2.1. Liouville manifolds with boundary and orientations

When $(M,\lambda)$ is a Liouville sector with boundary $\partial M$, its ideal boundary is still well-defined by the Z-invariance requirement near infinity put on $\partial M$ in the definition of Liouville sectors [Reference Ganatra, Pardon and Shende10] and so is its completion $\overline M$. Then we have the formula for the topological boundary

\begin{equation*} \partial \overline M = \partial_\infty M \cup \partial M. \end{equation*}

To ease our exposition, we often abuse our notation

\begin{equation*} DM: = \partial_\infty M \cup \partial M \end{equation*}

for the coproduct $\partial_\infty M \coprod \partial M$ after the present section, as long as there is no danger of confusion. Likewise we also abuse the notation like

\begin{equation*} \partial_\infty M \cap H : = \partial_\infty M \cap \overline H \end{equation*}

for ideal boundary of σ-sectorial hypersurface H where the intersection is actually taken as a subset of $\overline M$. For the simplicity of notation, we will also use

(2.1)\begin{equation} H_\infty: = \partial_\infty M \cap \overline H \end{equation}

similarly, we denoted $F_\infty = \partial_\infty M \cap \partial M$ when $H = \partial M$.

2.1.1. Null foliation

We recall the well-known fact that each hypersurface $H \subset M$ in a symplectic manifold $(M,\omega)$ carries the canonical characteristic foliation $\mathcal D$. The definition of this foliation is based on the fact that any hypersurface S of $(M,\omega)$ is a coisotropic submanifold in that

1. (1) We have

   \begin{equation*} (T_x H)^{\omega_x} \subset T_x H, \end{equation*}

   for any $x \in H$, where $(T_x H)^{\omega_x}$ is the ω_x-orthogonal complement

   \begin{equation*} (T_x H)^{\omega_x}: = \{v \in T_x M \mid \omega_x(v,w) = 0 \, \forall w \in T_xH\}. \end{equation*}

2. (2) Let $\iota_H: H \to M$ be the inclusion map and

   \begin{equation*} \ker \iota_H^*\omega_x:= \{v \in T_x H \mid \omega_x(v, w) = 0 \, \forall w \in T_xH\} \end{equation*}

   has constant rank 1 for all $x \in H$.

Then we denote $\mathcal D = \ker \iota_H^*\omega$ which defines a 1-dimensional (integrable) distribution of H and call it the characteristic distribution or the null distribution of H. We denote by $\mathcal N_H$ the leaf space of the associated foliation. It is also well-known that $\mathcal D$ carries a transverse symplectic structure, which induces one on the leaf space

(2.2)\begin{equation} \mathcal N_{H}: = H /\sim \end{equation}

chart-wise. With slight abuse of notation, we will also denote by $\mathcal D$ the associated foliation. Of course, the quotient topology of a leaf space may not be Hausdorff in general. We will show that under the conditions laid out in Definition 1.3, the aforementioned transverse symplectic form, as well as its smooth structure, descends to the leaf space.

We denote the ideal boundary of H (relative to Z) by $\partial_\infty H=: H_\infty$. Then

\begin{equation*} H_\infty = \partial_\infty M \cap \overline H. \end{equation*}

At each point, $x \in \overline H \cap \mathcal{N}\text{bhd}(\partial_\infty M) \supset H_\infty$, we have a natural exact sequence

(2.3)\begin{equation} 0 \to \mathcal D_x \to T_x H \to T_x H/\mathcal D_x \to 0. \end{equation}

The quotient carries a canonical symplectic bilinear form and so carries a natural symplectic orientation.

Choice 2.3 (Orientation of $\mathcal D$)

Let $H \subset M$ be a proper σ-sectorial hypersurface. Make a choice of orientation on the trivial line bundle $\mathcal D \to H$.

Definition 2.4 (Presymplectic orientation on H)

Let $\mathcal D \to H$ be given an orientation $o_{\mathcal D}$ on a neighbourhood of $H_\infty$ in $\partial_\infty M$. We call the orientation on $TH|_{H\cap \mathcal{N}\text{bhd}(\partial_\infty M)}$ given by the direct sum orientation

\begin{equation*} T_x H|_{H_\infty} = (T_x H/\mathcal D_x) \oplus \mathcal D_x, \quad x \in H\cap \mathcal{N}\text{bhd}(\partial_\infty M) \end{equation*}

the presymplectic orientation of H relative to $o_{\mathcal D}$.

Example 2.5 ( $F^\pm_\infty$ on $T^*[0,1]$)

Now consider the case of the cotangent bundle $M = T^*[0,1]$ of the closed interval $[0,1]$ equipped with the Liouville form

(2.4)\begin{equation} \lambda = - p\, dq. \end{equation}

(This is the negative of the standard Liouville one-form pdq in the cotangent bundle.) The standard orientation of the interval induces a diffeomorphism $M \cong [0,1]_q \times \mathbb R_p$, which carries the symplectic orientation induced by the symplectic form

\begin{equation*} dq \wedge dp. \end{equation*}

(We alert the readers that this is the negative of the convention $dp \wedge dq$ used by [Reference Ganatra, Pardon and Shende10].) The boundary $\partial M \cong \{0,1\} \times \mathbb R_p$ has 2 connected components. The characteristic foliation’s orientation is compatible with the vector field $\frac{\partial}{\partial p}$. Note that the Liouville vector field of the Liouville form (2.4) on $T^*[0,1] \cong [0,1]_q \times \mathbb R_p$ is given by the Euler vector field

(2.5)\begin{equation} \vec E:= p\frac{\partial}{\partial p} \end{equation}

on $T^*M$ which vanishes at p = 0. So each leaf $\{q\} \times \mathbb R_p$ of the foliation consists of 3 different orbit sets of the Liouville vector field

\begin{equation*} \mathbb R_+ = (0,\infty), \quad \{0\}, \quad \mathbb R_- = (-\infty, 0). \end{equation*}

We may identify $\partial_\infty M$ with two disjoint copies of $[0,1]$ at ‘ $p= \pm \infty$’. $F_\infty$ consists of four points, which we will denote by $(0,\pm\infty)$ and $(1,\pm \infty)$ again using the informal notation allowing p to attain $\pm \infty$. Under this notation, we have that

(2.6)\begin{equation} F^+_\infty = \{(0,-\infty) , (1,\infty)\}, \qquad\text{and}\qquad F^-_\infty = \{(0,\infty), (1,-\infty)\}. \end{equation}

Example 2.6 ( $\mathop{\kern0pt{\rm dim}}\nolimits Q \geq 2$)

More generally, let $Q = Q^n$ be a connected n-manifold with boundary and let $M = T^*Q$. The inclusion $T(\partial Q) \hookrightarrow TQ$ induces a quotient map $T^*Q|_{\partial Q} \to T^*(\partial Q)$ of bundles on $\partial Q$; the kernel induces the characteristic foliation on

\begin{equation*} T^*Q|_{\partial Q} = \partial M. \end{equation*}

Informally: At a point $(q,p) \in \partial M$, the oriented vector defining the characteristic foliation is the symplectic dual to an inward vector normal to $\partial Q$. For example, identifying Q near $\partial Q$ with the right half plane with final coordinate p_n, in standard Darboux coordinate (q, p), the characteristic foliation is generated by ${\frac {\partial}{\partial p_n}}$.

2.2. Convexity of $H_\infty =\partial_\infty M \cap H$ and contact vector field

By applying the notion of σ-sectorial hypersurface from Definition 1.11 to the boundary $\partial M \subset M$, we introduce the following definition. This is the counterpart of the definition of sectorial hypersurface given in [Reference Ganatra, Pardon and Shende12, Definition 9.2].

Definition 2.7 (Liouville σ-sector)

Let M be a non-compact manifold with boundary such that its completion $\overline M$ has (topological) boundary given by the union

\begin{equation*} \partial_\infty M \cup \partial M = DM \end{equation*}

and $\partial_\infty M \cap \partial M$ is the codimension two corner of $\overline M$. M is called a Liouville σ-sector if its boundary $\partial M \subset M$ is a σ-sectorial hypersurface in the sense of Definition 1.11.

To avoid some confusion with the corners in $\partial M$, we call the intersection

\begin{equation*} \partial_\infty M \cap \partial \overline M \end{equation*}

the ceiling corner. This is the corner of the ideal completion $\overline M$ of M of codimension 2. (We will call the genuine corners of M the sectorial corners in § 4 when we consider the Liouville sectors with corners.)

Recall that $\partial_\infty M$ is naturally oriented as the ideal boundary of symplectic manifold M with Z pointing outward along $\partial_\infty M$.

We take a contact-type hypersurface $S_0 \subset M$ and identify a neighbourhood $\mathcal{N}\text{bhd}(\partial_\infty M)$ with the half $S_0 \times [0,\infty)$ of the symplectization of the contact manifold $(S_0, \iota_{S_0}^*\lambda)$. We denote

(2.7)\begin{equation} H_0 = S_0 \cap H. \end{equation}

Then considering the Liouville embedding $S_0 \times [0,\infty) \hookrightarrow M$, we can decompose M into

\begin{equation*} M = (M \setminus \mathcal{N}\text{bhd}(\partial_\infty M) )\cup \mathcal{N}\text{bhd}(\partial_\infty M) \end{equation*}

so that

• • $Z = \frac{\partial}{\partial s}$ for the symplectization form $d(e^s \pi^* \iota_{S_0}^*\lambda)$ of the contact manifold $(S_0, \iota_{S_0}^*\lambda)$ on $S_0 \times [0,\infty)$,

• • we may identify the one-form $\iota_{S_0}^*\lambda$ as a contact form of $\partial_\infty M$ by the natural diffeomorphism $S_0 \cong \partial_\infty M$ induced by this Liouville embedding $S_0 \times [0,\infty) \hookrightarrow M$.

By the convexity hypothesis of $H_\infty:= H \cap \partial_\infty M$ in $\partial_\infty M$, there exists a contact vector field η of the contact structure $(\partial_\infty M,\xi_\infty)$ on a neighbourhood of $H_\infty$ in $\partial_\infty M$ that is transverse to $H_\infty$.

Since there are different sign conventions in the literature in defining the contact Hamiltonian associated with a contact vector field, we set our sign convention as follows by adopting the one used by the present author in [Reference Oh17] and its sequels, which also coincides with that of [Reference de León and Valcázar8].

Definition 2.8 (Contact Hamiltonian)

We call the function

\begin{equation*} h: = - \theta(\eta) \end{equation*}

the contact Hamiltonian associated with the contact vector field η.

Remark 2.9.

It is well-known that a choice of contact vector field η transverse to $H_\infty$ in $\partial_\infty M$, gives rise to a decomposition of $H_\infty$ into

(2.8)\begin{equation} H_\infty = H_\infty^+ \sqcup \Gamma_\eta \sqcup H_\infty^- \end{equation}

where $H_\infty^\pm$ and $\Gamma_\eta$ are defined by

\begin{equation*} H_\infty^\pm = \{x \in H_\infty \mid \pm \theta(\eta(x)) \gt 0\}, \quad \Gamma_\eta = \{x \in H_\infty \mid \theta(\eta(x)) = 0\}. \end{equation*}

(Recall that $\Gamma_\eta$ is called the dividing set of η on $H_\infty$. See [Reference Giroux14] for a general study of convex hypersurface.) Other than the existence of the contact vector field transverse to the contact distribution, which is the defining property of the convexity of hypersurfaces, this contact geometry of ideal boundary $\partial_\infty M$ does not enter into our study of presymplectic geometry of coisotropic submanifold, $\partial M$, which makes our affirmative answer to the question [Reference Ganatra, Pardon and Shende10, Question 2.6] plausible. See Remark 3.3 below for a further elaboration.

3.1. The leaf space is a topological manifold

Before providing a smooth atlas on $\mathcal N_{H}$, our first order of business is to prove the existence of topological manifold structure thereon. This is the most technical step towards the goal of the section as common in the study of general topology argument. The proof of this proposition occupies the rest of this subsection.

We start with the following lemma.

