Real hypersurfaces of non--flat complex space forms in terms of the Jacobi structure operator

Abstract

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.

Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

“…Certain authors have studied real hypersurfaces under conditions which include the operator [6][7][8][9]. Other authors have studied real hypersurfaces under the condition = , equipped with one or two additional conditions [10][11][12][13][14][15], proving that these hypersurfaces are Hopf and classifying them as type .…”