On Characterizations of Hopf Hypersurfaces in a Nonflat Complex Space Form with Anti-commuting Operators.

Let M be a real hypersurface in a complex space form M ( c), $${c \neq 0}$$ . In this paper we prove that if $${R_{\xi}(\phi A - A\phi) + (\phi A - A\phi)R_{\xi} = 0}$$ holds on M, then M is a Hopf hypersurface, where $${R_{\xi}}$$ is the structure Jacobi operator, A is the shape operator of M in M ( c) and $${\phi}$$ is the tangential projection of the complex structure of M ( c). We characterize such Hopf hypersurfaces of M ( c). [ABSTRACT FROM AUTHOR]