Lemma 3.2. There exists a neighbourhood $\mathcal{N}\text{bhd} (\partial_\infty M \cap H)$ of the ceiling corner $\partial_\infty M \cap H$ in M and a smooth function

(3.1)\begin{equation} G: \mathcal{N}\text{bhd} (\partial_\infty M \cap H) \to [0,\infty) \end{equation}

on $\mathcal{N}\text{bhd} (\partial_\infty M \cap H)$ of M that has the following properties:

1. (1) $Z[G] = G$,

2. (2) its Hamiltonian vector field X_G is transverse to H and represents the given coorientation of H at each point $x \in H \cap \mathcal{N}\text{bhd} (\partial_\infty M \cap H)$.

Proof. By the defining data of Liouville σ-sectors, we have

• • $H_\infty$ is convex in $\partial_\infty M$,

• • Z is tangent to H near infinity.

the second requirement enables us to choose a contact-type hypersurface S ₀ far out close to $\partial_\infty M$ so that $S_0 \pitchfork H$. Write the smooth hypersurface $H_0: = S_0 \cap H$ of H.

We take a symplectization neighbourhood of $\partial_\infty M$ obtained by the Liouville embedding

(3.2)\begin{equation} \phi_{Z;S_0}: S_0 \times [0, \infty) \hookrightarrow M \end{equation}

defined by $\phi_{Z;S_0}(y,t): = \phi_Z^t(y)$. We denote by s the associated radial function defined by $s(y,t): = t$. Then we have the splitting

\begin{equation*} TM|_{S_0 \times [0,\infty)} \cong TS_0 \oplus {\mathbb R} \left\{\frac{\partial}{\partial s}\right\} \end{equation*}

and satisfies

(3.3)\begin{equation} s^{-1}(0) = S_0, \quad Z = \frac{\partial}{\partial s}, \quad S_0 \cong \partial_\infty M. \end{equation}

We also have the contact form $\theta \cong \iota|_{S_0}^*\lambda$ on S ₀ so that we can express the Liouville form as

\begin{equation*} \lambda = e^s \pi^* \theta \end{equation*}

on a neighbourhood $\mathcal{N}\text{bhd}(\partial_\infty M)$.

Using the convexity hypothesis of $H_\infty \subset \partial_\infty M$, we can take a contact vector field η on a neighbourhood of $H_\infty$ in $\partial_\infty M$ such that $\eta \pitchfork H_\infty$. Take its contact Hamiltonian $h = - \theta(\eta)$ on a neighbourhood of $H_\infty$ in $\partial_\infty M$. (Recall the sign convention from Definition 2.8 adopted in the present paper.) By considering the function $\pi^*h$ on a neighbourhood of $H_\infty$ in M, we take the associated homogeneous Hamiltonian function on the symplectization in a neighbourhood of $H_\infty$ in M, which we denote it by

\begin{equation*} G := e^s \pi^* h \end{equation*}

which is defined on a neighbourhood $H_\infty = H \cap \partial_\infty M$ in M, say, on

\begin{equation*} V \times [0,\infty) \subset s^{-1}([0, \infty)) \subset M, \end{equation*}

where $ V\subset \partial_\infty M$ is an open neighbourhood of $H_\infty$ in $\partial_\infty M$. Through the symplectization end Liouville embedding $S_0 \times [0,\infty) \hookrightarrow M$, we may identify the function $h: H_\infty \to {\mathbb R}$ with $\pi^*h|_{\{s=0\}}: H \cap S_0 \times \{0\} \to {\mathbb R}$. Then the Hamiltonian vector field $X_G|_{H \cap \{s \geq0\}}$ represents the coorientation of H compatible with the one on $H_\infty \subset \partial_\infty M$ given by η.

Clearly it satisfies $Z[G] = G$ since $Z = \frac{\partial}{\partial s}$ thereon. This finishes the proof.

We fix a Riemannian metric g on M that is Z-invariant near infinity $\partial_\infty M$. More explicitly, we require the metric to satisfy

• • Near H, we require it to have the form

  \begin{equation*} g = g_H \oplus dv^2 \end{equation*}

  on the neighbourhood $H \times (-\epsilon, \epsilon) \hookrightarrow M$ where v is the coordinate of $(-\epsilon,\epsilon)$.

• • Near $\partial_\infty M$ on the symplectization end, we require the metric to satisfy

  \begin{equation*} g = g_{S_0} \oplus ds^2 \end{equation*}

  on $S_0 \times [0, \infty)$ where $g_{S_0}$ is any Riemannian metric on S ₀, recalling $Z = \frac{\partial}{\partial s}$ on this region.

• • Near $H_\infty = H \cap \partial_\infty M$, we require that the above two choices are compatible in that $g_H|_{H_\infty} = g_{S_0}|_{H_\infty}$ and has the form

  \begin{equation*} g = g_{H_\infty} \oplus du^2 \oplus ds^2. \end{equation*}

In addition, using the coorientation hypothesis on $H \subset M$, we fix a coorientation. (For the case of $H = \partial M$, we use the canonical outward coorientation.) Then we choose the aforementioned contact vector field η so that it defines the same coorientation as that of the coorientation on $H \subset M$ induced by $X_G|_H$. With the above Riemannian metric equipped with the neighbourhood of H, we require

(3.4)\begin{equation} d\lambda\left(\mathcal D_H,\frac{\partial}{\partial v}\right) \gt 0 \end{equation}

with $\mathcal D_H$ equipped with the one given in Definition 2.3: Note that we have the exact sequence of symplectic vector bundle

(3.5)\begin{equation} 0 \to {\rm span}\left\{\mathcal D_H, \frac{\partial}{\partial v} \right\} \to TM|_H \to TM \Big/{\rm span}\left\{\mathcal D_H, \frac{\partial}{\partial v} \right\} \to 0 \end{equation}

where we have

\begin{equation*} TM\Big/{\rm span}\left\{\mathcal D_H, \frac{\partial}{\partial v} \right\} \cong TH/ \mathcal D_H. \end{equation*}

Proof. We first recall that the Liouville vector field Z is tangent to H near infinity. We define the radial coordinate s as in the proof of Lemma 3.2.

We start with defining the vector field $W \in \mathcal D \subset TH$ on H along the hypersurface $s^{-1}(N) = S_0 \times \{N\} \cong S_0$ for a sufficiently large N > 0 in the given symplectization end. We can express it as the sum

\begin{equation*} W= Y' + a \frac{\partial}{\partial s}, \end{equation*}

for some function $a = a(y)$ on S ₀, and Yʹ tangent to $s^{-1}(N) \cap H$ for all $N \geq 0$. Then we have

\begin{equation*} 0 \lt d\lambda \left(Y' + a \frac{\partial}{\partial s},\frac{\partial}{\partial v}\right) \end{equation*}

and

\begin{eqnarray*} d\lambda \left(Y' + a \frac{\partial}{\partial s},X\right) & = & 0 \quad \text{for all}\ X\in TH,\\ d\lambda \left(Y' + a \frac{\partial}{\partial s},\frac{\partial}{\partial s}\right) & = & 0 \quad \text{on}\ s^{-1}(N) \cap H \end{eqnarray*}

The second equation also implies $d\lambda \left(Y', \frac{\partial}{\partial s}\right) = 0$ since $Z = \frac{\partial}{\partial s}$ is tangent to H for all sufficiently large N > 0.

Using the property that Z is tangent to H near infinity, we will choose W near infinity, say for $s \geq N$ for a sufficiently large N > 0 so that

(3.6)\begin{equation} W(y,s) := d\phi_Z(W(y,N)) = Y'(y) \oplus a(y) \frac{\partial}{\partial s} \end{equation}

i.e. it is just the s-translation of the initial vector $W(y,0)$. By normalizing W to $W/|W|$, we may assume that W has unit norm which makes its choice unique among the vector fields tangent to $\mathcal D$ in the orientation given in Choice 2.3.

Next, we would like to extend the vector field W to everywhere on H that still satisfies the standing requirements (1) - (3). For this purpose, we consider equation for $W \in TH$ in the orientation from Choice 2.3 to satisfy

(3.7)\begin{equation} W \in \mathcal D \subset TH, \quad d\lambda(W, TH) = 0, \quad d\lambda\left(W, \frac{\partial}{\partial v}\right) \gt 0. \end{equation}

By further requiring $|W| = 1$, the equation is uniquely solvable at each point of $y \in H$. This finishes the proof of Proposition 3.4.

In the course of proving the above proposition, we have also proved the following.

Proof. Statement (1) is obvious since W is nowhere vanishing section of the line bundle $\mathcal D \to H$. For Statement (2), we note that X_R satisfies $d\lambda(X_R, TH) = dR(TH) dt(TH) \equiv 0$ and hence is tangent to $\mathcal D$. Furthermore, we have

\begin{equation*} d\lambda\left(X_R,\frac{\partial}{\partial u}\right) \equiv 1 \gt 0 \end{equation*}

Then by the aforementioned uniqueness, we derive $W = \frac{X_R}{|X_R|_g}$.

This corollary will be useful for the later study of intrinsic characterization of Liouville sectors with corners. (Of course this is a tautological property with the original definition of Liouville sectors from [Reference Ganatra, Pardon and Shende10].)

The next lemma states that the leaf space $\mathcal N_H$ is Hausdorff with respect to the quotient topology of $\pi: H \to \mathcal N_H$. This should be classical which can be derived from the property of the quotient topology and the existence of continuous section $\sigma_{{\rm ref}}: \mathcal N_H \to H$. For readers’ convenience, we give its proof in Appendix B.

The next lemma shows that the presence of a continuous section implies the triviality of the fibration $\pi: H \to \mathcal N_H$.

Lemma 3.8. Take a continuous section $\sigma_{{\rm ref}}: \mathcal N_{H} \to H$ guaranteed by Definition 1.3. We write

(3.8)\begin{equation} F_{{\rm ref}}: = \operatorname{Image} \sigma_{{\rm ref}} \subset H. \end{equation}

Then the flow map

(3.9)\begin{equation} \Phi_{{\rm ref}}: F_{{\rm ref}} \times \mathbb R \to H; \quad \Phi_{{\rm ref}}(y,t) = \phi_{W}^t(\sigma_{{\rm ref}}(\pi(x)) \end{equation}

is a homeomorphism.

Proof. We will first show

1. (1) Any trajectory of W eventually exits from any given compact subset $K \subset M$ both forward and backward.

2. (2) Moreover, every leaf is a flow orbit of W and vice versa.

It is a standard fact that each leaf is second countable because the manifold M is assumed to be second countable. (This rules out the possibility for a leaf becomes a ‘Long line’ [Reference Steen and Seebach26, pp. 71-72].) Note that since W is regular, each leaf of H of the characteristic foliation is a flow line of the regular vector field W. (See [Reference Candel and Conlon5, Section 2.1].) Furthermore no leaf can be a point. By the condition stated in Definition 1.3 (c), W cannot have a non-trivial periodic orbit either. Therefore, each flow trajectory $t \mapsto \Phi_{{\rm ref}}^t(y)$ in H defined on $\mathbb R$ is one-to-one, and hence $\Phi_{{\rm ref}}$ is a one-one map.

Furthermore, there is a uniquely defined $T \in \mathbb R$ such that $\phi_{W}^T(\sigma_{{\rm ref}}(\pi(x)) = x$ for each $x \in H$. We define a function $T: M \to \mathbb R$ by

(3.10)\begin{equation} T(x): = \unicode{x2018}\text{the reaching time of the flow of}\ W\ \text{issued at}\ \sigma_{{\rm ref}}(\pi(x))\text{'}. \end{equation}

Lemma 3.9. The function T is continuous.

Proof. Let $x \in M$ and set $x_0: = \sigma_{{\rm ref}}(\pi(x))$. Choose a foliation chart $(U_\alpha, \varphi_\alpha)$ with $\varphi_\alpha = (y_1, \ldots, y_{2n-1}, t)$ with $x_0 = (0,0)$ so that $W = \frac{\partial}{\partial t}$. It follows from the Hausdorff property of $\mathcal N_H$, we may assume that the restriction σ _ref to $\pi(U_\alpha)$ defines a (local) section of π. Since the map $\{t = 0\}$ is also the image of continuous section on the same domain $\pi(U_\alpha)$, we can express

\begin{equation*} F_{{\rm ref}}\cap U_\alpha = \{(y, t) \in U_\alpha \mid t = \sigma'(y)\} \end{equation*}

with $y = (y_1, \cdots, y_{2n-1})$ for some continuous one-to-one map $\sigma: \{t = 0\} \cap U_\alpha \to U_\alpha$.

