Action-angle coordinates on a saturated proper submersion neighbourhood of a compact connected regular fibre, with the two-form primitive supplied locally

Regular-fibre geometry: by the projection charts of [F6], ${\pi }^{-1}(b)$ is an embedded $n$-manifold with tangent space $\ker d{\pi }_m$. Involution gives $d{F}_j(X_{F_i})=\omega (X_{F_j},X_{F_i})=0$, so the $X_{F_i}$ are tangent. If ${\sum }_i{c}_i{X}_{F_i}=0$, contraction with $\omega$ gives ${\sum }_i{c}_{i}d{F}_i=0$, and regularity makes every $c_i=0$. These $n$ fields therefore span $\ker d{\pi }_m$. Involution makes their span isotropic; it equals its symplectic orthogonal because any vector orthogonal to all $X_{F_i}$ lies in $\ker d{\pi }_m$. Thus each fibre is Lagrangian. Vertical fields and the fibrewise action: for $\alpha \in {\mathrm{\Omega }}^1(B)$ define the smooth field $X_{\alpha }$ on $U$ by ${\iota }_{X_{\alpha }}\omega ={\pi }^{\ast }\alpha$, which exists uniquely by [F8]; writing $\alpha ={\sum }_i{\alpha }_{i}d{F}_i$ with $C^{\mathrm{\infty }}$ coefficients ${\alpha }_i$ (the $F_i$ are the coordinate functions of the ball $B$), the uniqueness in [F1] gives $X_{\alpha }={\sum }_i{\alpha }_i{X}_{F_i}$; by [F2] the fields $X_{F_i}$ are tangent to every fibre, and for $m\in {\pi }^{-1}(b)$ the evaluation map $T_b^{\ast }B\to T_m{\pi }^{-1}(b)$, ${\alpha }_b\mapsto X_{{\alpha }_b}(m)$, is a linear isomorphism: it is injective because ${\pi }^{\ast }{\alpha }_b=0$ at $m$ forces ${\alpha }_b=0$ ($d{\pi }_m$ is surjective, $\pi$ being a submersion), and both spaces have dimension $n$. By [F3] the restricted fields are complete on every compact fibre, so their commuting flows combine into a global ${\mathbb{R}}^n$-action on each fibre, and by [F4] and [F2] each fibre ${\pi }^{-1}(b)$ is a single transitive orbit with stabilizer a full lattice.

The action map: shrink $B$ if necessary so that there is a smooth section $\sigma :B\to U$ through a point of $N$ ([F6]; a local section exists by the local projection form and can be restricted to a smaller ball). Define $a:T^{\ast }B\to U$, $a({\alpha }_b):={\mathrm{\Phi }}_{{\alpha }_b}^1(\sigma (b))$, the time-one flow of the vertical field $X_{{\alpha }_b}$ starting at $\sigma (b)$; it is defined because the flow preserves the compact fibre ${\pi }^{-1}(b)$ (step 1.1) and the restricted field is complete there. In the base coordinates $F_i$, writing ${\alpha }_b={\sum }_i{c}_{i}d{F}_i{\mathrm{\mid }}_b$, this map is the composite ${\mathrm{\Phi }}_{X_{F_n}}^{c_n}\circ \cdots \circ {\mathrm{\Phi }}_{X_{F_1}}^{c_1}(\sigma (b))$. Each smooth field has a smooth flow on its open domain by The fundamental theorem on flows, and each indicated finite time is admissible by fibrewise completeness; thus near every $(b,c)$ the displayed composite is smooth, proving smoothness of $a$. Then $Da({\alpha }_b)$ is invertible for every ${\alpha }_b$: the base component of $Da$ is the identity because $\pi \circ a=\pi$ pointwise, and the vertical component at ${\alpha }_b$ is the derivative in $s$ at $s=0$ of ${\mathrm{\Phi }}_{{\alpha }_b+s{\beta }_b}^1(\sigma (b))$, which by commutativity of the flows equals $X_{{\beta }_b}(a({\alpha }_b))$; the map $T_b^{\ast }B\to T_{a({\alpha }_b)}{\pi }^{-1}(b)$, ${\beta }_b\mapsto X_{{\beta }_b}(a({\alpha }_b))$, is the isomorphism of step 1.1, so the vertical component is an isomorphism and hence by [F6] $a$ is a local diffeomorphism.

