Abstract
In this paper a new condition for the Lie derivative of the structure Jacobi operator of real hypersurfaces in non-flat complex space forms is introduced. A classification theorem for real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator satisfies this condition and at the same time the commuting condition of the shape operator with the structure Jacobi operator, is presented. Furthermore, three-dimensional real hypersurfaces in non-flat complex space forms, whose structure Jacobi operator satisfies the new condition, are classified.
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