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The conformal condition is a quantum worldsheet Weyl anomaly cancellation, not the classical metric equation of the Polyakov action. Assume a smooth target metric, no antisymmetric background field, and curvature/gradient scales large compared with $\sqrt{{\alpha }^{\prime }}$. Use $[{\nabla }_a,{\nabla }_b]v^c=R^c{}_{dab}{v}^d$ and $R_{ab}=R^c{}_{acb}$, consistent with the Ricci-scalar Weyl transformation in the hint.

A sign convention in the printed action must be made explicit. With worldsheet signature $(-,+)$, continuation $\tau =-i{\tau }_E$ and $e^{i{S}_L}=e^{-S_E}$ sends its negative kinetic term to a positive Euclidean kinetic term, while its positive Lorentzian curvature coupling becomes a negative Euclidean curvature coupling. Define the conventional Euclidean dilaton by $\phi =-\Phi$ for this literal action. Then $S_{E,\mathrm{dil}}=(4\pi {)}^{-1}\int \sqrt{h}{R}^{(2)}\phi$. The Lorentzian dilaton coupling sign convention is therefore important: a convention with a negative Lorentzian curvature coupling would instead use $\phi =\Phi$ and reverse every term linear in the printed $\Phi$ below.

For completeness, the metric part of the one-loop calculation can be obtained by a geodesic background field expansion of a string sigma model. Write $X={\exp }_{\bar{X}}Y$. Its quadratic Euclidean action containsHere $D_{\mu }{Y}^a={\partial }_{\mu }{Y}^a+{\Gamma }^a{}_{bc}{\partial }_{\mu }{\bar{X}}^b{Y}^c$. In dimension $2-\epsilon$, the ultraviolet coincident contraction has pole $⟨Y^a{Y}^b{⟩}_{\mathrm{div}}={\alpha }^{\prime }{g}^{ab}/\epsilon$ with an infrared regulator. Contracting the curvature term gives a divergence $-(4\pi \epsilon {)}^{-1}\int \sqrt{h}{R}_{cd}\partial {\bar{X}}^c\partial {\bar{X}}^d$. It is cancelled by the metric counterterm $\delta g_{ab}={\alpha }^{\prime }{R}_{ab}/\epsilon$, giving the one-loop sigma-model beta function ${\beta }_{ab}^g={\alpha }^{\prime }{R}_{ab}+O({\alpha }^{\prime 2})$ before the dilaton improvement. Equivalently its local Euclidean Weyl variation is $-(4\pi {)}^{-1}\int \sqrt{h}\omega R_{ab}\partial X^a\partial X^b$, with terms proportional to the embedding equations understood as field redefinitions.

Set $\Omega =e^{\omega }$ in the supplied curvature transformation. On a closed string worldsheet, integration by parts givesThe chain rule and leading string embedding map equation implyThe kinetic embedding equation suffices for extracting the leading metric coefficient. More precisely, the full Euclidean embedding equation makes the parenthesis $({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$, giving a curvature contribution $-({\alpha }^{\prime }/4\pi )\int \sqrt{h}\omega R^{(2)}|\nabla \phi {|}^2$ to the Weyl variation. This contributes to the scalar coefficient below rather than to the leading metric tensor coefficient. Combining the curvature counterterm anomaly with this variation gives the leading metric-dilaton Weyl condition:This is the gravitational equation in the string frame, rather than an ordinary Einstein equation with a minimally coupled scalar. In terms of the field and signs literally printed in this Lorentzian action it readsThe frequently used $R_{ab}+2{\nabla }_a{\nabla }_b\Phi =0$ is obtained if the printed $\Phi$ is identified with the conventional Euclidean dilaton, which requires the opposite Lorentzian curvature-coupling sign. Both conventions describe the same mathematics after $\Phi \mapsto -\Phi$, but one cannot change only the field equation silently. Also, vanishing of this tensor coefficient is the metric part of Weyl invariance; the dilaton curvature coefficient and central-charge condition remain to be checked.

Now derive the scalar consequence without presupposing its integration constant. The contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. The Ricci identity applied to the gradient of a scalar givesDiverging the metric equation and then substituting $R_{ab}=-2{\nabla }_a{\nabla }_b\phi$ yieldsThus the bracket is constant on each connected target component. Its trace equation is $R+2{\square }_g\phi =0$, so the dilaton equation from contracted Bianchi identity becomesThe second equation uses the literal printed sign convention. These are nonlinear scalar wave equations analogous to the Klein-Gordon equation. For the exponentiated dilaton wave equation, put $F=e^{-2\phi }=e^{2\Phi }$; differentiating twice givesHere ${\square }_g={\nabla }^a{\nabla }_a$ is the Lorentzian Laplace-Beltrami operator, and $|\nabla \phi {|}^2=g^{ab}{\partial }_a\phi {\partial }_b\phi$ is a Lorentzian contraction, not necessarily nonnegative.