Write $T_x: = T(x)$ and consider the pair

\begin{equation*} (U_\beta, \varphi_\beta) = (\phi_W^{T_x}(U_\alpha), \varphi_\alpha \circ \phi_W^{T_x}) \end{equation*}

which defines a foliation chart on U_β at x. Write $\varphi_\beta = (y', t')$. Then $\phi_W^{T_x} \circ \sigma_{\rm ref}$ defines another local continuous section on $\pi(U_\alpha)$, and we have

\begin{eqnarray*} \phi_W^{T_x}(F_{{\rm ref}} \cap U_\alpha) & = & \left( \operatorname{Image} \phi_W^{T_x} \circ \sigma_{{\rm ref} }\right) \cap U_\beta \\ & = & \left \{(y',t') \in U_\beta \, \Big| \, t' = \sigma\left( \left( \phi_W^{T_x}\right)^{-1}(x')\right) \right\}. \end{eqnarray*}

By definition of $(U_\beta,\varphi_\beta)$, we have the relation

\begin{equation*} T_{x'} = T_x + t \left( (\phi_W^{T_x})^{-1}(x')\right) - t(x) \end{equation*}

for all $x' \in U_\beta$. This formula clearly shows that tʹ is a continuous function on U_β, and satisfies

\begin{equation*} T(x') - T(x) = T_{x'} - T_x = t \left( (\phi_W^{T_x})^{-1}(x')\right) - t(x) \end{equation*}

for all $x' \in U_\beta$. Since $x' \mapsto t\left((\phi_W^{T_x})^{-1}(x')\right)$ is continuous at any point xʹ where the map is defined and $ t\left(\phi_W^{T_x})^{-1}(x) \right)= t(x)$, this explicit formula shows that T is continuous at x. Since continuity is local, this proves continuity of T.

Therefore, the inverse image $T^{-1}(-C,C)$ is an open subset and H is an increasing union

\begin{equation*} H = \bigcup_{C \in \mathbb N} T^{-1}(-C,C) \end{equation*}

of open subset $T^{-1}(-C,C)$.

Let $K \subset H$ be any compact subset. Then $K \subset T^{-1}(-C,C)$ for some C > 0. Since $|W| = 1$ and W is tangent to the leaf $\ell_y$ through y of the characteristic foliation, any point $y \in F_{{\rm ref}}$ we have

\begin{equation*} \phi_W^{\pm(2\max\{C,|T(y)|\})}(y) \cap K = \emptyset. \end{equation*}

This proves the aforementioned claims.

Then, combining this with the aforementioned completeness, we can define another map

(3.11)\begin{equation} \Psi_{{\rm ref}}: H \to F_{{\rm ref}} \times \mathbb R; \quad \Psi_{{\rm ref}}(x) = (\sigma_{{\rm ref}}(\pi(x)), T(x)). \end{equation}

By construction, $\Psi_{{\rm ref}}$ is continous and satisfies

\begin{equation*} \Psi_{{\rm ref}} \circ \Phi_{{\rm ref}} = id|_H, \quad \Phi_{{\rm ref}} \circ \Psi_{{\rm ref}} = id|_{F_{{\rm ref}} \times {\mathbb R}} \end{equation*}

This finishes the proof of Lemma 3.8.

Consider the leaf map $\pi_{{\rm ref}}: F_{{\rm ref}} \to {\mathcal N}_H$, where $F_{{\rm ref}}$ equipped with the subspace topology of H and ${\mathcal N}_H$ is the quotient topology of the projection $\pi:H \to {\mathcal N}_H$.

Proof. Since $\sigma_{{\rm ref}}: \mathcal N_H \to H$ is a section, we have $\pi_{{\rm ref}} \circ \sigma_{{\rm ref}} = id_{\mathcal N_H}$ which shows π _ref is surjective.

On the other hand, if $y_1 \neq y_2$ in $F_{{\rm ref}}$, then $\ell_{y_1} \neq \ell_{y_2}$ since otherwise we would have

\begin{equation*} y_1 = \sigma_{{\rm ref}}(\ell_{y_1}) = \sigma_{{\rm ref}}(\ell_{y_2}) = y_2 \end{equation*}

which is a contradiction. This shows that π _ref is a bijective continuous map.

By construction, the map

\begin{equation*} \ell \mapsto \sigma_{{\rm ref}}(\ell); \, {\mathcal N}_H \to F_{{\rm ref}} \end{equation*}

defines a continuous map which also satisfies $\pi_{{\rm ref}} \circ \sigma_{{\rm ref}} = id|_{{\mathcal N}_H}$, and $\sigma_{{\rm ref}}\circ \pi_{{\rm ref}} = id|_{F_{{\rm ref}}}$. This proves that the map σ _ref is a continuous inverse of π _ref. Therefore, the map π _ref is a homeomorphism.

Now we go back to the proof of Theorem 3.1.

Proof. Wrap-up of the proof of Theorem 3.1.

First we show the following.

Lemma 3.11. $F_{{\rm ref}}$ with the subspace topology of H is locally Euclidean (and in particular, locally compact).

Proof. To see the locally Euclidean property of $F_{{\rm ref}}$, let $x_0 \in F_{{\rm ref}}$ be any given point. We have only to note that (3.11) induces a homeomorphism

\begin{equation*} U/\sim \,\, \to F_{{\rm ref}}\cap U \end{equation*}

for a sufficiently small foliation chart U containing x ₀, where ∼ is the orbit equivalence with respect to W. Since $U/\sim$ is homeomorphic to ${\mathbb R}^{2n-1}$, so is $F_{{\rm ref}}\cap U$. This proves that $F_{{\rm ref}}\cap U$ is locally Euclidean.

Now the combination of Corollary 3.10 and Lemma 3.11 finishes the proof of Theorem 3.1.

3.2. Smooth structure on the leaf space

When the leaf space is Hausdorff and locally Euclidean, the well-known construction of coisotropic reduction (or symplectic reduction) applies to prove existence of the symplectic structure on the leaf space once the smooth structure on the leave space is equipped. (See [Reference Abraham and Marsden1] for example.) Since we also need to construct the map Ψ appearing in the statement of Theorem 1.5 and will also use the details of the proof later, we provide the full details of the existence proofs of both structures below along the way partly for readers’ convenience.

The goal of this section is to prove the first item of Theorem 1.5. We start with the following proposition whose proof will occupy the entirety of this subsection.

Proposition 3.12. The leaf space $\mathcal N_{H}$ carries a canonical smooth manifold structure such that

1. (1) $\pi: H \to \mathcal N_H$ is a smooth submersion and

2. (2) there is a smooth diffeomorphism $\Psi: H \to \mathcal N_H \times {\mathbb R}$ which makes the following diagram commute

(3.12)

We follow the standard notation of [Reference Candel and Conlon5] in our discussion of foliations. It follows from a well-known result in foliation theory that the foliation $\mathcal F$ is determined by its holonomy cocycle $\gamma = \{\gamma_{\alpha\beta}\}_{\alpha, \beta \in \mathfrak U}$ with

\begin{equation*} \gamma_{\alpha\beta}: y_\beta(U_\alpha \cap U_\beta) \to y_\alpha(U_\alpha \cap U_\beta), \end{equation*}

arising from the transverse coordinate map $y_\alpha: U_\alpha \to \mathbb F^{2n-2} = \mathbb R^{2n-2}$ or $\mathbb H^{2n}$.

Each y_α is a submersion and $\gamma_{\alpha\beta}$ is given by $y_\alpha = y_\alpha(y_\beta)$ in coordinates. (See e.g. [Reference Candel and Conlon5, Definition 1.2.12].) Furthermore, for the null foliation $\mathcal F$ of the coisotropic submanifold H, we can choose a foliated chart $\mathcal U = \{(U_\alpha,\varphi_\alpha)\}_{\alpha \in \mathfrak U}$ so that the associated cocycle elements $\gamma_{\alpha\beta}$ become symplectic, i.e. the foliation $\mathcal F$ carries a transverse symplectic structure. We refer readers to the proof of Proposition 3.12 below for the details.

We will first show that the above holonomy cocycle naturally descends to a smooth atlas on $\mathcal N_{H}$ under the defining condition of σ-sectorial hypersurface above, especially in the presence of a continuous section of the projection $\pi_H:H \to \mathcal N_{H}$.

For this purpose, we consider a coherent regular foliated atlas $\{\varphi_{\alpha}: U_\alpha \to \mathbb R^{2n-1}\}$, and its associated foliation cocycle $\gamma = \{\gamma_{\alpha\beta}\}$ (see e.g. [Reference Candel and Conlon5, Section 1.2.A]).

By considering a refinement $\{U_{\alpha'}\}$ of the given covering, we can choose a collection of foliated charts $\varphi_{\alpha'}: U_{\alpha'} \to \mathbb R^{2n-2} \times \mathbb R$ of the form

(3.13)\begin{equation} ( y_1, \ldots, y_{2n-2},t) \end{equation}

whose transverse coordinate $(y_1, \ldots, y_{2n-2})$ satisfies

(3.14)\begin{equation} dt(W) \equiv 1. \end{equation}

We take a maximal such collection which we denote by

(3.15)\begin{equation} \mathcal O' = \{(\varphi_{\alpha'},U_{\alpha'})\}. \end{equation}

By the definition of transverse coordinates $(y_1,\cdots, y_{2n-2})$ of the foliated chart, it follows that the collection thereof defines a smooth atlas of $\mathcal N_{H}$. We write the resulting atlas of $\mathcal N_{H}$ by

(3.16)\begin{equation} [\mathcal O']:= \{[\varphi_{\alpha'}]: [U_{\alpha'}]\ \to \mathbb R^{2n-2}\}. \end{equation}

Proof. To show smoothness of π, we will show that for any smooth function $f: \mathcal N_{H} \to \mathbb R$ the composition $f\circ \pi$ is smooth. For this purpose, at any point x, we consider the foliated chart $\varphi_\alpha: U_\alpha \to \mathbb R^{2n-1}$ given above in (3.13).

Let $f: \mathcal N_{H} \to \mathbb R$ be any smooth function on $\mathcal N_{H}$. With respect to the aforementioned foliated atlas of H, we will show that $f \circ \pi$ is smooth at every point $x \in H$.

If x is contained in $U_{\alpha'}$, we have

\begin{equation*} (f\circ \pi) \circ (\varphi_{\alpha'})^{-1}(y_1, \ldots, y_{2n-2},t) = f \circ [\varphi_{\alpha'}]^{-1}(y_1,\ldots, y_{2n-2}) \end{equation*}

The right-hand side is smooth in the variables $y_1, \ldots, y_{2n-2}$ by the hypothesis on f and does not depend on t-variable. This, in particular, implies that the left-hand map $(f\circ \pi) \circ (\varphi_{\alpha'})^{-1}$ is smooth at x.

Otherwise, let $(\varphi_\beta, U_\beta)$ be a foliation chart at x. We take a flow map $\phi_{W}^T$ satisfying $y: = \phi_{W}^T(x) \in U_\beta'$ for some chart $(\varphi_{\beta'}, U_{\beta'}) \in \mathcal O'$ at y given by

\begin{equation*} \left(U_{\beta'} = \phi_{W}^T(U_{\beta}), \quad \varphi_{\beta'} = \varphi_\beta \circ (\phi_{W}^T)^{-1}\right) \end{equation*}

which is contained in $\mathcal O'$ by the maximality of the collection $\mathcal O'$.

Therefore, the map $(f\circ \pi) \circ \varphi_{\beta'}^{-1}$ is smooth at $y = \phi_{W}^T(x) \in U_{\beta'}$. We can factorize $f \circ \pi$ into

\begin{equation*} f\circ \pi = \left((f\circ \pi) \circ \varphi_{\beta'}^{-1}\right) \circ \left((\varphi_{\beta'} \circ \phi_{W}^T|_{U_\beta})\right) \end{equation*}

which is a composition of two smooth maps and so smooth at x. This implies $f\circ \pi$ is smooth at x. This finishes the proof of smoothness $\pi \circ f$ for all smooth function $f: \mathcal N_{H} \to \mathbb R$ and hence proves that π is smooth.

