Exponential map of a Lie group

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$, let $G$ be a finite-dimensional real Lie group with identity $e$, and write $\mathfrak{g}=T_{e}G$. For each $X\in \mathfrak{g}$, the existence-and-uniqueness theorem One-parameter subgroups are integral curves of left-invariant fields supplies a unique one-parameter subgroup ${\gamma }_X:\mathbb{R}\to G$ with ${\gamma }_X^{\mathrm{\prime }}(0)=X$. The exponential map of $G$ is the well-defined total map

${\exp }_G:\mathfrak{g}⟶G,{\exp }_G(X):={\gamma }_X(1)\mathrm{.}$

Here ${\mathrm{A}\mathrm{C}}_{\omega }$ is the axiom of countable choice. The same theorem identifies ${\gamma }_X$ with the integral curve through $e$ of the left-invariant field $X^L$; the left translate $t\mapsto g{\gamma }_X(t)$ is therefore the corresponding integral curve through $g$. The countable-choice assumption is used exactly through that supplied smooth invariant-field and completeness result, and evaluation at the single time $1$ adds no choice.

A Lie group is nonempty and boundaryless. If $\dim G=0$, then $\mathfrak{g}=0$ and ${\exp }_G(0)=e$; the definition is unchanged in dimension one. No metric or nondegeneracy condition occurs, and $1$ is not an endpoint of the global parameter domain $\mathbb{R}$. This item defines a map and asserts no biconditional characterization.