The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space TpM of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξl)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξl = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.
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 (
The aim of the present paper is the classification of real hypersurfaces M, whose Jacobi structure Operator is generalized ξ−parallel. The notion of generalized ξ−parallel Jacobi structure Operator is rather new and much weaker than ξ− parallel Jacobi structure Operator which has been studied so far.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.