New conditions on two tensors associated to the structure Jacobi operator of a real hypersurface in complex projective space

Abstract A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space that allows us to define, for any nonnull real number k , the so-called k th generalized Tanaka-Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R ξ of M . Following Tachibana, [“Analytic tensor and its generalization,” Tohoku Math. J. , vol. 12, pp. 208–221, 1960] we classify real hypersurfaces in complex projective space for which any of such tensors is either pure or hybrid with respect to the structure operator ϕ .