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It is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.
Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type
$\left( A \right)$
in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.