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We consider ${\cal N}=1$ supersymmetric gauge theories in the conformalwindow. By applying a suitable matter superfield rescaling and aWeyl-transformation the renormalisation group running (matter and gauge field$Z$-factors) are absorbed into the metric. The latter becomes a function of the$Z$-factors. The Euler flow $\Delta a \equiv a_{\rm UV} - a_{\rm IR} |_{{\calN}=1}$ is then obtained by free field theory computation with the non-trivialdynamics coming from expanding the Euler invariant in the flow dependentmetric. The result is therefore directly obtained in terms of the infraredanomalous dimension confirming an earlier result using the matching ofconserved currents.

${\cal N}$ = $1$ Euler Anomaly from RG-dependent metric-Background