Submersivity of π is obvious by the above construction.

3.3. Construction of a smooth section: smoothing

Finally, we would like to improve the existence of continuous section of $\pi: H \to \mathcal N_H$ to a smooth one $\sigma^{{\rm sm}}: \mathcal N_H \to H$. For this purpose, we apply the ‘standard mollifer smoothing and a partition of unity’. However, a priori the set of sections of the fibration $H \to \mathcal N_H$ is not a linear space which prevents us from directly implementing the smoothing of the sections.

The first order of business for our purpose is to reduce the problem of smoothing to that of smoothing a section of certain smooth line bundle. For this purpose, we need to choose a collection of the atlas of foliated atlases of $\mathcal N_H$ that is compatible with the flow of the leaf-generating vector field W on H.

For the simplicity of notation and exposition, we write the maximal atlas $\mathcal O'$ chosen in the previous subsection back as $\mathcal O$ and the atlas of $\mathcal N_H$ given in (3.16) without prime.

Thanks to the property (3.14), the transition map

\begin{equation*} \varphi_{{\alpha\beta}} = \varphi_\alpha \circ \varphi_\beta^{-1}: \varphi_\beta(U_\alpha \cap U_\beta)\to \varphi_\alpha(U_\alpha \cap U_\beta) \end{equation*}

associated with the covering $\mathcal O$ has the form

(3.17)\begin{equation} \varphi_{\alpha\beta}(y,t) = (\psi_{\alpha\beta}(y),t + c_{\alpha\beta}(y)) \end{equation}

where $y = (y_1, \ldots, y_{2n-2})$ on $\varphi_\alpha(U_\alpha \cap U_\beta)$ for some smooth functions $c_{\beta\alpha}$ and $\psi_{\beta\alpha}$ on $\varphi^\alpha(V_\alpha \cap V_\beta)$.

A direct translation of the cocycle condition of $\{\varphi_{\alpha\beta}\}$ gives rise to the following identities for $c_{\alpha\beta}$.

Lemma 3.15. Let $\psi_{\alpha\beta} $ be the transition map for the transverse coordinate charts of φ_α which is given by

\begin{equation*} \psi_\alpha \circ \psi_\beta^{-1}: \psi_\beta([U_\alpha] \cap [U_\beta]) \to \psi_\alpha([U_\alpha] \cap [U_\beta]). \end{equation*}

Then the collection $c_{\alpha\beta}$ satisfies

(3.18)\begin{equation} c_{\alpha\gamma} = c_{\beta\gamma} + c_{\alpha\beta} \circ \psi_{\beta\gamma} \end{equation}

In particular, $c_{\alpha\alpha} \equiv 0$ for all α.

The rest of this subsection will be occupied by the proof of the following.

Proposition 3.16. There exists a smooth section $\sigma^{{\rm sm}}: \mathcal N_H \to H$ and a diffeomorphism $\Psi: H \to \mathcal N_H \times {\mathbb R}$ such that

(3.19)\begin{equation} \sigma^{{\rm sm}}(\ell) = \Psi^{-1}(\ell,0). \end{equation}

which makes the diagram (3.12) commute.

We first provide some general discussion on the coordinate representation of sections of $\pi:H \to \mathcal N_H$. Let $\sigma: \mathcal N_H \to H$ be a continuous section of π and $T_\sigma: H \to {\mathbb R}$ be the continuous function associated with σ given in (3.10). Then we have

(3.20)\begin{equation} F_{{\rm ref}} := \operatorname{Image} \sigma = T_\sigma^{-1}(0) \end{equation}

and a homeomorphism $\Psi_\sigma: H \to \mathcal N_H \times {\mathbb R}$ of the type

\begin{equation*} \Psi_\sigma(x) = (\pi_H(x), T_\sigma(x)) \end{equation*}

whose inverse $\Phi_\sigma: \mathcal N_H \times {\mathbb R} \to H$ is given by the flow map

\begin{equation*} \Phi_\sigma(\ell,t) = \phi_{W}^t(\sigma(\ell)) \end{equation*}

such that $T_\sigma(\phi_{W}^t(x)) = t$ for all $x \in F_{{\rm ref}}$.

We take a collection $\{(U_{\alpha},\varphi^\alpha)\}$ with $\varphi^\alpha: U_\alpha \to {\mathbb R}^{2n-1}$ of foliated charts of H that covers $F_{{\rm ref}} = T^{-1}(0)$ each element of which is centred at a point in $F_{{\rm ref}}$. We write

\begin{equation*} \varphi^\alpha = (y^\alpha, t^\alpha) = ( y_1^\alpha, \ldots, y_{2n-2}^\alpha, t^\alpha). \end{equation*}

Thanks to the requirement (3.14), we must have

(3.21)\begin{equation} t^\alpha = t^\beta + c_{\alpha\beta}(y^\beta) \end{equation}

on $U_\alpha \cap U_\beta$. (See (3.17).)

Let σ be the given continuous section. On each such a chart $(U_{\alpha}, \varphi_\alpha)$ with $\varphi_\alpha = (y^\alpha, t^\alpha)$, the level set $T_\sigma^{-1}(0)$ of the continuous function T_σ can be locally represented as

\begin{equation*} F_{{\rm ref}} \cap U_{\alpha} = \{x \in U_{\alpha} \mid t^\alpha = f_\alpha(y^\alpha)\}, \quad y^\alpha = (y_1, \ldots, y_{2n-2}) \in V_\alpha \subset {\mathbb R}^{2n-2} \end{equation*}

for some continuous function $f_\alpha = f_\alpha(y_1, \ldots, y_{2n-2})$ that satisfies

(3.22)\begin{equation} \begin{cases} T\circ \varphi_{\alpha\beta}^{-1}(y,t) = t -f_\alpha(y),\\ f_\alpha(0,\ldots, 0) = 0 \end{cases}. \end{equation}

The transverse coordinates $(V_\alpha, \psi_\alpha)$ induce a smooth chart on $[U_\alpha] \subset \mathcal N_H$, and the function f_α induces a continuous function $f_\alpha'$ thereon. Note that the section σ can be expressed in terms of its local representatives $\{\sigma_\alpha: = \sigma|_{[U_\alpha]}\}$: we require them to satisfy

\begin{equation*} \varphi_\alpha(\sigma_\alpha(\ell)) = (\psi^\alpha(\ell), f_\alpha(\ell)) \end{equation*}

in terms of the coordinate charts $([U_\alpha], \psi^\alpha)$ of $\mathcal N_H$ and $(U_\alpha, (y^\alpha,t_\alpha))$ of H. It follows from the above discussion that to define a global section out of the collection $\{\sigma^\alpha\}$, the collection should satisfy

(3.23)\begin{equation} g_\alpha \circ \psi_{\alpha\beta}= g_\beta + c_{\alpha\beta} \end{equation}

by (3.21).

We summarize the above discussion into the following.

Lemma 3.17. A section of $\pi_H: H \to \mathcal N_H$ is characterized by the collection of maps $\{g_\alpha\}$ and $\{c_{\alpha\beta}\}$ with $g_\alpha: \psi_\alpha([U_\alpha]) \to {\mathbb R}$, $ \, c_{\alpha\beta}: \psi_\alpha([U_\alpha] \cap [U_\beta]) \to {\mathbb R}$ that satisfy (3.23), or equivalently

(3.24)\begin{equation} g_\beta = g_\alpha \circ \psi_{\alpha\beta} - c_{\alpha\beta} \end{equation}

on $\psi_\beta([U_\alpha] \cap [U_\beta])$ and vice versa.

Proof. For the proof of (3.24), we apply Lemma 3.15 to (3.23) and get

\begin{equation*} c_{\beta\alpha} \circ \psi_{\beta\alpha}^{-1} = c_{\beta\alpha} \circ \psi_{\alpha\beta} = c_{\beta\beta} - c_{\alpha\beta}= - c_{\alpha\beta}. \end{equation*}

Then we rewrite (3.23) into

\begin{eqnarray*} g_\beta & = & (g_\alpha + c_{\beta\alpha}) \circ \psi_{\beta\alpha}^{-1} = (g_\alpha + c_{\beta\alpha}) \circ \psi_{\alpha\beta} \\ & = & g_\alpha \circ \psi_{\alpha\beta} + c_{\beta\alpha}\circ \psi_{\alpha\beta}\\ & = & g_\alpha \circ \psi_{\alpha\beta}- c_{\alpha\beta} \end{eqnarray*}

which finishes the proof.

By exponentiating (3.24), we get $ e^{g_\alpha} \circ \psi_{\alpha\beta} = e^{c_{\alpha\beta}} e^{g_\beta} $ which is equivalent to

(3.25)\begin{equation} e^{g_\alpha} \circ \psi_\alpha = e^{c_{\alpha\beta} \circ \psi_\beta} e^{g_\beta} \circ \psi_\beta. \end{equation}

If we set $s_\alpha = e^{g_\alpha}\circ \psi_\alpha$ and $ g_{\alpha\beta} = e^{c_{\alpha\beta} \circ \psi_\beta}, $ the equation becomes $s_\alpha = g_{\alpha\beta} s_\beta$ on $[U_\alpha] \cap [U_\beta]$.

Proof. By definition of $c_{\alpha\beta}$, it is a smooth function. Equation (3.18) is equivalent to

\begin{equation*} c_{\alpha\gamma} \circ \psi_\gamma = c_{\beta\gamma} \circ \psi_\gamma + c_{\alpha\beta}\circ \psi_\beta. \end{equation*}

By exponentiating this equation, we obtain

\begin{equation*} g_{\alpha\gamma} = g_{\beta\gamma} g_{\alpha\beta} = g_{\alpha\beta}g_{\beta\gamma}. \end{equation*}

Furthermore, since $c_{\alpha\alpha} = 0$, we have $g_{\alpha\alpha} = 1$. This finishes the proof.

This shows that the collection $\{g_{\alpha\beta}\}$ defines a real-oriented smooth line bundle on $\mathcal N_H$, and $\{s_\alpha\}$ associated with the local representatives $\{f_\alpha\}$ of the given section σ defines a nowhere vanishing continuous section thereof.

We denote this smooth oriented line bundle by ${\mathcal L}$. Lemma 3.17 shows that this collection also provides ${\mathcal L}$ with a trivializing cover and hence defines a smooth trivialization

\begin{equation*} {\mathcal L} \to \mathcal N_H \times {\mathbb R}. \end{equation*}

We summarize the above discussion into the following.

Lemma 3.20. Consider the collections $\{g_{\alpha\beta}\}$ and $\{s_\alpha\}$ defined by

\begin{equation*} g_{\alpha\beta} = e^{c_{\alpha\beta}\circ \psi_\beta}, \quad s_\alpha = e^{f_\alpha\circ \psi_\alpha} \end{equation*}

of continuous ${\mathbb R}_+$-valued functions, respectively. Then the collection $\{s_\alpha\}$ defines a nowhere vanishing continuous section of the smooth oriented line bundle ${\mathcal L}$. We denote the associated global section of $\mathcal L$ by s_σ.

We are now ready to complete the proof of Proposition 3.16.

Proof. Wrap-up of the proof of Proposition 3.16.

We would like to construct a smooth section or the collection $\{g_\alpha\}$ satisfying (3.24), knowing the existence of this continuous section σ. For this purpose, we have only to find a smooth approximation of the section s_σ of the line bundle ${\mathcal L}$, which is a standard process by taking the mollifier smoothing whose details are now in order.

