A study of real hypersurfaces with Ricci operators in 2-dimensional complex space forms

Abstract

We prove that a real hypersurface M in complex projective space P 2 ‫)ރ(‬ or complex hyperbolic space H 2 ‫,)ރ(‬ whose Ricci operator is η-parallel and commutes with the structure tensor on the holomorphic distribution, is a Hopf hypersurface. We also give a characterization of this hypersurface.

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

“…Here we remark that main results shown in [10, 26] are true only for real hypersurfaces of dimension >3. In this paper, by means of Theorem 1.3, we obtain some new characterization results for type ( A ) and ( B ) real hypersurfaces and extend those in [14, 18]. Moreover, in terms of Theorem 1.3 we obtain some non‐existence results of three‐dimensional real hypersurfaces with recurrent Ricci tensor, extending Kon and Loo's result (see [17]) for real hypersurfaces of dimension three and answering partially a question proposed in [4].…”

“…In view of this, many authors studied Ricci η‐parallelism together with some other conditions. We refer the reader to [10, 14, 18, 23, 26] for many results related with such topic. Here we remark that main results shown in [10, 26] are true only for real hypersurfaces of dimension >3.…”

The property of real hypersurfaces in 2-dimensional complex space form with Ricci operator

“…Additionally, with respect to the Ricci operator and η -parallel, Song and 2 of the present authors [3] proved the following.…”

“…, where S and η(X) denote the Ricci operator and the dual 1-form of the unit vector field ξ . Additionally, with respect to the Ricci operator and η -parallel, Song and 2 of the present authors [3] proved the following.…”