No abstract

No abstract

No abstract

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

We study 3-τ -α-manifolds having Qξ parallel to characteristic field ξ and the scalar curvature constant along the geodesic foliation generated by e or φe. We find out a new class of contact metric 3-manifolds and we give non-flat examples of this class. (2000): 53D10. Mathematics Subject Classification