We denote by $s_{\sigma;\alpha}$ the local representative of s_σ determined by $e^{f_\alpha} \circ \psi_\alpha$, i.e. we will characterize the section s_σ by the collection $\{s_\alpha: [U_\alpha] \to {\mathbb R} \}$ that satisfy

\begin{equation*} s_\alpha = g_{\alpha\beta}\, s_\beta. \end{equation*}

For this purpose, without loss of any generality, we assume $\psi_\alpha([U_\alpha]) = I^{2n-2}$ with $I = (-1,1)$ for all α and take a family of mollifier $\{\rho_\delta\}_{\delta \gt 0}$ supported in $I^{2n-2}$. We then take the collection $\{s_\alpha\}$ by setting

\begin{equation*} s_\alpha = h_\alpha^\delta\circ \psi_\alpha \end{equation*}

for the mollifier smoothing of the functions $\{e^{f_\alpha}\}$ which are defined by

\begin{equation*} h_\alpha^\delta = e^{f_\alpha} * \rho_\delta \end{equation*}

for all α. Here $*$ is the standard convolution product defined by

\begin{equation*} a* b(x) : = \int_{{\mathbb R}^{2n-2}} a(x-y) b(y)\, dy \end{equation*}

for two real-valued functions $a, \, b: {\mathbb R}^{2n-2} \to {\mathbb R}$. Then we take the sum

\begin{equation*} s^{{\rm sm}}: = \sum_\alpha \chi_\alpha s_{\sigma;\alpha} \end{equation*}

for a partitions of unity $\{\chi_\alpha\}$ subordinate to $\{[U_\alpha]\}$ which defines a global smooth section of ${\mathcal L}$.

It follows from the general property of the mollifier smoothing that $h^\delta_\alpha \to e^{f_\alpha}$ as δ → 0 in compact open topology or in C ⁰ topology. This is easy to check (or see [Reference Gelfand and Shilov13] for example). Therefore $h_\alpha^\delta$ is nowhere vanishing for a sufficiently small $\delta = \delta_\alpha \gt 0$, and so we can take the logarithm $g_\alpha = \log h_\alpha^{\delta_\alpha}$ so that $h_\alpha^\delta = e^{g_\alpha}$ unambiguously.

Reading back the above explicit correspondence between a section of $H\to \mathcal N_H$ and a nowhere-vanishing section of the line bundle $\mathcal L$, we conclude that the collection $\{g_\alpha \circ \psi_\alpha\}$ associated with $\{[U_\alpha]\}$ represents a smooth section of the projection $\pi: H \to \mathcal N_H$. We denote by σ ^sm the corresponding smooth section.

Now we consider the flow map of the vector field W

\begin{equation*} \Phi_H^{\sigma^{{\rm sm}}}: \mathcal N_H \times {\mathbb R} \to H \end{equation*}

given by $\Phi_H^{\sigma^{{\rm sm}}}(\ell,t) = \phi_{W}^t(\sigma^{{\rm sm}}(\ell))$, and define the map $\Psi: H \to \mathcal N_H \times {\mathbb R}$ to be its inverse

(3.26)\begin{equation} \Psi(x) = (\pi_H(x), T_{\sigma^{{\rm sm}}}(x)). \end{equation}

By construction, Ψ now satisfies all the properties required in Proposition 3.12. This finally completes the proof of Proposition 3.12.

This will finish the proof of the diagram (3.28) required in the proof of Theorem 1.5.

3.4. Symplectic structure on the leaf space

Now we turn to the construction of symplectic structure. Using Proposition 3.12, we fix a smooth section $\sigma^{{\rm sm}} : \mathcal N_H \to H$ and write $F: = \operatorname{Image} \sigma^{{\rm sm}}$.

When we choose the above-used coherent atlas, we can choose it so that the associated cocycle $\gamma_{\alpha\beta}$ becomes symplectic by requiring the chart $(U_\alpha, \varphi_\alpha)$ also to satisfy the defining equation

(3.27)\begin{equation} (y^\alpha)^*\omega_0 = \iota_{H}^*\omega, \quad \omega = d\lambda \end{equation}

of the general coisotropic reduction (see [Reference Abraham and Marsden1, Theorem 5.3.23] for example) where $\iota_{H}: H \to M$ is the inclusion map and ω ₀ is the standard symplectic from on $\mathbb R^{2n-2}$. (See also [Reference Gotay15, Reference Oh and Park20].) By using such a foliated chart satisfying (3.27), the associated holonomy cycles define symplectic atlas and so a symplectic structure on $\mathcal N_{H}$, when the holonomy is trivial as in our case where we assume the presence of smooth section. This will then finish construction of reduced symplectic structures on $\mathcal N_H$. (We refer to [Reference Oh and Park20, Section 5] for a detailed discussion on the construction of transverse symplectic structure for the null foliation of general coisotropic submanifolds.)

An immediate corollary of the above construction of diffeomorphism $\Psi: H \to \mathcal N_H \times {\mathbb R}$ is that any Liouville σ-sector is a Liouville sector in the sense of [Reference Ganatra, Pardon and Shende10].

Remark 3.21.

On the other hand, the converse is almost a tautological statement in that [Reference Ganatra, Pardon and Shende10, Lemma 2.5] shows that any of their three defining conditions given in [Reference Ganatra, Pardon and Shende10, Definition 2.4] is equivalent to the condition

• • There exists a diffeomorphism $\Psi: H \to F \times {\mathbb R}$ making (3.12) commute

Once this is in our disposal, Ψ induces a diffeomorphism $ [\Psi]: \mathcal N_H \to F. $ Therefore, we can choose a continuous section $ \sigma_{{\rm ref}}: \mathcal N_H \to H $ required for the definition of σ-sectorial hypersurface to be

\begin{equation*} \sigma_{{\rm ref}}(\ell): = [\Psi]^{-1}(\ell), \quad \ell \in \mathcal N_H. \end{equation*}

Now we wrap up the proof of Theorem 1.5 as the special case $H = \partial M$ of the following theorem. We will postpone the proof of Statement (3) till the next subsection.

3.5. Induced Liouville structure on the leaf space

Finally, we prove Statement (4) of Theorem 3.22 by extracting some consequences on the above-constructed symplectic structure on $\mathcal N_H$ derived from the given property of the characteristic foliation $\mathcal D$ near infinity. Recall the definitions $F = \operatorname{Image} \sigma^{{\rm sm}}$ and the smooth flow map $\Phi_H := (\Psi_H^{{\rm sm}})^{-1}$

(3.29)\begin{equation} \Phi_H: F \times [0,\infty) \to H \end{equation}

where $\Psi_H^{{\rm sm}}$ is given in (3.26). By the convexity hypothesis on $H_\infty$, we have a contact vector field η on $\partial_\infty M$ that is transverse to $H_\infty$.

Proof. Note that $F = \operatorname{Image} \sigma^{{\rm sm}}$ is a symplectic submanifold of M and the symplectic form $d\lambda$ induces an exact symplectic form $d(\iota_F^*\lambda) = \iota_F^*d\lambda $ for the inclusion map

\begin{equation*} \iota_F: F \hookrightarrow H \hookrightarrow (M,\lambda). \end{equation*}

Therefore, it follows from (3.27) and $\pi_F^*\circ \sigma^{{\rm sm}} = id_{\mathcal N_H}$

\begin{eqnarray*} \omega_{\mathcal N_H} & = & (\pi_{F }\circ \sigma^{{\rm sm}})^*\omega_{\mathcal N_H} = (\sigma^{{\rm sm}} )^*(\pi_{F }^*\omega_{\mathcal N_H})\\ & = & (\sigma^{{\rm sm}} )^*(\iota_{F }^*d\lambda) = (\sigma^{{\rm sm}} )^*d(\iota_F^*\lambda) = d((\sigma^{{\rm sm}} )^*\iota_F^*\lambda) \\ & = & d((\iota_F \circ \sigma^{{\rm sm}} )^*\lambda)\end{eqnarray*}

which proves exactness of $\omega_{\mathcal N_H}$: Here the third equality follows from the defining equation (3.27) and the equalities

\begin{equation*} \pi_{F } = \pi_H \circ \Phi_H , \quad \iota_{F } = \iota_H \circ \Phi_H \end{equation*}

with the map $\Phi_H $ given in (3.29).

This leads us to the following reduced Liouville structure on $\mathcal N_H$.

4.1. Gotay’s coisotropic embedding theorem of presymplectic manifolds

For a finer study of the neighbourhood structure of the sectorial corner C, we first recall below some basic properties of the coisotropic submanifolds and the coisotropic embedding theorem of Gotay [Reference Gotay15]. See also [Reference Oh and Park20, Reference Weinstein28] for relevant material on the geometry of coisotropic submanifolds. We will mostly adopt the notations used in [Reference Gotay15], [Reference Oh and Park20, Section 3].

Let $(Y,\omega_Y)$ be any presymplectic manifold. The null distribution on Y is the vector bundle

\begin{equation*} E: = (TY)^{\omega_Y} \subset TY, \, \quad E_y = \ker \omega_Y|_y. \end{equation*}

This distribution is integrable since ω_Y is closed. We call the corresponding foliation the null foliation on Y and denote it by

\begin{equation*} \mathcal F = \mathcal F_Y. \end{equation*}

(Then E is nothing but the total space of the foliation tangent bundle $T\mathcal F$.) We now consider the dual bundle $\pi: E^* \to Y$, which is the foliation cotangent bundle

\begin{equation*} E^* = T^*\mathcal F. \end{equation*}

The tangent bundle $TE^*$ of the total space $E^*$ has its restriction to the zero section $Y \hookrightarrow E^*$; this restriction carries a canonical decomposition

\begin{equation*} TE^*|_Y \cong TY \oplus E^*. \end{equation*}

Example 4.3. A typical example of a presymplectic manifold is given by

\begin{equation*} (Y,\omega_Y) = (H, \omega_H), \quad \omega_H: = \iota_H^*\omega \end{equation*}

arising from any coisotropic submanifold $H \subset^{\iota_H} (X,\omega)$. Then $E = \mathcal D_H$, the null distribution of $(H,\omega_H)$. It is easy to check that the isomorphism

\begin{equation*} TX \to T^*X \end{equation*}

maps $TY^\omega$ to the conormal $N^*Y \subset T^*X$ and induces an isomorphism between $NY = (TX)|_Y/TY$ and $E^*$.

Gotay [Reference Gotay15] takes a transverse symplectic subbundle G of TY and associates to each splitting

(4.2)\begin{equation} \Gamma: \quad TY = G \oplus E, \quad E = T \mathcal F \end{equation}

the zero section map

\begin{equation*} \Phi_\Gamma: Y \hookrightarrow T^*\mathcal F = E^* \end{equation*}

as a coisotropic embedding with respect to a ‘canonical’ two-form $\omega_{E^*}$ on $E^*$ which restricts to a symplectic form on a neighbourhood of the zero section of $E^*$ such that

\begin{equation*} \omega_Y = \Phi_\Gamma^*\omega_{E^*}. \end{equation*}

We now describe the last symplectic form closely following [Reference Gotay15].

We denote the aforementioned neighbourhood by

\begin{equation*} V \subset T^*\mathcal F = E^*. \end{equation*}

Using the splitting Γ, which may be regarded as an ‘Ehresmann connection’ of the ‘fibration’

\begin{equation*} T \mathcal F \to Y \to \mathcal N_Y, \end{equation*}

we can explicitly write down a symplectic form $\omega_{E^*}$ as follows.

First note that as a vector bundle, we have a natural splitting

\begin{equation*} TE^*|_Y \cong TY \oplus E^* \cong G \oplus E \oplus T^*\mathcal F \end{equation*}

on Y, which can be extended to a neighbourhood V of the zero section $Y \subset E^*$ via the ‘connection of the fibration’ $T^*\mathcal F \to Y$. (We refer readers to [Reference Oh and Park20] for a complete discussion on this.)

We denote

\begin{equation*} p_\Gamma : TY \to T\mathcal F \end{equation*}

the (fibrewise) projection to $E=T\mathcal F$ over Y with respect to the splitting (4.2). We have the bundle map

\begin{equation*} TE^* \stackrel{T\pi}\longrightarrow TY \stackrel{p_\Gamma} \longrightarrow E \end{equation*}

over Y.

