= Solution
The target metric is a coupling of the <string nonlinear sigma model>. Quantum consistency requires the gauge-fixed worldsheet theory to preserve <Weyl transformation>[Weyl invariance], so its <sigma-model beta functions> must vanish. With no B-field or varying dilaton, the metric beta function begins as
$$
\beta^G_{\mu\nu}=\alpha'R_{\mu\nu}+O(\alpha'^2).
$$
Therefore vanishing of the <worldsheet Weyl anomaly> requires
$$
\boxed{R_{\mu\nu}=0}
$$
to leading order in $\alpha'$: the target-space metric must be Ricci-flat at this order.
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