  In a previous work we obtained a set of necessary conditions for the linear
approximation in cosmology. Here we discuss the relations of this approach with
the so called covariant perturbations. It is often argued in the literature
that one of the main advantages of the covariant approach to describe
cosmological perturbations is that the Bardeen formalism is coordinate
dependent. In this paper we will reformulate the Bardeen approach in a
completely covariant manner. For that, we introduce the notion of pure and
mixed tensors, which yields an adequate language to treat both perturbative
approaches in a common framework. We then stress that in the referred covariant
approach one necessarily introduces an additional hyper-surface choice to the
problem. Using our mixed and pure tensors approach, we were able to construct a
one-to-one map relating the usual gauge dependence of the Bardeen formalism
with the hyper-surface dependence inherent to the covariant approach. Finally,
through the use of this map, we define full non-linear tensors that at first
order correspond to the three known gauge invariant variables $\Phi$, $\Psi$
and $\Xi$, which are simultaneously foliation and gauge invariant. We then
stress that the use of the proposed mixed tensors allows one to construct
simultaneously gauge and hyper-surface invariant variables at any order.