Definition 4.5 (Canonical one-form $\theta_\Gamma$ on $E^*$)

Let $\zeta \in E^*$ and $\xi \in T_\zeta E^*$. We define the one form $\theta_\Gamma$ on $E^*$ whose value is to be the linear functional

\begin{equation*} \theta_{\Gamma}|_\zeta \in T_\zeta^*E^* \end{equation*}

at ζ that is determined by its value

(4.3)\begin{equation} \theta_\Gamma|_{\zeta}(\xi): = \zeta(p_\Gamma \circ T\pi(\xi)) \end{equation}

against $\xi \in T_\zeta(T^*\mathcal F)$.

(We remark that this is reduced to the canonical Liouville one-form θ on the cotangent bundle $T^*L$ for the case of Lagrangian submanifold L in which case $\omega_Y = 0$ and the splitting is trivial and not needed.)

Then we define the closed (indeed exact) two-form on $E^* = T^*\mathcal F$ by

\begin{equation*} - d\theta_\Gamma. \end{equation*}

Together with the pull-back form $\pi^*\omega_Y$, we consider the closed two-form $\omega_{E^*,\Gamma}$ defined by

(4.4)\begin{equation} \omega_{E^*,\Gamma}:= \pi^*\omega_Y - d\theta_\Gamma \end{equation}

on $E^* = T^*\mathcal F$. It is easy to see that $\omega_{E^*,\Gamma}$ is non-degenerate in a neighbourhood $V \subset E^* $ of the zero section. (See the coordinate expression [Reference Oh and Park20, Equation (6.6)] of $d\theta_\Gamma$ and ω_V.)

Definition 4.6 (Gotay’s symplectic form [Reference Gotay15])

We denote the restriction of $\omega_{E^*,\Gamma}$ to V by ω_V, i.e.

\begin{equation*} \omega_V: = (\pi^*\omega_Y - d\theta_\Gamma)|_V. \end{equation*}

We call this two-form Gotay’s symplectic form on $V \subset E^*$.

The following theorem ends the description of Gotay’s normal form for the neighbourhood of a coisotropic submanifold $C \subset (M,\omega)$ of any symplectic manifold $(M,\omega)$ as a neighbourhood V of the zero section of $T^*\mathcal F_C$ of its null foliation $\mathcal F_C$ on C equipped with the symplectic form.

Theorem 4.7 (See [Reference Gotay15, Reference Oh and Park20])

Let $Y \subset (X,\omega_X)$ be any coisotropic submanifold. Fix a splitting Γ in (4.2). Then there is a neighbourhood $\mathcal{N}\text{bhd}(Y):= U \subset X$ and a diffeomorphism

\begin{equation*} \Phi_\Gamma : U \to V \subset E^* \end{equation*}

such that the following hold:

1. (1) $\omega_X = \Phi_\Gamma^*\omega_{E^*,\Gamma}$ on $U \subset X$.

2. (2) For two different choices, Γ and $\Gamma'$, of splitting of TY, the associated two forms $\omega_{E^*,\Gamma}$ and $\omega_{E^*,\Gamma'}$ are diffeomorphic relative to the zero section $Y \subset E^*$, on a possibly smaller neighbourhood $V' \subset E^*$ of Y.

We have the natural projection map

(4.5)\begin{equation} \widetilde \pi_Y: \mathcal{N}\text{bhd}(Y) \to Y \end{equation}

defined by

(4.6)\begin{equation} \widetilde \pi_Y := \pi_{E^*}\circ \Phi_\Gamma \circ \iota_Y, \end{equation}

for the inclusion map $\iota_Y: Y \hookrightarrow \mathcal{N}\text{bhd}(Y)=:U \subset X$, which is induced by restricting the canonical projection $E^* \to Y$ to the neighbourhood $V \subset E^*$ of the zero section Y. In particular, we have

\begin{equation*} \ker d_x\pi_Y = E_x = \mathcal D_Y|_x. \end{equation*}

4.2. Structure of the null foliations of σ-sectorial corners

We apply the discussion in the previous subsection to general transverse coisotropic collection

\begin{equation*} \{H_1, \ldots, H_m\}. \end{equation*}

For any given subset $I \subset \{1, \ldots, m\}$, we denote

\begin{equation*} H_I = \bigcap_{i \in I} H_i \end{equation*}

and $\pi_{H_I}: H_I \to \mathcal N_{H_I}$ be the canonical projection. We also denote the full intersection by

\begin{equation*} C= \bigcap_{i=1}^m H_i. \end{equation*}

Furthermore, by the transverse intersection property of the coisotropic collection, we can choose the collection $\{\sigma_{C,1}, \ldots, \sigma_{C,m}\}$ to have the complete intersection property in that their images form a collection of transverse intersection. More precisely, we fix the following choice of smooth sections for a finer study of the neighbourhood structure of further constructions we will perform

Choice 4.8 (Choice of sections $\sigma_i: \mathcal N_{H_i} \to H_i$)

For each $i = 1, \ldots, m$, we choose a smooth section

\begin{equation*} \sigma_i: \mathcal N_{H_i} \to H_i \end{equation*}

for each $i = 1, \ldots, m$. Denote the set of sections $\sigma_i: \mathcal N_{H_i} \to H_i$ by

(4.7)\begin{equation} \sigma = \{\sigma_1, \ldots, \sigma_m\}. \end{equation}

Recall from § 3 that for each i a choice of smooth section

\begin{equation*} \sigma_i: \mathcal N_{H_i} \to H_i \end{equation*}

provides the trivialization map

\begin{equation*} \Psi_i^{\sigma_i}: H_i \to \mathcal N_{H_i} \times \mathbb R, \quad \Psi_i^{\sigma_i}(x) = (\pi_{H_i}(x), t_i^{\sigma_i}(x)) \end{equation*}

given in (3.12). We choose each σ_i to be $\sigma_i = \sigma_{H_i}$ as defined in (3.19) For the given choice of $\sigma = \{\sigma_1, \ldots, \sigma_m\}$, we collectively write

(4.8)\begin{equation} \Psi_i^\sigma: = \Psi_i^{\sigma_i}, \quad i = 1, \ldots, m. \end{equation}

The following theorem is the generalization of Theorem 1.5 whose proof also extends the one used in § 3 to the case with corners. The main task for this extension is to establish compatibility of the null foliations of various coisotropic intersections arising from taking a sub-collection $I \subset \{1, \ldots, m\}$: This compatibility condition and construction of relevant strata is in the same spirit as the combinatorial construction of a toric variety out of its associated fan. (See [Reference Fulton9] for example.)

Theorem 4.9. Let $(M,\lambda)$ be a Liouville σ-sector with corners, and let Z be the Liouville vector field of $(M,\lambda)$. Let

\begin{equation*} \sigma = \{\sigma_1, \ldots, \sigma_m\} \end{equation*}

be a collection of smooth sections $\sigma_i: \mathcal N_{H_i} \to H_i$ for $i=1, \ldots, m$. Then the leaf space $\mathcal N_{C}$ carries a canonical structure $\lambda_{\mathcal N_{C}}$ of a Liouville manifold with boundary and corners.

We also define the function $t_i^{C,\sigma}: C \to \mathbb R$ to be the restriction

(4.9)\begin{equation} t_i^{C,\sigma}= t_i^{\sigma_i}|_C \end{equation}

where $t_i^{\sigma_i}$ is the function appearing in (3.13). The collection $\sigma = \{\sigma_i\}$ also induces a surjective map $\Psi_C: C \to \mathcal N_C \times \mathbb R^m$,

(4.10)\begin{equation} \Psi_C^\sigma(x): = \left(\pi_C(x), \left(t_1^{C,\sigma}(x),\ldots, t_m^{C,\sigma}(x)\right)\right) \end{equation}

which is also smooth with respect to the induced smooth structure on $\mathcal N_C$. (The functions $t_i^{C,\sigma}$ correspond to t_i appearing in [Reference Arnol’d2, Section 49] in the discussion below.)

Proposition 4.10. There is an $\mathbb R^m$-action on C that is free, proper and discontinuous and such that C is foliated by the $\mathbb R^m$-orbits. In particular, the map

\begin{equation*} \Psi_C^\sigma: C \to \mathcal N_C \times \mathbb R^m \end{equation*}

is an $\mathbb R^m$-equivariant diffeomorphism with respect to the $\mathbb R^m$-action on C and that of linear translations on $\mathbb R^m$.

Proof. Let $(s_1, \ldots, s_m)$ be the standard coordinates of $\mathbb R^m$. We set

(4.11)\begin{equation} Z_i : = (\Psi_C^\sigma)^*\left(\vec 0_{\mathcal N_C} \oplus \frac{\partial}{\partial s_i}\right). \end{equation}

Then $Z_i \in \mathcal D_C$, and $[Z_i,Z_j] = 0$ since $[\frac{\partial}{\partial s_i},\frac{\partial}{\partial s_j}] = 0$. On C, we also have

\begin{equation*} t_j^{C,\sigma}(Z_i) = d(s_j \circ \Psi_C^\sigma)\left((\Psi_C^\sigma)^*\left(\vec 0_{\mathcal N_C} \oplus \frac{\partial}{\partial s_i}\right)\right) = ds_j(\frac{\partial}{\partial s_i}) = \delta_{ij}. \end{equation*}

In particular, Z_i is tangent to all level sets of $t_i^{C,\sigma}$ with j ≠ i and is transverse to the level sets of $t_i^{C,\sigma}$ for each i.

The so-constructed global frame $\{Z_1, \ldots, Z_m\}$ of TC on C are commuting vector fields. Therefore, we have an $\mathbb R^m$-action on C induced by the flows of commuting vector fields $\{Z_1, \ldots, Z_m\}$.

Lemma 4.11. This $\mathbb R^m$-action is also proper and discontinuous. In particular, its isotropy subgroup is a discrete subgroup of $\mathbb R^m$.

Proof. The Liouville vector field Z is tangent to every H_i near infinity. Since Z is tangent to H_i for all i near infinity, the flag

\begin{equation*} H_1 \cap \cdots \cap H_m \subset H_1 \cap \cdots \cap H_{m-1} \subset \cdots \subset H_1 \end{equation*}

is Z-invariant near infinity, and in particular, we have

\begin{equation*} Z \in TC \end{equation*}

near infinity of C. Since $Z[s] = 1$, Z is also transverse to $s^{-1}(r)$ for all sufficiently large r > 0. Therefore the $\mathbb R^m$-action induces a free $\mathbb R^m/\mathbb R$-action on the set $\partial_\infty C = \partial_\infty M \cap C$ of asymptotic Liouville rays tangent to C. Since the latter set is compact, it follows that the $\mathbb R^m/\mathbb R$-action is proper and discontinuous. Since the flow of Z or the $\mathbb R$-action induced by Z moves the level of s by 1 as time varies by 1, we conclude that the $\mathbb R^m$-action on C is proper and discontinuous.

Once the action is proved to be proper and discontinuous, the second statement of the lemma follows e.g. from the proof in [Reference Arnol’d2, Section 49, Lemma 3], to which we refer. This finishes the proof.

With Lemma 4.11 in our disposal, the standard argument in the construction of action-angle coordinates proves that each orbit of the $\mathbb R^m$-action is homeomorphic to $\mathbb R^{n_1} \times T^{n_2}$ for some $n_1, \, n_2$ with $n_1 + n_2 = n$. (See [Reference Arnol’d2, Section 49, Lemma 3] and its proof.)

Now we immediately conclude the following

Corollary 4.12. Suppose $\pi_C: C \to \mathcal N_C$ has contractible fibres. Then

1. (1) The $\mathbb R^m$-action is free and its fibre is naturally diffeomorphic to $\mathbb R^m$, i.e. it is a principle $\mathbb R^m$ bundle over $\mathcal N_C$.

2. (2) The map Ψ is an $\mathbb R^m$-equivariant diffeomorphism with respect to the translations of $\mathbb R^m$.

