  A model of SU(2) gauge theory in the space-time $R\times S^3$ is constructed
in terms of local gauge-invariant variables. A metric tensor $g_{\mu \nu}$ is
defined starting with the components of the strength tensor $F_{\mu \nu}^k$ and
of its dual $\widetilde{F}_{\mu \nu}^k$. It is shown that the components
$g_{\mu \nu}$ determine the gauge field equations if some supplementary
constraints are imposed. Two families of analitical solutions of the field
equations are also obtained.
