  Explicit construction of the second order left differential calculi on the
quantum group and its subgroups are obtained with the property of the natural
reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to
contain the 3-dimensional differential calculi on the quantum subgroup
$SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$,
$B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the
quantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$,
$T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$,
$Z_{+}^{(2)}(C)$ and on the their real forms. The classical limit ($q\to 1$) of
the left differential calculus is the nondeformed differential calculus. The
differential calculi on the Borel subgroups $B_{L}(C)$, $B_{U}(C)$ of the
$SL_{q}(2,C)$ coincide with two solutions of Wess-Zumino differential calculus
on the quantum plane $C_q(2|0)$.
  The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$
quantum group symmetry over two-dimensional nondeformed Minkovski space and in
the $\sigma$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum group
symmetry is considered. The Lagrangian formalism over the quantum group
manifolds is discussed. The variational calculus on the $SL_{q}(2,R)$ group
manifold is obtained. The classical solution of $C_{q}(2|0)$ {$\sigma$}-model
is obtained.
