ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

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 of the flat connection

Example

For a smooth map $f : N \to M$ , the pullback of the flat connection on $M \times R^{r}$ is the componentwise derivative on the canonically identified product $f^{*} (M \times R^{r}) = N \times R^{r}$ .

Facts & Assumptions

Given: A smooth map and the specified product trivialization.

[F1]

Pullback connections have the pulled-back local connection matrix, on arbitrary pullback sections (Pullback connection is well defined and functorial).

[F2]

The product flat connection has zero matrix in the constant frame (The flat connection on a trivial vector bundle).

Verification

1.1

The identification sends $(q, (f (q), v))$ to $(q, v)$ , with smooth inverse $(q, v) \mapsto (q, (f (q), v))$ . It takes the pullback frame to the constant frame. By [F1] and [F2] the new matrix is $f^{*} 0 = 0$ , hence $\nabla_{X}^{f} (\sum_{a} u^{a} e_{a}) = \sum_{a} X (u^{a}) e_{a}$ for arbitrary smooth functions on $N$ .

F1F2given

2.1

In particular, even if $f : R \to M$ is constant, the pullback section $s (t) = t e_{1}$ has $\nabla_{\partial_{t}}^{f} s = e_{1}$ when $r \geq 1$ . It need not be a section pulled back from $M$ . Constant coefficients, in contrast, have zero derivative. Rank zero gives the unique zero operator and an empty source gives the empty bundle; no immersion, injectivity or nonzero differential is required.

step 1.1∎

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

Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)