The inverse of $\Psi_C^\sigma$ denoted by

(4.12)\begin{equation} \Phi_C^\sigma: \mathcal N_C \times \mathbb R^m \to C \end{equation}

is also easy to explicitly write down as follows. First we note

\begin{equation*} t_i^{C,\sigma}(\sigma_{C,i}(\pi_C(x))) = 0 \end{equation*}

for all $i=1,\ldots, m$ by the definitions of $\sigma_{C,i}$ and $t_i^{C,\sigma}$. Now let a point

\begin{equation*} (\ell, (t_1, \ldots, t_m)) \in \mathcal N_C \times \mathbb R^m \end{equation*}

be given. Then there is a unique point $x \in C$ satisfying

(4.13)\begin{equation} \begin{cases} \pi_C(x) = \ell\\ x = \bigcap_{i=1}^n (t_i^{C,\sigma})^{-1}(t_i). \end{cases} \end{equation}

(See (4.9) for the definition of $t_i^{C,\sigma}$ and Proposition 3.4 for the definition of the vector field $Z_i'$, respectively.) Then we define $\Phi_C^\sigma(\ell, (t_1, \ldots, t_m))$ to be this unique point. It is easy to check from definition that $\Phi_C^\sigma$ is indeed the inverse of $\Psi_C^\sigma$. This finishes the proof of Proposition 4.10.

By applying the above proof and Proposition 4.10 to any sub-collection $I \subset \{1, \ldots, m\}$ including the full collection itself, we also obtain the following stronger form of Theorem 4.9

Theorem 4.13. Let $I \subset \{1, \ldots, m\}$ be any sub-collection and define

\begin{equation*} H_I = \bigcap_{i \in I} H_i. \end{equation*}

Assume $\pi_{H_I}: H_I \to \mathcal N_{H_I}$ has contractible fibres. Let $\lambda_{\mathcal N_{H_I}}$ be the canonical induced Liouville form as before. Then the following hold:

1. (1) There is an $\mathbb R^{|I|}$-action on H_I that is free, proper and discontinuous and such that H_I is foliated by the $\mathbb R^{|I|}$-orbits. In particular, the map

   \begin{equation*} \Psi_{H_I}^\sigma: H_I \to \mathcal N_{H_I} \times \mathbb R^{|I|} \end{equation*}

   is an $\mathbb R^{|I|}$-equivariant diffeomorphism with respect to the $\mathbb R^{|I|}$-action on H_I and that of linear translations on $\mathbb R^{|I|}$.

2. (2) The leaf space $\mathcal N_{H_I}$ carries a canonical structure of a Liouville manifold with boundary and corners.

By applying the above to the full collection $C = H_{\{1,\ldots, m\}}$, we have finished the proof of Theorem 4.9.

4.3. Compatibility of null foliations of transverse coisotropic intersections

Let $C_\delta = C$ as in the previous section and let $\{\sigma_1, \ldots, \sigma_m\}$ a collection of sections $\sigma_i:\mathcal N_{H_i} \to H_i$ made in Choice 4.8. For each subset $I \subset \{1, \ldots, m\}$, we have the following section

\begin{equation*} \sigma_i: \mathcal N_{H_i} \to H_i \end{equation*}

defined by

(4.14)\begin{equation} \sigma_I([\ell]): = \Phi_{H_I}^\sigma([\ell], (0,\ldots, 0)) = (\Psi_{H_I}^\sigma)^{-1}([\ell], (0,\ldots, 0)) \end{equation}

for the diffeomorphism $\Phi_{H_I}$ given in (4.12) applied to $C = H_I$.

Then for each pair of subsets $I \subset J $ of $\{1,\ldots, n\}$, we have $H_J \subset H_I$ and the map

\begin{equation*} \psi_{JI}^\sigma: \mathcal N_{H_J} \to \mathcal N_{H_I} \end{equation*}

given by

(4.15)\begin{equation} \psi_{JI}^\sigma([\ell]): = \pi_{\mathcal N_{H_I}}(\Phi_{H_J}^\sigma([\ell], (0,\ldots,0)). \end{equation}

In particular, consider the cases with $I = \{i\}$, $J = \{i, j\}$ and $K = \{i,j,k\}$. Then we prove the following compatibility of the collection of maps ψ_IJ: For each i ≠ j, we consider the maps

\begin{equation*} \psi_{ij,i}^\sigma: \mathcal N_{H_i\cap H_j} \to \mathcal N_{H_i} \end{equation*}

defined by $\psi_{ij,i}^\sigma := \psi_{\{ij\}\{i\}}$, and the inclusion maps

\begin{equation*} \iota_{ij,i}: H_i \cap H_j \to H_i. \end{equation*}

Proposition 4.14. Let $\{H_1,\ldots,H_m\}$ be a collection of hypersurfaces satisfying only (S1) and (S2). Then the maps $\psi_{ij,i}^\sigma$ satisfy the following:

1. (1) They are embeddings.

2. (2) The diagram

   commutes for all pairs $1 \leq i, \, j \leq n$.

3. (3) The diagrams are compatible in the sense that we have

   \begin{equation*} \psi_{ij,i}^\sigma \circ \psi_{ijk,ij}^\sigma = \psi_{ijk,i}^\sigma. \end{equation*}

   for all triples $1 \leq i,\, j, \, k \leq n$.

(4.16)

Proof. We first show that the map $\psi_{ij,i}^\sigma$ is an embedding. Let $\ell_1, \, \ell_2$ be two leaves of the null-foliation of $H_i \cap H_j$ such that

\begin{equation*} \ell_1 \cap H_i = \ell_2 \cap H_i. \end{equation*}

By definition of leaves, we have only to show that $\ell_i \cap \ell_j \neq \emptyset$.

Let $x \in H_i$ be in the above two common intersection which obviously implies

\begin{equation*} x \in \ell_1 \cap \ell_2 \subset H_i \cap H_j. \end{equation*}

This proves $\psi_{ij,i}^\sigma$ is a one-one map. Then smoothness and the embedding property of $\psi_{ij,i}^\sigma$ follow from the definition of smooth structures given on the leaf spaces.

For the commutativity, we first note

(4.17)\begin{equation} \psi_{ij,i}^\sigma (\pi_{ij}(x)) = \pi_i(\Phi_{ij}^\sigma((\pi_{ij}(x), 0,0))) \end{equation}

by the definition of the maps $\psi_{ij,i}^\sigma$. But by the definition (4.12) of $\Phi_{ij}^\sigma$, the point

\begin{equation*} y := \Phi_{ij}^\sigma((\pi_{ij}(x), 0,0)) \end{equation*}

is the intersection point

\begin{equation*} y \in \operatorname{Image} \sigma_i \cap \operatorname{Image} \sigma_j. \end{equation*}

Since $x \in H_i \cap H_j$, we can express it as

\begin{equation*} x = \Phi_{ij}^\sigma(\pi_{ij}(x), t_1, t_2) \end{equation*}

for some $t_1, \, t_2 \in \mathbb R$. In other words, it is obtained from y by the characteristic flows of H_i and H_j by definition of $\Phi_{ij}^\sigma$ in (4.12). In particular, we have

\begin{equation*} \pi_i(\iota_{ij,i}(x)) = \pi_i(y). \end{equation*}

On the other hand, the definition of the null foliation of $\mathcal N_{H_i}$ implies

(4.18)\begin{equation} \pi_i(y) = \psi_{ij,i}^\sigma (\pi_{ij}(x)) \end{equation}

for all $x \in H_i \cap H_j$. Combining the last two equalities and commutativity of the diagram $\pi_i \circ \iota_{ij,i} = \psi_{ij,i}^\sigma \circ \pi_{ij}$, we have proved the commutativity of (4.16).

Finally, we show that $\psi_{ij,i}^\sigma$ is a symplectic map. Consider the pull-back

\begin{equation*} \omega_{ij}^\sigma: = (\psi_{ij,i}^\sigma)^*(\omega_{\mathcal N_{H_i}}). \end{equation*}

We will show that $\omega_{ij}^\sigma$ satisfies the defining property

\begin{equation*} \pi_{H_i \cap H_j}^*\omega_{ij}^\sigma = \iota_{H_i \cap H_j}^*\omega, \quad \omega = d\lambda \end{equation*}

of the reduced form on $\mathcal N_{H_i \cap H_j}$ under the coisotropic reduction on the coisotropic submanifolds $H_i \cap H_j \subset M$. We compute

\begin{eqnarray*} \pi_{H_i \cap H_j}^*\omega_{ij}^\sigma & = & \pi_{H_i \cap H_j}^*(\psi_{ij,i}^\sigma)^*(\omega_{\mathcal N_{H_i}})\\ & = & (\psi_{ij,i}^\sigma \circ \pi_{H_i \cap H_j})^*(\omega_{\mathcal N_{H_i}}) \\ & = & (\pi_{H_i}\circ \iota_{H_i\cap H_j,H_i})^* \omega_{\mathcal N_{H_i}} \\ & = & (\iota_{H_i\cap H_j,H_i})^*(\pi_{H_i}^*\omega_{\mathcal N_{H_i}}) \\ & = & (\iota_{H_i\cap H_j,H_i})^*(\iota_{H_i}^*\omega) = \iota_{H_i\cap H_j}^*\omega \end{eqnarray*}

where we use the defining condition of the reduced form $\omega_{\mathcal N_{H_i}}$ of $\omega_{\partial H_i}$

\begin{equation*} \pi_{H_i}^*\omega_{\mathcal N_{H_i}} = \iota_{H_i}^*\omega \end{equation*}

for the penultimate equality. Therefore we have proved

\begin{equation*} \pi_{H_i \cap H_j}^*\omega_{ij}^\sigma = \iota_{H_i \cap H_j}^*\omega. \end{equation*}

This shows that the form $\omega_{ij}^\sigma$ satisfies the defining equation (3.27) of the reduced form $\omega_{H_i \cap H_j}$. Then by the uniqueness of the reduced form, we have derived

\begin{equation*} \omega_{ij}^\sigma = \omega_{H_i \cap H_j}. \end{equation*}

This proves $(\psi_{ij,i}^\sigma)^*\omega_{H_i} = \omega_{H_i \cap H_j}$, which finishes the proof of Statement (1).

Statement (2) also follows by a similar argument this time from the naturality of the coisotropic reduction by stages: Consider $H_i, \, H_j, \, H_k$ in the given coisotropic collection and consider the two flags

(4.19)\begin{equation} H_i \cap H_j \cap H_k \subset H_i \cap H_j \subset H_i \end{equation}

and

(4.20)\begin{equation} H_i \cap H_j \cap H_k \subset H_i. \end{equation}

The composition $\psi_{ij,i}^\sigma \circ \psi_{ijk,ij}^\sigma$ is the map obtained by the coisotropic reductions in two stages and $\psi_{ijk,i}^\sigma$ is the one obtained by the one-stage reduction performed in the proof of Statement 1 with the replacement of the pair $(H_i \cap H_j, H_i)$ by $(H_i\cap H_j \cap H_k, H_i)$. Then by the naturality of the coisotropic reduction, we have proved Statement 2. This finishes the proof of the proposition.

The following is an immediate corollary of the above proposition and its proof. (See Remark 1.17 for the relevant remark on the stratified presymplectic manifolds.)

Corollary 4.15. The collection of maps

\begin{equation*} \{\psi_{I}\}_{I \subset \{1, \ldots, m\}} \end{equation*}

are compatible in that the leaf space $\mathcal N_{H_I}$ carries the structure of symplectic manifold with boundary and corners.

5.1. Definition of Liouville σ-sectors with corners

Denote by $\iota_{CH_i}: C \to H_i$ the inclusion map, and $\sigma = \{\sigma_1, \ldots, \sigma_m\}$ be the collection as before. This induces the diagram

(5.1)

for all i which are compatible in the sense of Statement (2) of Proposition 4.14. In fact, we have

(5.2)\begin{equation} \mathcal D_C = \mathcal D_{H_1}|_C + \mathcal D_{H_2}|_C + \cdots + \mathcal D_{H_m}|_C \end{equation}

which canonically induces the leaf map $\psi_{CH_i}^\sigma$ in the bottom arrow that makes the diagram commute.

With these preparations, we are finally ready to provide the sectional characterization of Liouville sectors with corners.

Definition 5.1 (Liouville σ-sectors with corners)

Let M be a manifold with corners equipped with a Liouville one-form λ. We call $(M,\lambda)$ a Liouville σ-sector with corners if at each sectorial corner δ of $\partial M$, the corner can be expressed as

\begin{equation*} C_\delta := H_{\delta,1} \cap \ldots \cap H_{\delta,m} \end{equation*}

for a transverse coisotropic collection

\begin{equation*} \{H_{\delta,1}, \ldots, H_{\delta,m}\} \end{equation*}

of σ-sectorial hypersurfaces such that fibres of the map

\begin{equation*} \pi_{C_\delta}: C_\delta \to \mathcal N_{C_\delta} \end{equation*}

are contractible. We call such a corner C_δ a σ-sectorial corner of codimension m.

