Comparing Lie Derivatives on Real Hypersurfaces in Complex Projective Spaces

On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka-Webster connection. Therefore, we can also consider their associated Lie derivatives. We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka-Webster connection either in the direction of the structure vector field xi or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M.