The Bianchi and Ricci identities alone do not imply $C=0$. The linear dilaton counterexample to zero integration constant is flat target space with $\phi =q_a{X}^a$: the tensor equation holds for every constant $q$, while $C=-2q^2$. A non-null $q$ disproves any deduction of the zero-constant scalar equation from that tensor equation alone.

The leading dilaton Weyl anomaly coefficient is obtained as follows. The matter and reparameterization-ghost central charges give the constant $(D-26)/6$. Expanding the curvature coupling along the same geodesic fluctuation gives the quadratic term $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$; its coincident contraction is cancelled by $\delta \phi =-{\alpha }^{\prime }{\square }_g\phi /(2\epsilon )$, giving the term $-{\alpha }^{\prime }{\square }_g\phi /2$. The full embedding-equation contribution identified above supplies $+{\alpha }^{\prime }|\nabla \phi {|}^2$. Therefore full leading-order bosonic-string Weyl invariance also requiresIt fixes $C=(D-26)/(3{\alpha }^{\prime })$. Hence in the critical $D=26$ theory, or under boundary conditions making the constant vanish,This explains precisely the extra input needed for a zero-mass Klein-Gordon-type equation. In noncritical dimension the corresponding equation has the central-charge-deficit constant, subject to the usual controlled-background assumptions.

The Euler-Lagrange field equation obtained by varying the nonminimally coupled scalar field isOne can check stress-energy conservation directly. The divergence of the kinetic contribution is $(\square \Phi ){\nabla }_b\Phi$, because second derivatives of a scalar commute. With $f={\Phi }^2$, the contracted Bianchi identity gives ${\nabla }^a{G}_{ab}=0$, while the Ricci identity givesConsequentlySince ${\nabla }_{b}f=2\Phi {\nabla }_b\Phi$, the full stress-energy tensor satisfies

The general reason is the diffeomorphism Noether identity for a scalar field. Under the Lie derivative along a compactly supported vector field $X$, $\delta g_{ab}=2{\nabla }_{(a}{X}_{b)}$ and $\delta \Phi =X^a{\nabla }_a\Phi$. Invariance of the action under diffeomorphisms gives, after integration by parts,Arbitrariness of $X$ gives the same divergence identity. Conservation requires the scalar equation of motion, with no need to impose the Einstein field equations for the background metric.

For the torsion-free Levi-Civita connection, cyclically summing the displayed coordinate expression and using ${\Gamma }^a{}_{bc}={\Gamma }^a{}_{cb}$ gives the algebraic first Bianchi identityApply the Jacobi identity to three covariant-derivative commutators, or differentiate the coordinate formula in normal coordinates. The third-derivative terms and the derivatives of quadratic connection terms cancel cyclically, giving the differential second Bianchi identityContracting the first and third curvature indices and then contracting once more yieldsor the contracted Bianchi identity

For the metric variation $h_{ab}=\delta g_{ab}$,while the two double-divergence terms in the supplied metric variation of scalar curvature become a boundary term after integration by parts for tensor fields. The Einstein-Hilbert action therefore contributes $G_{ab}+\Lambda g_{ab}$. Varying the Proca action with respect to the metric gives the Proca stress-energy tensor, and the full Einstein-Proca theory equation is

Since $\delta F_{ab}=2{\nabla }_{[a}\delta A_{b]}$, variation with respect to $A_b$ and one integration by parts give the covariant Proca equationTaking its covariant divergence givesbecause the resulting contractions pair the symmetric Ricci tensor with the antisymmetric electromagnetic field tensor. Since $m\ne 0$, the Lorenz constraint in Proca theory follows:Moreover, $F=dA$ and $d^2=0$, so the Electromagnetic Bianchi identity is

Finally, use that identity to differentiate the matter tensor. The result groups naturally asBoth terms vanish by the Proca equation and its Lorenz constraint in Proca theory. Thus ${\nabla }^a{T}_{ab}=0$, exactly as required by the contracted Bianchi identity applied to the Einstein field equations. The vector equation is therefore consistent with the gravitational equation.

