PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

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.

Local coordinate formula for a bundle connection

Statement

In a local frame $e$ , write $s = e u$ with coefficient column $u$ . Then $\nabla_{X} s = e (X (u) + ω (X) u), \nabla s = e (d u + ω u) .$ Here the last expression means an $E$ -valued one-form, and $X (u)$ and $d u$ act entrywise.

Facts & Assumptions

Given: A connection restricted to an open frame domain, a local section $s = \sum_{j} u^{j} e_{j}$ , and a local vector field $X$ .

[F1]

The connection matrix is defined by the derivatives of the frame sections (Connection one form in a local frame).

[F2]

Directional differentiation is real-linear and obeys the section Leibniz rule (Connection laws in directional form).

Proof

1.1

Apply the Leibniz rule to each of the finitely many summands: $\nabla_{X} s = \sum_{j} X (u^{j}) e_{j} + \sum_{j} u^{j} \nabla_{X} e_{j}$ . Inserting the frame derivatives yields $\sum_{i} (X (u^{i}) + \sum_{j} ω^{i}_{j} (X) u^{j}) e_{i}$ .

F1F2

2.1

This is exactly the first matrix formula. At each point, every tangent vector is the value of a local coordinate vector-field combination; equality upon all such evaluations therefore gives the one-form formula. The calculation is valid on an empty frame domain, with an empty sum in rank zero, and with one summand in rank one. On a zero-dimensional base both differentiated functions and one-forms vanish. No choice of a global frame is used.

step 1.1∎

Depends on

Used by

Dual connection Definition
The difference of two connections is an endomorphism valued one form Proposition
Connection one form transformation law Theorem

Dependency tree · two levels

9 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)