Lie and generalized Tanaka–Webster derivatives on real hypersurfaces in complex projective spaces

On a real hypersurface M in complex projective space we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. We classify real hypersurfaces such that both the Lie derivative associated to the Levi-Civita connection and the kth g-Tanaka–Webster derivative in the direction of the structure vector field ξ coincide when we apply them to either the shape operator or the structure Jacobi operator of M.