form is complex analytically isometric to a projective space CP n if c > 0, a hyperbolic space CH n if c < 0, or a Euclidean space C n if c = 0. The induced almost contact metric structure of a real hypersurface M of M n (c) will be denoted by (φ, ξ, η, g). The vector field ξ is defined by ξ = −J N where J is the complex structure of M n (c) and N is a locally defined unit normal vector field.Real hypersurfaces in CP n which are homogeneous, were classified by Takagi (1973). Berndt (1989) classified real hypersurfaces with principal structure vector Th. Theofanidis · Ph. J. Xenos (
Motivated by the work done in [4], [5], [12] and [15], we classify real hypersurfaces in CP 2 and CH 2 equipped with pseudo-parallel structure Jacobi operator.
The ξ-parallelness condition of the structure Jacobi operator of real hypersurfaces has been studied in combination with additional conditions. In the present paper we study three dimensional real hypersurfaces in CP 2 or CH 2 equipped with ξ-parallel structure Jacobi operator. We prove that they are Hopf hypersurfaces and if additional η(Aξ) = 0, we give the classification of them.
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