Intrinsic properties of real hypersurfaces in complex space forms (Q2735679)

Let \(M^n(c)\), \(c\neq 0\), be a complex space form of complex dimension \(n\). In this paper, the author focusses on existence and classification results for hypersurfaces subjected to several conditions which are of an intrinsic nature. Several researchers treated such problems but in many cases complete results are only known under additional conditions: \(n\geq 3\) or the hypersurface satisfies the extrinsic condition of being a Hopf hypersurface. NEWLINENEWLINENEWLINEThe main aim of the author is to give a non-existence theorem for Einstein or Ricci-parallel hypersurfaces by means of a single simple proof. Earlier, the cases \(n=2\) and \(n\geq 3\) were treated separately.NEWLINENEWLINENEWLINEThe rest of the paper provides a survey on results about other classes of hypersurfaces such as hypersurfaces with cyclic-parallel or Codazzi Ricci tensor, semi-symmetric, Ricci-semi-symmetric and cyclic-Ryan hypersurfaces and finally, hypersurfaces which satisfy several recurrence conditions on the curvature. In all these cases, he states what has been settled yet (mainly when \(n\geq 3\) or for Hopf hypersurfaces) and points out the problems which are left open.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].