  We consider the Yang-Mills equations with a matrix gauge group $G$ on the de
Sitter dS$_4$, anti-de Sitter AdS$_4$ and Minkowski $R^{3,1}$ spaces. On all
these spaces one can introduce a doubly warped metric in the form $d s^2 =-d
u^2 + f^2 d v^2 +h^2 d s^2_{H^2}$, where $f$ and $h$ are the functions of $u$
and $d s^2_{H^2}$ is the metric on the two-dimensional hyperbolic space $H^2$.
We show that in the adiabatic limit, when the metric on $H^2$ is scaled down,
the Yang-Mills equations become the sigma-model equations describing harmonic
maps from a two-dimensional manifold (dS$_2$, AdS$_2$ or $R^{1,1}$,
respectively) into the based loop group $\Omega G=C^\infty (S^1, G)/G$ of
smooth maps from the boundary circle $S^1=\partial H^2$ of $H^2$ into the gauge
group $G$. From this correspondence and the implicit function theorem it
follows that the moduli space of Yang-Mills theory with a gauge group $G$ in
four dimensions is bijective to the moduli space of two-dimensional sigma model
with $\Omega G$ as the target space. The sigma-model field equations can be
reduced to equations of geodesics on $\Omega G$, solutions of which yield
magnetic-type configurations of Yang-Mills fields. The group $\Omega G$
naturally acts on their moduli space.
