Notes on diffeomorphisms symmetry of $f(R)$ gravity in the cosmological
  context
  We study the metric perturbations in the context of restricted $f(R)$
gravity, in which a parameter for deviation from the full diffeomorphisms of
space-time is introduced. We demonstrate that one can choose the parameter to
remove the induced anisotropic stress, which is present in the usual $f(R)$
gravity. Moreover, to prevent instability for the vector and tensor metric
perturbations, some constraints on the model are obtained.
