= Dilaton equation from contracted Bianchi identity
{title2=$\Box_g\varphi-2|\nabla\varphi|^2=C$}
Diverging $R_{ab}+2\nabla_a\nabla_b\varphi=0$ and using the <contracted Bianchi identity> and <Ricci identity> gives $\nabla_b(R+4\Box_g\varphi-4|\nabla\varphi|^2)=0$. The trace equation $R+2\Box_g\varphi=0$ then gives $\Box_g\varphi-2|\nabla\varphi|^2=C$, with $C$ constant on each connected component. Zero does not follow from the tensor equation alone. The additional dilaton anomaly coefficient fixes $C=(D-26)/(3\alpha')$ at leading order for the bosonic string without an antisymmetric background. In critical dimension it is zero; noncritical backgrounds have a central-charge-deficit term.
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