DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Pullback comparison of a Lorentzian metric

Definition

Let $M$ be a smooth four-manifold, let $g$ be a smooth symmetric covariant $2$ -tensor on $M$ , and let $Φ : M \to M$ be a smooth diffeomorphism. The pullback tensor $Φ^{*} g$ is the smooth covariant $2$ -tensor defined at each $p \in M$ by

$(Φ^{*} g)_{p} (X, Y) : = g_{Φ (p)} (d Φ_{p} X, d Φ_{p} Y), X, Y \in T_{p} M .$

When $g$ is Lorentzian, this formula defines the metric components seen in the coordinate representative obtained by $Φ$ . Equality of $Φ^{*} g$ and $g$ means equality of the bilinear forms on each tangent space, not equality of their component arrays in a chosen chart.

Definition references

Smooth covariant 2-tensor fields on a smooth manifold

The differential of a smooth map

$C^{r}$ and smooth maps between smooth manifolds

Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality

Depends on

Dependency tree · 2 levels

Sources

J. Sbierski, On the Existence of a Maximal Cauchy Development for the Einstein Equations — a Dezornification, arXiv:1309.7591v3 (2015), Section 2, Definition 2.1 and Remark 2.5: pullback of induced metric data and diffeomorphism-invariance convention