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

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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.

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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

1.1F2

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

2.1F2F3step 1.1

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.

3.1F1F2F3step 1.1step 2.1∎

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.

The exponential map is a local diffeomorphism at zero

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources