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.
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.

Depends on

Used by

Local logarithm on a Lie group Definition
Roots of a compact connected Lie group Definition
Hyperbolic space as so zero n one mod so n Example
Peter–Weyl decomposition of L2(SU(2)) Example
Rational highest-weight modules from adjoint Plücker vectors Lemma
An ideal integrates to a connected immersed normal subgroup Proposition
Central quotients and intermediate character lattices Proposition
Differentiation and integration of highest weights Proposition
For connected groups, equivariance is equivalent to the moment-map Poisson bracket identity Proposition
Root and weight lattice sandwich Proposition
A homomorphism from a connected Lie group is determined by its differential at the identity Theorem
Analytic and root-system Weyl groups agree Theorem
Cartan closed subgroup theorem Theorem
Cayley transforms connect theta-stable Cartans in the classification Theorem
Compact connected Lie groups are classified by root data Theorem
Compact roots form a reduced crystallographic root system Theorem
Conjugacy of Cartan subalgebras Theorem
Conjugacy of maximal tori Theorem
Highest weights for compact connected groups Theorem
Semisimple compact groups up to isogeny Theorem
Structure of compact connected abelian Lie groups Theorem
The compact Weyl group is finite Theorem
Vogan diagram for a fixed Cartan involution is well defined up to equivalence Theorem

Dependency tree · two levels

21 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)