Konstantina Panagiotidou

Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

Abstract In this paper three dimensional real hypersurfaces in non-flat complex space forms whose k-th Cho operator with respect to the structure vector field ξ commutes with the structure Jacobi operator are classified. Furthermore, it is proved that the only three dimensional real hypersurfaces in non-flat complex space forms, whose k-th Cho operator with respect to any vector field X orthogonal to structure vector field commutes with the structure Jacobi operator, are the ruled ones. Finally, results concerning real hypersurfaces in complex hyperbolic space satisfying the above conditions are also provided.

REAL HYPERSURFACES IN A NON-FLAT COMPLEX SPACE FORM WITH LIE RECURRENT STRUCTURE JACOBI OPERATOR

The aim of this paper is to introduce the notion of Lie recurrent structure Jacobi operator for real hypersurfaces in non-flat complex space forms and to study such real hypersurfaces. More precisely, the non-existence of such real hypersurfaces is proved.

ON THE LIE DERIVATIVE OF REAL HYPERSURFACES IN ℂP 2 AND ℂH 2 WITH RESPECT TO THE GENERALIZED TANAKA-WEBSTER CONNECTION

Abstract. In this paper the notion of Lie derivative of a tensor fieldT of type (1,1) of real hypersurfaces in complex space forms with re-spect to the generalized Tanaka-Webster connection is introduced andis called generalized Tanaka-Webster Lie derivative. Furthermore, threedimensional real hypersurfaces in non-flat complex space forms whosegeneralized Tanaka-Webster Lie derivative of 1) shape operator, 2) struc-ture Jacobi operator coincides with the covariant derivative of them withrespect to any vector field X orthogonal to ξ are studied. 1. IntroductionA complex space form is an n-dimensional Kahler manifold of constant holo-morphic sectional curvature c. A complete and simply connected complex spaceform is analytically isometric to a complex projective space CP n if c > 0, or toa complex Euclidean space C n if c = 0, or to a complex hyperbolic space CH n if c < 0. The complex projective and complex hyperbolic spaces are callednon-flat complex space forms, since c 6= 0 and the symbol M

The normal Jacobi operator of real hypersurfaces in complex hyperbolic two-plane Grassmannians

The aim of this paper is to introduce the notion of normal Jacobi operator of real hypersurfaces in complex hyperbolic two-plane Grassmannians. Furthermore, results concerning real hypersurfaces in complex hyperbolic two-plane Grassmannians whose normal Jacobi operator satisfies conditions of parallelism will be given.

The *-Ricci Tensor of Real Hypersurfaces in Symmetric Spaces of Rank One or Two

Complex projective and hyperbolic spaces, i.e. non-flat complex space forms, are symmetric spaces of rank one. Complex two-plane Grassmannians are symmetric spaces of rank two. Let M be a real hypersurface in a symmetric space of rank one or two. Many geometers, such as Berndt, Jeong, Kim, Ortega, Perez, Santos, Suh, Takagi and others have studied real hypersurfaces in above spaces in terms of their operators and tensor fields. This paper will be divided into two parts. Firstly, results concerning real hypersurfaces in non-flat complex space forms in terms of their∗-Ricci tensor, S ∗, which in case of real hypersurfaces was first studied by Hamada (Real hypersurfaces of complex space forms in terms of Ricci *-tensor. Tokyo J. Math. 25, 473–483 (2002)), will be presented. More precisely, it will be answered if there exist or not real hypersurfaces, whose∗-Ricci tensor is parallel, semi-parallel, i.e. R ⋅ S ∗ = 0, or pseudo-parallel, i.e. \(R(X,Y ) \cdot S^{{\ast}} = L\{(X \wedge Y ) \cdot S^{{\ast}}\}\) with L ≠ 0 (Kaimakamis and Panagiotidou, Parallel∗-Ricci tensor of real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Taiwan. J. Math., to appear, DOI 10.11650/tjm.18.2014.4271; Kaimakamis and Panagiotidou, Conditions of parallelism of∗-Ricci tensor of real hypersurfaces in \(\mathbb{C}P^{2}\) and \(\mathbb{C}H^{2}\). Preprint). Secondly, the formula of∗-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians will be provided (Panagiotidou, The∗-Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, work in progress).