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

Vector bundle transition functions satisfy the cocycle identities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources