ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

The Euclidean metric as a symmetric two-tensor

Example

On $R^{n}$ , the Euclidean metric

g = \sum_{i = 1}^{n} d x^{i} \otimes d x^{i}

is a smooth section of the symmetric subbundle of $T_{2}^{0} R^{n}$ .

Facts & Assumptions

Given: The Euclidean metric $g$ on $R^{n}$ .

[F1]

The symmetric two-tensors form a fibrewise subbundle of the covariant tensor bundle (Symmetric and alternating covariant tensor subbundles).

[L1]

That fibrewise symmetric part is a smooth vector subbundle (Symmetric and alternating images are smooth subbundles).

Verification

technique · direct

1.1

The coefficients of $g$ in the standard coordinates are constant, so $g$ is smooth.

given

2.1

For vectors $u, v$ , one has $g (u, v) = g (v, u)$ , so each fibre value $g_{p}$ is symmetric. Hence [F1] places $g$ in the symmetric fibrewise part, and [L1] identifies that part as a smooth subbundle.

F1L1step 1.1

3.1

Therefore the Euclidean metric is a symmetric smooth two-tensor.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)