Real hypersurfaces in nonflat complex space forms with Reeb recurrence condition. (English) Zbl 07815339
\( M^{n}(c) \), \( c \in \mathrm{R} \) is a complete simply connected complex space form which is complex analytically
isometric to three classical manifolds. The authors define a real hypersurface in \( M^{n}(c) \) and it turns out that
there exists an almost contact metric structure on the real surface denoted by \((\Phi, \xi, \eta,g) \).The notion of real
hypersurface of type \( (A) \) is given by Cecil-Ryan (2015) and Niebergall-Ryan (1997). The \( (1,1) \) tensor field
\( L = \Phi A - A \Phi \), where \( A \) stands for the shape operator of the hypersurface is called the structure Lie operator
of a real hypersurface. Then the Reeb recurrence condition for \( L \), i.e. (1) \( \nabla_{\xi} L = \omega(\xi) L \) for certain
one form \( \omega(\xi) \) and the Lie Reeb recurrence condition (2) \( \mathcal{L}_{\xi} L = \omega(\xi) L \)for \( L \) are introduced.
According to Theorem 1.4 (main result) on a real hypersurface in a non-flat complex form the hypersurface is of type \( (A)
\iff L \) is Reeb recurrent operator (1) \( \iff L \) is Lie Reeb recurrent operator (2).
Reviewer: Angela Slavova (Sofia)