LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (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 of a conformally Euclidean three-metric

Physical scope

A local differential-geometric identity for a Riemannian metric; it asserts no Einstein constraint or physical model.

Statement

Let $E \subset R^{3}$ be a nonempty open set with Cartesian coordinates $x^{1}, x^{2}, x^{3}$ , standard Euclidean metric $δ = δ_{i j} d x^{i} d x^{j}$ with $δ_{i j} = 1$ for $i = j$ and $0$ otherwise, inverse components $δ^{i j}$ , and Laplacian $Δ_{δ} = \sum_{i = 1}^{3} \partial_{i}^{2}$ . Let $u \in C^{\infty} (E, (0, \infty))$ and set $h = u^{4} δ$ . Form the product Lorentzian manifold $\hat{M} = R_{τ} \times E$ with metric $g = - d τ^{2} + h$ and future orientation $\partial_{τ}$ ; its slice $Σ = {0} \times E$ is a smooth spacelike hypersurface with induced metric $h$ . Use the induced connection $D$ and curvature convention supplied for this slice by the earlier normal-decomposition theorem. In Cartesian coordinates define $R^{ℓ}_{b c d}$ by $R (\partial_{c}, \partial_{d}) \partial_{b} = R^{ℓ}_{b c d} \partial_{ℓ}$ , ${Ric}_{i j}^{h} : = R^{k}_{i k j}$ , and $S_{h} : = h^{i j} {Ric}_{i j}^{h}$ . For $f = 2 \log u$ , write $∣ d f ∣_{δ}^{2} : = δ^{i j} (\partial_{i} f) (\partial_{j} f)$ . Then $S_{h} = - 8 u^{- 5} Δ_{δ} u .$

This statement concerns a smooth positive conformal metric and its induced scalar curvature. The product construction only fixes the geometric convention; no Einstein equation, energy condition, or completeness hypothesis is imposed.

Statement references

Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form

Facts & Assumptions

Given: The open Euclidean coordinate domain $E$ , its metric $δ$ , the positive smooth function $u$ , and the metric $h = u^{4} δ$ in the Statement.

[F1]

For an induced Riemannian metric $h$ on a smooth spacelike hypersurface, $D$ is its Levi-Civita connection, $R^{h}$ has the ambient curvature operator convention, and $S_{h}$ is its scalar curvature (Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form, proof step 1.1 and Statement clause (c)).

[A1]

$u$ is smooth and strictly positive, so $f : = 2 \log u$ is smooth and $h = e^{2 f} δ$ is a smooth Riemannian metric.

Proof

Given: [F1], [A1], and the Cartesian coordinates in the Statement.

1.1F1A1algebra

Write $Γ^{k}_{i j}$ for the Christoffel coefficients of $D$ in the coordinate frame. Torsion-freedom gives $Γ^{k}_{i j} = Γ^{k}_{j i}$ , and metric compatibility gives $\partial_{i} h_{j k} = h_{a k} Γ^{a}_{i j} + h_{j a} Γ^{a}_{i k}$ . Add this identity to the one with $i, j$ interchanged and subtract the one with $i, k$ interchanged; using symmetry of $h$ and $Γ$ yields $2 h_{k a} Γ^{a}_{i j} = \partial_{i} h_{j k} + \partial_{j} h_{i k} - \partial_{k} h_{i j}$ . Multiplying by $h^{ℓ k} / 2$ gives the coordinate Levi-Civita formula $Γ^{ℓ}_{i j} = \frac{1}{2} h^{ℓ k} (\partial_{i} h_{j k} + \partial_{j} h_{i k} - \partial_{k} h_{i j})$ .

2.1step 1.1A1algebra

Since $h_{i j} = e^{2 f} δ_{i j}$ and $h^{i j} = e^{- 2 f} δ^{i j}$ , substitution into step 1.1 gives $Γ^{k}_{i j} = δ_{i}^{k} f_{j} + δ_{j}^{k} f_{i} - δ_{i j} f^{k}$ , where $f_{i} = \partial_{i} f$ and $f^{k} = δ^{k ℓ} f_{ℓ}$ .

3.1F1step 2.1algebra

In a coordinate frame the definition $R (X, Y) Z = D_{X} D_{Y} Z - D_{Y} D_{X} Z - D_{[X, Y]} Z$ gives $R^{ℓ}_{b c d} = \partial_{c} Γ^{ℓ}_{d b} - \partial_{d} Γ^{ℓ}_{c b} + Γ^{ℓ}_{c e} Γ^{e}_{d b} - Γ^{ℓ}_{d e} Γ^{e}_{c b}$ ; contraction in the first and third slots gives $({R i c}_{h})_{i j} = \partial_{k} Γ^{k}_{i j} - \partial_{j} Γ^{k}_{k i} + Γ^{k}_{k ℓ} Γ^{ℓ}_{i j} - Γ^{k}_{j ℓ} Γ^{ℓ}_{k i}$ . Substituting step 2.1 and applying the product rule gives $\partial_{k} Γ^{k}_{i j} = 2 f_{i j} - δ_{i j} Δ_{δ} f$ , $\partial_{j} Γ^{k}_{k i} = 3 f_{i j}$ , $Γ^{k}_{k ℓ} Γ^{ℓ}_{i j} = 6 f_{i} f_{j} - 3 δ_{i j} ∣ d f ∣_{δ}^{2}$ , and $Γ^{k}_{j ℓ} Γ^{ℓ}_{k i} = 5 f_{i} f_{j} - 2 δ_{i j} ∣ d f ∣_{δ}^{2}$ . Therefore $({R i c}_{h})_{i j} = - f_{i j} - δ_{i j} Δ_{δ} f + f_{i} f_{j} - δ_{i j} ∣ d f ∣_{δ}^{2}$ .

4.1F1A1step 3.1algebra∎

Contract the Ricci tensor in step 3.1 with $h^{i j} = e^{- 2 f} δ^{i j}$ to obtain $S_{h} = e^{- 2 f} (- 4 Δ_{δ} f - 2 ∣ d f ∣_{δ}^{2})$ . For $f = 2 \log u$ , the chain rule gives $Δ_{δ} f = 2 u^{- 1} Δ_{δ} u - 2 u^{- 2} ∣ d u ∣_{δ}^{2}$ and $∣ d f ∣_{δ}^{2} = 4 u^{- 2} ∣ d u ∣_{δ}^{2}$ . The two $∣ d u ∣_{δ}^{2}$ contributions cancel, and $e^{- 2 f} = u^{- 4}$ , so $S_{h} = - 8 u^{- 5} Δ_{δ} u$ .

Remarks

The identity is proved here from the library's curvature sign convention. It is used only for explicit smooth conformal metrics; no scalar-curvature sign condition or completeness assertion is assumed.

Depends on

Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form