  The algebraic formulation of the quantum group covariant noncommutative
geometry in the framework of the $R$-matrix approach to the theory of quantum
groups is given. We consider structure groups taking values in the quantum
groups and introduce the notion of the noncommutative connections and
curvatures transformed as comodules under the "local" coaction of the structure
group which is exterior extension of $GL_{q}(N)$. These noncommutative
connections and curvatures generate $ GL_{q}(N)$-covariant quantum algebras.
For such algebras we find combinations of the generators which are invariants
under the coaction of the "local" quantum group and one can formally consider
these invariants as the noncommutative images of the Lagrangians for the
topological Chern-Simons models, non-abelian gauge theories and the Einstein
gravity. We present also an explicit realization of such covariant quantum
algebras via the investigation of the coset construction
$GL_{q}(N+1)/(GL_{q}(N)\otimes GL(1))$.
