PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Translations are diffeomorphisms and their differentials trivialize the tangent bundle

Statement

Assume ${A C}_{ω}$ . For every $g$ in a Lie group $G$ , the maps $L_{g}$ and $R_{g}$ are diffeomorphisms with respective inverses $L_{g^{- 1}}$ and $R_{g^{- 1}}$ . Moreover,

$Φ_{L} : G \times T_{e} G ⟶ T G, Φ_{L} (g, X) = d (L_{g})_{e} X,$

and

$Φ_{R} : G \times T_{e} G ⟶ T G, Φ_{R} (g, X) = d (R_{g})_{e} X,$

are smooth vector-bundle isomorphisms over ${id}_{G}$ .

The countable-choice assumption is used exactly through the supplied theorem that equips tangent bundles and global differentials with their smooth structures.

Facts & Assumptions

Given: ${A C}_{ω}$ , a Lie group $G$ with identity $e$ , and $g \in G$ .

[F1]

${A C}_{ω}$ is countable choice. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F2]

Left and right translations are $L_{g} (h) = g h$ and $R_{g} (h) = h g$ , and both are smooth. Left and right translations on a Lie group.

[F3]

Assuming ${A C}_{ω}$ , the global differential of a smooth map is smooth between the canonical smooth tangent bundles. Assuming countable choice, the global differential of a smooth map is smooth.

[F4]

The differential of a diffeomorphism is a linear isomorphism on every tangent space. The differential of a diffeomorphism is an isomorphism.

[F5]

A fibrewise bijective smooth bundle map over a diffeomorphism is a vector-bundle isomorphism. A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism.

Proof

technique · direct

1.1F2algebra

The group laws give $L_{g} \circ L_{g^{- 1}} = L_{g^{- 1}} \circ L_{g} = {id}_{G}$ and $R_{g} \circ R_{g^{- 1}} = R_{g^{- 1}} \circ R_{g} = {id}_{G}$ . All four translations are smooth by [F2], so $L_{g}$ and $R_{g}$ are diffeomorphisms with the asserted inverses.

1.2F1F2F3algebra

Let $m : G \times G \to G$ be multiplication. Near an arbitrary $g_{0}$ , choose product coordinates in which $m$ is represented by a smooth map $μ (a, b)$ . The local matrix of $d (L_{g})_{e}$ is the second-variable Jacobian $D_{2} μ (a, b_{e})$ , whose entries are smooth in $a$ . Equivalently, this is the restriction of the smooth global differential $d m$ supplied by [F3]. Therefore $Φ_{L}$ is a smooth bundle map. The same calculation with the variables reversed gives smoothness of $Φ_{R}$ .

2.1F4step 1.1

By [F4] and step 1.1, each fibre map $d (L_{g})_{e} : T_{e} G \to T_{g} G$ and $d (R_{g})_{e} : T_{e} G \to T_{g} G$ is a linear isomorphism. Hence $Φ_{L}$ and $Φ_{R}$ are fibrewise linear bijections over ${id}_{G}$ .

3.1F5step 1.1step 2.1step 1.2

Apply [F5] to the smooth fibrewise bijections from steps 2.1 and 1.2 over ${id}_{G}$ . Both $Φ_{L}$ and $Φ_{R}$ are vector-bundle isomorphisms. Fibrewise, their inverses are $(g, V) \mapsto (g, d (L_{g^{- 1}})_{g} V)$ and $(g, V) \mapsto (g, d (R_{g^{- 1}})_{g} V)$ , respectively.

4.1F1F2F3F4F5step 1.1step 2.1step 1.2step 3.1∎

A Lie group is nonempty. If $\dim G = 0$ , all tangent fibres are zero spaces and the displayed maps are the unique fibre maps; if $\dim G = 1$ , the same proof applies. No metric or nondegeneracy condition occurs, and the group is boundaryless by the page convention. The only choice assumption is the stated ${A C}_{ω}$ , used through [F3] for the canonical smooth tangent bundles/global differential; all group operations and local computations are supplied or pointwise and add no choice. The item asserts explicit inverse identities but no biconditional.

Translations are diffeomorphisms and their differentials trivialize the tangent bundle

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources