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

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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.
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The exponential map is a local diffeomorphism at zero

Statement

Assume ${A C}_{ω}$ . Let $G$ be a finite-dimensional real Lie group with identity $e$ and Lie algebra $g$ . There are open neighborhoods $V \subseteq g$ of $0$ and $U \subseteq G$ of $e$ such that

$\exp_{G} ∣_{V} : V ⟶ U$

is a diffeomorphism. The countable-choice assumption is inherited exactly from the supplied smoothness and identity-differential theorem.

Facts & Assumptions

Given: ${A C}_{ω}$ and a finite-dimensional real Lie group $G$ with identity $e$ and Lie algebra $g$ .

[F1]

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

[F2]

Assuming ${A C}_{ω}$ , $\exp_{G}$ is smooth, $\exp_{G} (0) = e$ , and $d (\exp_{G})_{0} = {id}_{g}$ . The Lie-group exponential map is smooth with identity differential at zero.

[F3]

A smooth map whose differential at a point is an isomorphism restricts to a diffeomorphism between neighborhoods of that point and its image. The smooth inverse function theorem on manifolds.

Proof

technique · direct

1.1

By [F2], $\exp_{G}$ is smooth, sends $0$ to $e$ , and its differential at $0$ is the identity of $g$ , hence a linear isomorphism.

2.1

Apply [F3] to $F = \exp_{G}$ at $0$ . Using step 1.1, obtain open neighborhoods $V$ of $0$ and $U$ of $e = \exp_{G} (0)$ such that $\exp_{G} ∣_{V} : V \to U$ is a diffeomorphism.

F2F3step 1.1

3.1

Lie groups are nonempty and boundaryless. If $\dim G = 0$ , then $V = {0}$ and $U = {e}$ may be chosen open and the restriction is the unique diffeomorphism; in dimension one the same inverse function theorem applies. The neighborhoods are open and contain their named points, so there is no endpoint issue. No metric or nondegeneracy condition occurs. The only choice use is the stated ${A C}_{ω}$ , inherited through [F2]; applying [F3] once adds no family choice. No biconditional is asserted.

F1F2F3step 1.1step 2.1∎

Depends on

Used by

Local logarithm on a Lie group Definition
An ideal integrates to a connected immersed normal subgroup Proposition
A homomorphism from a connected Lie group is determined by its differential at the identity Theorem
Cartan closed subgroup theorem Theorem

Dependency tree · two levels

19 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

Brian Conrad and Aaron Landesman, Compact Lie Groups (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)