The period structure: put $\mathrm{\Lambda }:=a^{-1}(\sigma (B))$; near each of its points $\mathrm{\Lambda }$ is an $n$-dimensional smooth submanifold of $T^{\ast }B$ because $a$ is a local diffeomorphism there and $\sigma (B)$ is a smooth submanifold ([F6]); moreover, since $\pi \circ a=\pi$, the projection $\pi {\mathrm{\mid }}_{\mathrm{\Lambda }}$ has invertible differential, so $\mathrm{\Lambda }$ is locally the graph of a smooth section of $T^{\ast }B\to B$. The fibre ${\mathrm{\Lambda }}_{b_0}$ is the stabilizer of the point $\sigma (b_0)$ of the ${\mathbb{R}}^n$-action of step 1.1, hence by [F4] a full lattice in the fibre $T_{b_0}^{\ast }B$; choose a $\mathbb{Z}$-basis ${\lambda }_1,\dots ,{\lambda }_n$ of it, extend each to a smooth local sheet ${\beta }_i$ of $\mathrm{\Lambda }$ near $b_0$, and shrink $B$ so that all ${\beta }_i:B\to T^{\ast }B$ are defined, are sections of $\mathrm{\Lambda }$, and their values stay linearly independent.

A Lagrangian section: the pullback $\tau :={\sigma }^{\ast }\omega$ is a smooth closed two-form on $B$ (closed because $d\tau ={\sigma }^{\ast }d\omega =0$); shrink $B$ to a metric ball, which is star-shaped, and by [F7] applied to the two-form $\tau$ choose a smooth one-form $\alpha$ on this ball with $d\alpha =\tau$. Define the translated section ${\sigma }^{\mathrm{\prime }}:={\mathrm{\Phi }}_{-\alpha }^1\circ \sigma$. Then by [F5], $({\mathrm{\Phi }}_{-\alpha }^1{)}^{\ast }\omega =\omega +{\pi }^{\ast }d(-\alpha )=\omega -{\pi }^{\ast }d\alpha$, so $({\sigma }^{\mathrm{\prime }}{)}^{\ast }\omega ={\sigma }^{\ast }({\mathrm{\Phi }}_{-\alpha }^1{)}^{\ast }\omega =\tau -d({\sigma }^{\ast }{\pi }^{\ast }\alpha )=\tau -d\alpha =0$, where ${\sigma }^{\ast }{\pi }^{\ast }\alpha =\alpha$; hence ${\sigma }^{\mathrm{\prime }}$ is a Lagrangian section. In the cotangent-bundle trivialization define $T_{-\alpha }({\eta }_b)={\eta }_b-\alpha (b)$, a smooth diffeomorphism with inverse $T_{\alpha }$. By commutativity of the fibre action the map based on the new section is $a^{\mathrm{\prime }}({\eta }_b):={\mathrm{\Phi }}_{{\eta }_b}^1({\sigma }^{\mathrm{\prime }}(b))=a({\eta }_b-\alpha (b))=a\circ T_{-\alpha }({\eta }_b)$. Consequently $a^{\mathrm{\prime }}$ is smooth and is a local diffeomorphism everywhere, by step 2.1. The stabilizer of a transitive abelian fibre action is independent of the basepoint, so the period lattices computed using $\sigma$ and ${\sigma }^{\mathrm{\prime }}$ agree.