The Killing equation gives ${\nabla }_{\mu }{K}^{\mu }=0$ and ${\nabla }_{[\mu }{K}_{\lambda ]}={\nabla }_{\mu }{K}_{\lambda }$. Commuting derivatives once and contracting the Ricci identity givesTherefore, using the contracted Bianchi identity and symmetry of $R_{\mu \beta }$,Since ${\nabla }_{[\mu }{K}_{\lambda ]}$ is antisymmetric, swapping the two contracted index names shows that the left side equals its derivative-antisymmetrized form:Now set $A_{\mu \lambda }={\nabla }_{[\mu }{K}_{\lambda ]}$. The right side is the contracted commutator $[{\nabla }^{\mu },{\nabla }^{\lambda }]A_{\mu \lambda }$. By the curvature commutator on a covariant tensor, it is a sum of contractions of the symmetric Ricci tensor with the antisymmetric tensor $A$, and hence vanishes. Thusso the Killing vector preserves scalar curvature.

At an arbitrary point choose normal coordinates, so the Christoffel symbols vanish there. Torsion freedom and commutation of partial derivatives then give the algebraic first Bianchi identityDifferentiate the coordinate curvature expression and cyclically antisymmetrize. Third derivatives cancel, giving the second Bianchi identityBoth statements are tensorial and hence hold in every coordinate system. Contracting the differential identity, using the curvature symmetries and metric compatibility, yieldsthe contracted Bianchi identity.

Contract the isotropic-curvature formula on $a,c$. In dimension $n$ it givesSubstitution into the contracted Bianchi identity givesSince $n>2$, ${\nabla }_{b}K=0$, so $K$ is constant on each connected component. Thus this is constant sectional curvature, withThis argument is Schur theorem in pseudo-Riemannian geometry.

Contract the Riemann tensor to obtainand define the Einstein tensorContracting the differential Bianchi identity twice, using the curvature symmetries and metric compatibility, gives the contracted Bianchi identityIt follows immediately thatThe Einstein equations with cosmological constant areSince $\nabla g=0$, the left side is divergence-free, so consistency requires and implies the local conservation law ${\nabla }^{\nu }{T}_{\mu \nu }=0$.

At an arbitrary point choose normal coordinates, so ${\Gamma }^a{}_{bc}=0$ there. Torsion freedom and commuting partial derivatives immediately give the algebraic first Bianchi identityDifferentiating the coordinate curvature formula at that point and cyclically antisymmetrizing gives the differential identityBoth equations are tensorial, so validity in normal coordinates at every point proves them in every coordinate system.

Contract the differential identity on its first and third curvature indices and use the algebraic symmetries of the Riemann tensor. One obtains the contracted Bianchi identity

Define the Ricci tensor by contracting the first and third Riemann indices:Contract the first Bianchi identity in its upper index and third lower index. The middle term vanishes by antisymmetry in the first pair, while antisymmetry in the final pair turns the last term into $-R_{\gamma \alpha }$. Thusso the Ricci tensor is symmetric.

For the stated constant-curvature form,The contracted Bianchi identity isUsing $\nabla g=0$ givesorIn four dimensions,More generally, $K$ is constant for every $n>2$. In dimension $n=2$, the contracted Bianchi identity gives no such conclusion, and the Gaussian curvature may vary from point to point.

Expand the Einstein tensor about the Minkowski metric. The quadratic term in the metric perturbation defines $t_{\mu \nu }=-G_{\mu \nu }^{(2)}[h,h]/(8\pi )$, a symmetric tensor relative to the background. For a first-order vacuum solution, the expanded contracted Bianchi identity gives ${\partial }^{\mu }{t}_{\mu \nu }=0$. Equivalently, a second-order correction $j$ obeys $G^{(1)}[j]=8\pi t$, whose flat divergence vanishes identically. This conservation is an on-shell statement. The local $t$ depends on perturbative gauge and the background split; conservation alone does not make it a gauge-independent local gravitational stress-energy tensor.

For the Levi-Civita connection, the cyclic covariant derivative of curvature vanishes: ${\nabla }_{[e}{R}^a{}_{|b|cd]}=0$. Contracting this identity gives the contracted Bianchi identity.

Rescaling the spatial metric by the squared static lapse function gives a useful three-dimensional quotient metric $\gamma$. In Vacuum Einstein equations, its Ricci tensor obeys $R_{ij}(\gamma )=2{\partial }_{i}U{\partial }_{j}U$, while $U$ is harmonic. The contracted Bianchi identity recovers the harmonic equation without dividing at critical points.

For action $S=\int \sqrt{-g}[-\frac{1}{2}(\nabla \Phi {)}^2-\xi R{\Phi }^2]d^{4}x$, metric variation defines $\delta S=\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$ and gives the displayed tensor. The contracted Bianchi identity and the scalar derivative commutator yield ${\nabla }^a{T}_{ab}=(\square \Phi -2\xi R\Phi ){\nabla }_b\Phi$, so the stress-energy tensor is conserved on the field equation.