LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Scalar Curvature Under Conformal Rescaling

Statement

Let $M$ be a smooth $n$ -manifold with $n \geq 2$ , let $g$ be a smooth Lorentzian metric, and let $σ \in C^{\infty} (M, R)$ . Define $\tilde{g} = e^{2 σ} g$ and $□_{g} f : = g^{M N} \nabla_{M} \nabla_{N} f$ . Use the local Christoffel coefficients $Γ^{P}_{M N} = \frac{1}{2} g^{P Q} (\partial_{M} g_{N Q} + \partial_{N} g_{M Q} - \partial_{Q} g_{M N})$ , curvature $R^{P}_{Q M N} = \partial_{M} Γ^{P}_{N Q} - \partial_{N} Γ^{P}_{M Q} + Γ^{P}_{M R} Γ^{R}_{N Q} - Γ^{P}_{N R} Γ^{R}_{M Q}$ , Ricci tensor $R_{Q N} = R^{P}_{Q P N}$ , and scalar curvature $R_{g} = g^{Q N} R_{Q N}$ , with analogous definitions for $\tilde{g}$ . Then the pointwise identity is

R_{\tilde{g}} = e^{- 2 σ} (R_{g} - 2 (n - 1) □_{g} σ - (n - 1) (n - 2) g^{M N} (\partial_{M} σ) (\partial_{N} σ)) .

No field equation, compactness, boundary condition, or physical model is assumed.

Statement references

Sterckx, §3.2.1, equation (3.12)

Facts & Assumptions

Given: $M$ , $g$ , $σ$ , $\tilde{g} = e^{2 σ} g$ , and the curvature convention in the Statement.

[A1]

$g$ is nondegenerate and Lorentzian, so its inverse and Levi-Civita connection are defined; $e^{2 σ} > 0$ , so $\tilde{g}$ is also nondegenerate with the same signature.

Proof

1.1givenA1algebra

Write $σ_{M} = \partial_{M} σ$ , $σ^{P} = g^{P Q} σ_{Q}$ , and $C^{P}_{M N} : = {\tilde{Γ}}^{P}_{M N} - Γ^{P}_{M N}$ . Substituting ${\tilde{g}}_{M N} = e^{2 σ} g_{M N}$ and ${\tilde{g}}^{M N} = e^{- 2 σ} g^{M N}$ into the Christoffel formula in the Statement, the derivatives of $g$ reproduce $Γ^{P}_{M N}$ , while the derivatives of $e^{2 σ}$ give $C^{P}_{M N} = δ_{M}^{P} σ_{N} + δ_{N}^{P} σ_{M} - g_{M N} σ^{P}$ .

2.1step 1.1A1algebra

Because $C$ is a tensor, the Ricci definitions give ${\tilde{R}}_{M N} = R_{M N} + \nabla_{P} C^{P}_{M N} - \nabla_{N} C^{P}_{M P} + C^{P}_{P Q} C^{Q}_{M N} - C^{P}_{N Q} C^{Q}_{M P}$ . Using $\nabla g = 0$ and symmetry of the Hessian, the derivative terms are $\nabla_{P} C^{P}_{M N} - \nabla_{N} C^{P}_{M P} = - (n - 2) \nabla_{M} \nabla_{N} σ - g_{M N} □_{g} σ$ . Direct contraction of the displayed $C$ terms gives $C^{P}_{P Q} C^{Q}_{M N} = n (2 σ_{M} σ_{N} - g_{M N} ∣ d σ ∣_{g}^{2})$ and $C^{P}_{N Q} C^{Q}_{M P} = (n + 2) σ_{M} σ_{N} - 2 g_{M N} ∣ d σ ∣_{g}^{2}$ , where $∣ d σ ∣_{g}^{2} : = g^{P Q} σ_{P} σ_{Q}$ . Thus ${\tilde{R}}_{M N} = R_{M N} - (n - 2) \nabla_{M} \nabla_{N} σ - g_{M N} □_{g} σ + (n - 2) (σ_{M} σ_{N} - g_{M N} ∣ d σ ∣_{g}^{2})$ .

3.1givenA1step 2.1algebra∎

Contract the Ricci identity in step 2.1 with ${\tilde{g}}^{M N} = e^{- 2 σ} g^{M N}$ . The Hessian terms contribute $- (n - 2) □_{g} σ - n □_{g} σ = - 2 (n - 1) □_{g} σ$ ; the gradient terms contribute $(n - 2) ∣ d σ ∣_{g}^{2} - n (n - 2) ∣ d σ ∣_{g}^{2} = - (n - 1) (n - 2) ∣ d σ ∣_{g}^{2}$ . The resulting scalar-curvature identity is exactly the one in the Statement.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Colin Sterckx, Modave Lecture notes: Introduction to Exceptional Field Theory, arXiv:2410.19600v1