Full local generation: after shrinking $B$, for every $b\in B$ the periods satisfy ${\mathrm{\Lambda }}_b={\sum }_i\mathbb{Z}{\beta }_i(b)$. If no such neighbourhood exists, Countable Choice selects $b_r\to b_0$ and a period outside the generated sublattice. Subtract an integer combination of the ${\beta }_i(b_r)$ so the remaining nonzero period is ${\gamma }_r={\sum }_i{c}_{ri}{\beta }_i(b_r)$ with $c_{ri}\in [-1\mathrm{/}2,1\mathrm{/}2]$. Work in the fixed trivialization $T^{\ast }B=B\times ({\mathbb{R}}^n{)}^{\ast }$, and choose a closed base ball about $b_0$ contained in the domain of all smooth sheets. Eventually every $b_r$ lies in this ball. The vectors $(b_r,c_r)$ lie in one fixed bounded subset of ${\mathbb{R}}^{2n}$; by For $n\ge 1$ every bounded sequence in ${\mathbb{R}}^n$ has a convergent subsequence pass to a subsequence with $c_r\to c_0$. The map $(b,c)\mapsto {\sum }_i{c}_i{\beta }_i(b)$ is continuous on the compact base ball times $[-1\mathrm{/}2,1\mathrm{/}2{]}^n$, so ${\gamma }_r\to {\gamma }_0\in T_{b_0}^{\ast }B$ in this same trivialization. Continuity of $a$ and $a({\gamma }_r)=\sigma (b_r)$ give $a({\gamma }_0)=\sigma (b_0)$, hence ${\gamma }_0\in {\mathrm{\Lambda }}_{b_0}$. Write ${\gamma }_0={\sum }_i{m}_i{\beta }_i(b_0)$ with integers $m_i$, and put ${\delta }_r={\gamma }_r-{\sum }_i{m}_i{\beta }_i(b_r)$. Then ${\delta }_r$ remains a nonzero period and tends to $0_{b_0}$, while $a({\delta }_r)=\sigma (b_r)=a(0_{b_r})$. Both arguments eventually lie in one neighbourhood of $0_{b_0}$ on which $a$ is injective, by step 2.1, contradiction.

The sheets are closed: for each $i$, the time-one flow ${\mathrm{\Phi }}_{{\beta }_i}^1$ of $X_{{\beta }_i}$ acts as the identity on every fibre, because ${\beta }_i(b)\in {\mathrm{\Lambda }}_b$ is in the stabilizer of every point of the transitive abelian action on ${\pi }^{-1}(b)$ (step 4.1); hence $({\mathrm{\Phi }}_{{\beta }_i}^1{)}^{\ast }\omega =\omega$; on the other hand, by [F5] and ${\iota }_{X_{{\beta }_i}}\omega ={\pi }^{\ast }{\beta }_i$, one has $\frac{d}{dt}({\mathrm{\Phi }}_{{\beta }_i}^t{)}^{\ast }\omega =({\mathrm{\Phi }}_{{\beta }_i}^t{)}^{\ast }d{\pi }^{\ast }{\beta }_i={\pi }^{\ast }d{\beta }_i$, so $\omega =({\mathrm{\Phi }}_{{\beta }_i}^1{)}^{\ast }\omega =\omega +{\pi }^{\ast }d{\beta }_i$ and ${\pi }^{\ast }d{\beta }_i=0$; since $\pi$ is a submersion, ${\pi }^{\ast }$ is injective on forms and $d{\beta }_i=0$.

Action coordinates: by [F7], applied on the ball $B$, each closed one-form ${\beta }_i$ has a smooth primitive $I_i$ with $d{I}_i={\beta }_i$; since ${\beta }_1(b),\dots ,{\beta }_n(b)$ are a basis of $T_b^{\ast }B$ for every $b\in B$ (step 3.1), the differential of $I=(I_1,\dots ,I_n):B\to {\mathbb{R}}^n$ is invertible everywhere, so by [F6] $I$ is a local diffeomorphism; restrict first to one inverse-function neighbourhood of $b_0$, choose an open ball $A$ about $I(b_0)$ in its image, and set $B_0:=I^{-1}(A)$. Then $I:B_0\to A$ is a diffeomorphism. Restrict all subsequent constructions to $B_0$ and its saturated preimage $U_0$; the primitives have already been constructed on the original ball, so no star-shapedness of $B_0$ is required.

Angle coordinates and the diffeomorphism: for $b\in B_0$ and $p\in {\pi }^{-1}(b)$ there is a unique $\theta (p)\in T^n=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n$ such that $p$ is obtained from ${\sigma }^{\mathrm{\prime }}(b)$ by the action of a vector $s\in {\mathbb{R}}^n$ with ${\theta }_j(p)\equiv s_j(\mathrm{m}\mathrm{o}\mathrm{d}1)$ through the covector ${\sum }_j{s}_j{\beta }_j(b)$: existence is transitivity of the fibre action (step 1.1 and [F4]), and uniqueness modulo ${\mathbb{Z}}^n$ is exactly the stabilizer statement ${\mathrm{\Lambda }}_b={\sum }_i\mathbb{Z}{\beta }_i(b)$ of step 4.1 read in the coefficient basis ${\beta }_j(b)$. This defines a smooth map $(I,\theta ):U_0\to A\times T^n$ which is bijective and, by the local diffeomorphism property of the shifted map $a^{\mathrm{\prime }}=a\circ T_{-\alpha }$ proved in step 3.2, composed with the smooth fibrewise basis map $(b,s)\mapsto {\sum }_j{s}_j{\beta }_j(b)$, a diffeomorphism onto $A\times T^n$; in particular ${\mathrm{\partial }}_{{\theta }_j}$ corresponds to the generator $X_{d{I}_j}$ of the action.

