= Polar-coordinate T-dual background
{title2=$d\widetilde s^2=dr^2+r^{-2}d\widetilde z^2,\ \widetilde\Phi=-\log r$}
The flat polar metric $dr^2+r^2dz^2$ has a circle isometry away from $r=0$. Its local <Buscher rules> give the displayed dual metric and <dilaton>. The dual <Ricci tensor> has components $R_{rr}=-2/r^2$ and $R_{\widetilde z\widetilde z}=-2/r^4$, while the dilaton Hessian has components $1/r^2$ and $1/r^4$. Thus the <leading metric-dilaton Weyl condition> $R_{ab}+2\nabla_a\nabla_b\Phi=0$ holds despite nonzero curvature. Flat spectators complete a critical target; the scalar anomaly condition also holds because $|\nabla\Phi|^2=1/r^2$, $\Box\Phi=2/r^2$ and $R=-4/r^2$. The construction is local on $r>0$, excluding the degenerating circle.
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