LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Vector bundle transition functions satisfy the cocycle identities

Statement

For vector bundle charts on a rank- $r$ bundle, the transition functions satisfy

$g_{α α} (p) = I_{r}, g_{γ α} (p) = g_{γ β} (p) g_{β α} (p)$

on every overlap where the expressions are defined.

Facts & Assumptions

Given: Three vector bundle charts $(U_{α}, Φ_{α})$ , $(U_{β}, Φ_{β})$ , and $(U_{γ}, Φ_{γ})$ on one rank- $r$ bundle.

[L1]

In a vector bundle chart overlap, $Φ_{β} \circ Φ_{α}^{- 1} (p, v) = (p, g_{β α} (p) v)$ with $g_{β α} (p) \in G L (r, R)$ (Vector bundle charts and transition functions).

Proof

technique · direct

1.1L1given

On $U_{α}$ , the map $Φ_{α} \circ Φ_{α}^{- 1}$ is the identity, so [L1] gives $(p, g_{α α} (p) v) = (p, v)$ for every $v \in R^{r}$ . Hence $g_{α α} (p) = I_{r}$ .

2.1L1step 1.1algebra∎

On a triple overlap, $Φ_{γ} \circ Φ_{α}^{- 1} = (Φ_{γ} \circ Φ_{β}^{- 1}) \circ (Φ_{β} \circ Φ_{α}^{- 1})$ . Applying [L1] to $(p, v)$ gives $(p, g_{γ α} (p) v) = (p, g_{γ β} (p) g_{β α} (p) v)$ for every $v$ , so $g_{γ α} (p) = g_{γ β} (p) g_{β α} (p)$ .

Depends on

Vector bundle charts and transition functions

Used by

Construction of a vector bundle from a smooth cocycle Theorem

Dependency tree · two levels

2 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)