The symplectic form in action-angle coordinates: let ${\omega }^{\mathrm{\prime }\mathrm{\prime }}:={\sum }_{i}d{\theta }_i\wedge d{I}_i$ be evaluated in the coordinate chart $(I,\theta )$ of step 7.1, which is the standard two-form of [F8]. In the coordinate basis $({\mathrm{\partial }}_{I_1},\dots ,{\mathrm{\partial }}_{I_n},{\mathrm{\partial }}_{{\theta }_1},\dots ,{\mathrm{\partial }}_{{\theta }_n})$ the two forms vanish on pairs of vertical vectors, because ${\mathrm{\partial }}_{{\theta }_j}$ corresponds to $X_{d{I}_j}$ (step 7.1), the fields $X_{d{I}_1},\dots ,X_{d{I}_n}$ span the tangent spaces of the Lagrangian fibres ([F2]), and the two-form ${\omega }^{\mathrm{\prime }\mathrm{\prime }}$ has the factor $d{I}_i$, which annihilates vertical vectors; they agree on vertical-horizontal pairs, because $\omega (X_{d{I}_j},{\mathrm{\partial }}_{I_k})={\iota }_{X_{d{I}_j}}\omega ({\mathrm{\partial }}_{I_k})={\pi }^{\ast }d{I}_j({\mathrm{\partial }}_{I_k})={\delta }_{jk}$ while ${\omega }^{\mathrm{\prime }\mathrm{\prime }}({\mathrm{\partial }}_{{\theta }_j},{\mathrm{\partial }}_{I_k})=d{I}_k({\mathrm{\partial }}_{I_j})={\delta }_{jk}$; and they both vanish on horizontal pairs: ${\omega }^{\mathrm{\prime }\mathrm{\prime }}$ because $d{\theta }_m({\mathrm{\partial }}_{I_j})=0$, and $\omega$ because on the slice $\theta =0$, which is the Lagrangian section ${\sigma }^{\mathrm{\prime }}(B_0)$ of step 3.2, the horizontal vectors ${\mathrm{\partial }}_{I_j}$ span the tangent space of ${\sigma }^{\mathrm{\prime }}$ where $({\sigma }^{\mathrm{\prime }}{)}^{\ast }\omega =0$, while for general $\theta$ the fibrewise translation by $\theta$ is a composition of flows of the fields $X_{d{I}_j}$ and each of those flows preserves $\omega$ by the computation of step 5.1 (with $d(d{I}_j)=0$) and maps the slice $\theta =0$ onto the slice $\theta =\text{const}$, hence carries horizontal vectors to horizontal vectors. Since these three families of pairs span all pairs of tangent vectors, $\omega ={\omega }^{\mathrm{\prime }\mathrm{\prime }}={\sum }_{i}d{\theta }_i\wedge d{I}_i$.

The functions $F_i$ and every smooth function on $U_0$ that is constant on the fibres of $\pi$ depend only on $I$: for such a function $g$, define $G:=g\circ {\sigma }^{\mathrm{\prime }}:B_0\to \mathbb{R}$, which is smooth by smoothness of $g$ and the section ${\sigma }^{\mathrm{\prime }}$; fibre constancy and $\pi \circ {\sigma }^{\mathrm{\prime }}={\mathrm{i}\mathrm{d}}_{B_0}$ give $g=G\circ \pi$, and $\pi =I^{-1}\circ (I\circ \pi )$ with $I:B_0\to A$ the diffeomorphism of step 6.1, so it is a function of $I$; in particular $F_i$ is. Together with steps 7.1 and 8.1 this exhibits the action-angle coordinates asserted in the statement, and the proof is complete.