Left Maurer--Cartan form

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), let $G$ be a Lie group Lie group with identity $e$, and write $\mathfrak{g}=T_{e}G$. A $\mathfrak{g}$-valued one-form on $G$ means a smooth map $\alpha :TG\to \mathfrak{g}$ whose restriction ${\alpha }_g:T_{g}G\to \mathfrak{g}$ is linear for every $g$. Equivalently, $V\mapsto ({\pi }_G(V),\alpha (V))$ is a smooth vector-bundle map $TG\to G\times \mathfrak{g}$ over ${\mathrm{id}}_G$ in the sense of Vector bundle maps over a smooth base map.

The left Maurer--Cartan form is the $\mathfrak{g}$-valued one-form $\theta :TG\to \mathfrak{g}$ whose fibre map at $g$ is

${\theta }_g=d(L_{g^{-1}}{)}_g:T_{g}G⟶T_{e}G=\mathfrak{g}\mathrm{.}$

This formula is well-defined because left translation by $g^{-1}$ sends $g$ to $e$. It is smooth and fibrewise linear because Translations are diffeomorphisms and their differentials trivialize the tangent bundle identifies

$TG⟶G\times \mathfrak{g},V_g⟼(g,{\theta }_g{V}_g)$

as the inverse vector-bundle isomorphism to left trivialization. In particular, ${\theta }_e=d(L_e{)}_e={\mathrm{id}}_{\mathfrak{g}}$.

The stated ${\mathrm{A}\mathrm{C}}_{\omega }$ is used exactly through that supplied smooth left-trivialization theorem; the pointwise formula and passage to its second component add no choice. A Lie group is nonempty. When $\dim G=0$, $\theta$ is the unique map between zero tangent fibres, and the definition is unchanged in dimension one. Lie groups are boundaryless by convention, and no metric, nondegeneracy, endpoint, or biconditional occurs.