= Solution
The string embedding coordinates form a <nonlinear sigma model> whose target-space metric is the spacetime metric $G_{\mu\nu}$. A consistent quantum string background must preserve worldsheet <Weyl transformation>[Weyl invariance], so all sigma-model beta functions must vanish, and the total matter-plus-ghost central charge must cancel. For a bosonic string with only a metric, this requires $D=26$ and, to leading order in $\alpha'$, $\beta^G_{\mu\nu}=\alpha'R_{\mu\nu}+O(\alpha'^2)=0$; thus the target metric must be Ricci-flat at leading order.
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