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

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Exponential map is natural for Lie-group homomorphisms

Statement

Assume ${A C}_{ω}$ . If $F : G \to H$ is a homomorphism of finite-dimensional real Lie groups, then for every $X \in g = T_{e} G$ ,

$F (\exp_{G} X) = \exp_{H} (d F_{e} X) .$

The countable-choice assumption is used exactly through the supplied one-parameter-subgroup/exponential characterization.

Facts & Assumptions

Given: ${A C}_{ω}$ , a Lie-group homomorphism $F : G \to H$ , and $X \in g = T_{e} G$ .

[F1]

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

[F2]

A one-parameter subgroup with initial velocity $Y$ is uniquely the curve $t \mapsto \exp (t Y)$ . One-parameter subgroups are exactly exponentials.

[F3]

The map $F$ is smooth, preserves identities, and satisfies $F (g h) = F (g) F (h)$ . Lie-group homomorphism, isomorphism, and automorphism.

[F4]

Differentials of smooth maps obey the chain rule. The chain rule for differentials of smooth maps.

Proof

technique · direct

1.1

By [F2], $a_{X} (t) = \exp_{G} (t X)$ is a one-parameter subgroup. Since [F3] makes $F$ a smooth group homomorphism, the composite $c (t) = F (a_{X} (t))$ is a one-parameter subgroup of $H$ .

F2F3

2.1

The same characterization [F2] gives $a_{X}^{'} (0) = X$ . Hence the chain rule [F4] yields $c^{'} (0) = d F_{e} (a_{X}^{'} (0)) = d F_{e} X .$

F2F3F4step 1.1

3.1

Apply [F2] in $H$ : the unique one-parameter subgroup with initial velocity $d F_{e} X$ is $t \mapsto \exp_{H} (t d F_{e} X)$ . Steps 1.1--2.1 identify $c$ with this curve. Evaluating at $t = 1$ gives $F (\exp_{G} X) = \exp_{H} (d F_{e} X) .$

F2step 1.1step 2.1

4.1

Both Lie groups are nonempty and boundaryless. Zero-dimensional source or target Lie algebras and dimension one require no change, including $X = 0$ . The one-parameter curves are global, so no endpoint issue occurs. No metric or nondegeneracy condition occurs. The only choice use is the stated ${A C}_{ω}$ , inherited through [F2]; composition with one supplied homomorphism and evaluation at one add no choice. No biconditional is asserted.

F1F2F3F4step 1.1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

16 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

Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)