  After adding auxiliary fields and integrating out the original variables, the
Yang-Mills action can be expressed in terms of local gauge invariant variables.
This method reproduces the known solution of the two dimensional $SU(N)$
theory. In more than two dimensions the action splits into a topological part
and a part proportional to $\alpha_s$. We demonstrate the procedure for $SU(2)$
in three dimensions where we reproduce a gravity-like theory. We discuss the
four dimensional case as well. We use a cubic expression in the fields as a
space-time metric to obtain a covariant Lagrangian. We also show how the
four-dimensional $SU(2)$ theory can be expressed in terms of a local action
with six degrees of freedom only.
