The aim of the present paper is to prove the non-existence of real hypersurfaces equipped with recurrent structure Jacobi operator in a non-flat complex space form.Mathematics Subject Classification (2010). Primary 53C40; Secondary 53D15.Keywords. Almost contact manifold, Jacobi operator, real hypersurface. 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 = −JN where J is the complex structure of M n (c) and N is a unit normal vector field.Real hypersurfaces in CP n which are homogeneous, were classified by Takagi ([7]). Berndt ([1]) classified real hypersurfaces with principal structure vector fields in CH n n > 2. In a real hypersurface M of a complex space form M n (c), c = 0, the Jacobi operator on M with respect to the structure vector field ξ, is called the structure Jacobi operator and is denoted by lX = R ξ (X) = R(X, ξ)ξ, where R(X, Y )Z denotes the curvature operator

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 classify real hypersurfaces with pseudo-D-parallel structure Jacobi operator, in non-flat complex space forms.

We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn (c) satisfying the two conditions φ l = l φ , $\varphi l=l\varphi ,$ where l = R ( ⋅ , ξ ) ξ  and  φ $l=R(\cdot ,\xi )\xi \text{ and }\varphi $ is the almost contact metric structure of M, and ( ∇ ξ l ) X = $\left( {{\nabla }_{\xi }}l \right)X=$ ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.