Abstract
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified.
Highlights
An n-dimensional Kähler manifold with constant holomorphic sectional curvature c called complex space form
Depending on the value of the holomorphic sectional curvature, a complete and connected complex space form can be analytically isometric to a complex projective space C Pn if c > 0, to a complex Euclidean space Cn if c = 0, or to a complex hyperbolic space C H n if c < 0
K the tensor field of type (1,1) FX, given by FX Y = F (k) ( X, Y ) for any Y ∈ TM can be considered. This operator is named the k-th Cho operator corresponding to X and is given by (k)
Summary
An n-dimensional Kähler manifold with constant holomorphic sectional curvature c called complex space form. Many problems of classification of real hypersurfaces in non-flat complex space forms are related to their Ricci tensor. We mention the classification of Hopf hypersurfaces in non-flat complex space forms with constant mean curvature and ξ-parallel Ricci tensor provided by Kimura and Maeda in [11]. More results on the study of real hypersurfaces in non-flat complex spaces forms in terms of their Ricci tensor are included in Section 6 of [14]. Associated to it, for any X tangent to M and any nonnull real number k the tensor field of type (1,1) FX , given by FX Y = F (k) ( X, Y ) for any Y ∈ TM can be considered This operator is named the k-th Cho operator corresponding to X and is given by FX Y = g(φAX, Y )ξ − η (Y )φAX − kη ( X )φY. At the end of the Section an open problem is stated
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