  We discuss the existence of Gribov ambiguities in $SU(m)\times U(1)$ gauge
theories over the $n-$spheres. We achieve our goal by showing that there is
exactly one conjugacy class of groups of gauge transformations for the theories
given above. This implies that these transformation groups are conjugate to the
ones of the trivial $SU(m)\times U(1)$ fiber bundles over the $n-$spheres. By
using properties of the space of maps $Map_{\ast}(S^n,G)$ where $G$ is one of
$U(1)$, $SU(m)$ we are able to determine the homotopy type of the groups of
gauge transformations in terms of the homotopy groups of $G$. The
non-triviality of these homotopy groups gives the desired result.
