Commuting complete flows give the global product action

The usual smooth real line is a Lie group under addition: addition and negation have polynomial coordinate formulas. Put $\mathrm{\Lambda }=2\pi \mathbb{Z}$. This is a subgroup, and it is normal because addition is abelian. It is closed: for $x\mathrm{\notin }\mathrm{\Lambda }$, choose a bounded open interval about $x$. Only finitely many integer multiples of the positive number $2\pi$ lie in that interval, and a smaller interval about $x$ avoids those finitely many points. Thus the complement of $\mathrm{\Lambda }$ is open. Countable choice follows from the supplied ZFC convention.

Apply the quotient theorems to obtain the quotient Lie group $C=\mathbb{R}\mathrm{/}\mathrm{\Lambda }$, with addition $[s]+[v]=[s+v]$, and the smooth surjective submersion $q:\mathbb{R}\to C$. Near each $c\in C$, the constant-rank coordinates express $q$ as a projection; fixing the discarded coordinates gives a smooth section $\sigma :U\to \mathbb{R}$ with $q\circ \sigma ={\mathrm{i}\mathrm{d}}_U$. Give $G=\mathbb{R}\times C$ the product smooth structure and componentwise addition. Its multiplication and inversion are smooth by the component and composition rules, so it is a Lie group with identity $(0,[0])$.

Periodicity gives ${\psi }_{s+2\pi n}={\psi }_s$ for every integer $n$, so the formula for $\alpha$ is independent of the representative of $[s]$. On $\mathbb{R}\times U\times M$, using a section from step 2.1, it is $\alpha ((t,c),p)={\varphi }_t({\psi }_{\sigma (c)}(p))$. The two flows are jointly smooth, and the component and composition rules make this local expression jointly smooth in all three variables. Such neighborhoods cover $G\times M$, proving joint smoothness of $\alpha$.

The zero-parameter flow identities give $\alpha ((0,[0]),p)=p$. For $a=(t,[s])$ and $b=(u,[v])$, commutation and the flow laws give ${\varphi }_t{\psi }_s{\varphi }_u{\psi }_v={\varphi }_{t+u}{\psi }_{s+v}$, hence $\alpha (a,\alpha (b,p))=\alpha (a+b,p)$ for every $p$. Together with joint smoothness, these are precisely the smooth left-action axioms. Each action map is an isometry: applying the metric-preservation identity successively to the differentials of ${\psi }_s$ and ${\varphi }_t$ preserves $g$ under their composite. Thus the claimed global smooth isometric action is obtained from the supplied global flows.