In the remaining section, we will derive the consequences of this definition. The following monoidal property is apparent from our definition of Liouville σ-sectors.

Proof. Let M ₁ and M ₂ be Liouville σ-sectors with corners. For the simplicity of exposition, we assume M_i without corners. The general case follows by similar arguments.

Recall that the set of manifolds with corners naturally forms a monoid and so $M_1 \times M_2$ is a manifold with corners with its boundary and corners given by

\begin{eqnarray*} \partial(M_1 \times M_2) & = & \partial M_1 \times M_2 \cup M_1 \times \partial M_2= : H_1 \cup H_2 \\ C_2 & = & \partial M_1 \times \partial M_2 = H_1 \cap H_2. \end{eqnarray*}

Obviously, both are coisotropic submanifolds of codimension 1 and 2, respectively, and C ₂ is a manifold without boundary. The required transversality hypothesis trivially holds.

It remains to show the property of their null foliations. Let $\mathcal F_i$ be the associated characteristic foliation of $\partial M_i$. Then the characteristic distribution of $\partial(M_1\times M_2)$ is given by

\begin{equation*} \mathcal D_1 \oplus \{0\}, \quad \{0\} \oplus \mathcal D_2 \end{equation*}

on $\partial M_1 \times M_2$ (resp. $M_1 \times \partial M_2$) whose leaf $\mathcal F_{\partial(M_1 \times M_2),(x,y)}$ is given by $(\mathcal F_1)_x \times \{y\}$ (resp. $\{x\} \times (\mathcal F_2)_y$) which is obviously trivial. Furthermore, the required section $\sigma_{\partial(M_1\times M_2)}$ is given by the map $\sigma_1 \times {\rm id}_Y$ and ${\rm id}_X \times \sigma_2$, respectively.

On the corner C ₂, its null distribution is given by

\begin{equation*} \mathcal D_{M_1 \times M_2;(x,y)} = \mathcal D_{1,x} \oplus \mathcal D_{2,y} \end{equation*}

whose associated leaves are $\mathcal F_{1,x} \times \mathcal F_{2,x}$ which are clearly contractible if so are $\mathcal F_{1,x}$ and $\mathcal F_{2,x}$. The product of Liouville sectors with corners can be treated similarly whose details are omitted. This finishes the proof.

5.2. Integrable systems and canonical splitting data

By applying Theorem 4.7 to the coisotropic submanifold C, we will obtain a neighbourhood $\mathcal{N}\text{bhd}(C) \subset M$ and the projection

\begin{equation*} \widetilde \pi_C: \mathcal{N}\text{bhd}(C) \to C. \end{equation*}

Choice 5.3 (Splitting $\Gamma_C^\sigma$)

Let $\sigma=\{\sigma_1,\ldots, \sigma_m\}$ be a choice of sections of transverse coisotropic collection $\{H_1, \ldots, H_m\}$. Then we associate the splitting

(5.3)\begin{equation} \Gamma = \Gamma_C^\sigma : \quad TC = G_C^\sigma \oplus \mathcal D_C \end{equation}

thereto given by the transverse symplectic subspace

(5.4)\begin{equation} G_C^\sigma |_x : = (d\Psi_C^\sigma|_x)^{-1}(T_{\pi_C(x)}\mathcal N_C \oplus \{0\}_{\mathbb R^m}). \end{equation}

Applying Theorem 4.7, we obtain a diffeomorphism

\begin{equation*} \Psi_\Gamma^\sigma: \mathcal{N}\text{bhd}(C) \to V \subset E^* = T^*\mathcal F_C \end{equation*}

where $\mathcal F_C$ is the null foliation of C. Furthermore, the pushforward of symplectic form $d\lambda$ on U is given by the canonical Gotay’s symplectic form on $V \subset E^*$

\begin{equation*} (\Psi_\Gamma^\sigma)_*(d\lambda) = \pi^*\omega_C - d\theta_\Gamma \end{equation*}

for the presymplectic form $\omega_C = \iota_C^*(d\lambda)$ on C. (See Theorem 4.7.)

Note that we have

\begin{equation*} \mathcal D_C|_x = {\rm span}_{\mathbb{R}}\{Z_1(x), \ldots, Z_m(x)\} \end{equation*}

by definition of Z_i above. Therefore, the aforementioned $\mathbb R^m$-action induces an $\mathbb R^m$-equivariant bundle isomorphism

\begin{equation*} \mathcal D_C \cong C \times \mathbb R^m \end{equation*}

over C. (This isomorphism does not depend on the choice of σ but depends only on the Liouville geometry of $\mathcal{N}\text{bhd}(C \cap \partial_\infty M)$. )

Then we have made the aforementioned splitting $ TC = G_C^\sigma \oplus \mathcal D_C $ given in (5.4) $\mathbb R^m$-equivariant. In other words, for each group element $\mathsf t = (t_1, \dots, t_m) \in \mathbb R^m$, we have the equality

\begin{equation*} d \mathsf t (G_x^\sigma ) = G_{\mathsf t\cdot x}^\sigma. \end{equation*}

For a fixed α > 0, we put

(5.5)\begin{equation} I_i^\sigma = \pm e^{\alpha t_i^{C,\sigma}} \end{equation}

which then satisfies $dI_i^\sigma(Z_i) = \alpha\, I_i^\sigma$ on C.

Noting that the induced $\mathbb R^m$-action on TC preserves the subbundle

\begin{equation*} T \mathcal F_C = \mathcal D_C \subset TC, \end{equation*}

the canonically induced action on $T^*C$ also preserves the subbundle

\begin{equation*} \mathcal D_C^\perp \subset T^*C \end{equation*}

for which we have the isomorphism

\begin{equation*} T^*\mathcal F \cong \mathcal D_C^{\perp}. \end{equation*}

Therefore, the $\mathbb R^m$-action on C can be lifted to $T^*\mathcal F$ which is the restriction of the canonical induced action on $T^*C$ of the one on C.

Proof. Let $\phi_{Z_j}^t$ be the flow of Z_j on C. Since the $\mathbb R^m$-action is abelian, the vector fields Z_j’s are pairwise commuting. Then the lifting $Z'_j$ is nothing but the vector field generating the isotopy of canonical derivative maps

\begin{equation*} ((d\phi_{Z_j}^t)^*)^{-1}: T^*C \to T^*C \end{equation*}

on $T^*C$. Since the flows $\phi_{Z_j}^t$ are commuting, their derivatives are also commuting. Then obviously their dual flows $((d\phi_{Z_j}^t)^*)^{-1}$ on $T^*C$ are also commuting and hence $Z'_j$’s too. The first condition also follows since any derivative maps zero vector to a zero vector. This finishes the proof.

We now define

\begin{equation*} \widetilde I_i^\sigma = I_i^\sigma \circ \pi_{T^*\mathcal F}. \end{equation*}

Then $\{d\widetilde I_1^\sigma,\ldots, d\widetilde I_m^\sigma\}$ are linearly independent on a neighbourhood of the zero section of $T^*\mathcal F$ if we choose the neighbourhood small enough. This is because $\{dI_1^\sigma, \ldots, dI_m^\sigma\}$ are linearly independent on C. By suitably adjusting the parametrization $t_i^{C,\sigma}$ of the $\mathbb R^m$-action, we can make the equation

(5.6)\begin{equation} d\widetilde I_i^\sigma(Z'_j) = \alpha \delta_{ij} \widetilde I_i^\sigma \end{equation}

hold.

This is precisely the situation of a completely integrable system to which we can apply the standard construction of action-angle coordinates. (See [Reference Arnol’d2, Section 49] for example.) Therefore, regarding $\{\widetilde I_1^\sigma, \ldots, \widetilde I_m^\sigma\}$ as the (fibrewise) angle coordinates, we can find a unique choice of (fibrewise) action coordinates

\begin{equation*} \{\widetilde R_1^\sigma, \ldots, \widetilde R_m^\sigma\} \end{equation*}

over $\mathcal N_C$ satisfying

\begin{equation*} \{\widetilde R_i^\sigma, \widetilde I_j^\sigma\} = \delta_{ij}, \quad \widetilde R_i^\sigma \circ \Phi_C^\sigma|_{H_j} = 0 \end{equation*}

on a neighbourhood $V \subset T^*\mathcal F_C$ of the zero section $0_{T^*\mathcal F_C} \cong C$. Now we define the pull-back functions

\begin{equation*} R_i^\sigma: = \widetilde R_i^\sigma \circ \Phi_C^\sigma, \quad I_j^\sigma: = \widetilde I_j^\sigma \circ \Phi_C^\sigma \end{equation*}

on $U= \mathcal{N}\text{bhd}(C)$. We also pull-back the vector fields $Z'_j$ to $\mathcal{N}\text{bhd}(C)$ by $\Phi_C^\sigma$ and denote them by Z_j. (Note that the notations $I_j^\sigma$ and Z_j are consistent in that their restrictions to C are nothing but the above already given $I_j^\sigma$ or Z_j, respectively on C.) Furthermore, we have the relationship

\begin{equation*} Z_j = X_{R_j^\sigma}. \end{equation*}

(See the definition (4.11) of Z_i on C.)

Then we have

\begin{equation*} \{R_i^\sigma,R_j^\sigma\} = \omega(X_{R_i^\sigma},X_{R_j^\sigma})= \omega(Z_i,Z_j) = 0 \end{equation*}

on $\mathcal{N}\text{bhd}(C)$. Since $ Z_i : = X_{R_i^\sigma}, $ we have

(5.7)\begin{equation} Z_i \rfloor \omega = dR_i^\sigma \end{equation}

on $U = \mathcal{N}\text{bhd}^Z(C)$. This is precisely the defining equation of the moment map $\phi_{G,C}^\sigma: \mathcal{N}\text{bhd}(C) \to \frak g^* \cong \mathbb R^m$ with $G = \mathbb R^m$ given by

\begin{equation*} \phi_{G,C}^\sigma(x) = (R_1^\sigma(x), \ldots, R_m^\sigma(x)) \end{equation*}

for the above $G=\mathbb R^m$-action. Recall that the hypersurfaces H_i are Z-invariant near infinity. Therefore, we can choose the neighbourhood $\mathcal{N}\text{bhd}(C)$ so that it is Z-invariant near infinity. Then by the requirement put on the Liouville vector field Z which is pointing outward along $\partial M$, we can choose the whole neighbourhood $\mathcal{N}\text{bhd}(C)$ Z-invariant. Together with the normalization condition of R_i’s

\begin{equation*} R_i^\sigma|_{H_i} = \widetilde R_i \circ \Phi_C^\sigma|_{H_i} = 0, \end{equation*}

it also implies $R_i^\sigma \geq 0$ on $\mathcal{N}\text{bhd}(C)$ for all i. We now take the neighbourhood $U \subset M$ to be this Z-invariant neighbourhood

\begin{equation*} U = \mathcal{N}\text{bhd}^Z(C). \end{equation*}

The content of the above discussion can be summarized into the following intrinsic derivation of the splitting data.

Theorem 5.5 (σ-Splitting data)

Let $C \subset \partial M$ be a sectorial corner of codimension n associated with the sectorial coisotropic collection $\{H_1, \ldots, H_m\}$ on $\partial M$. Then for each choice

\begin{equation*} \sigma = \{\sigma_1, \ldots, \sigma_m\} \end{equation*}

of sections $\sigma_i: \mathcal N_{H_i} \to H_i$ of $\pi_{H_i}$ for $i = 1, \ldots, n$, there is a diffeomorphism

\begin{equation*} \Psi_C^\sigma: \mathcal{N}\text{bhd}^Z(C) \cap \partial M \to F \times \mathbb R^m \end{equation*}

and

\begin{equation*} \psi_C^\sigma: \mathcal N_{C} \to F_C^\sigma \end{equation*}

such that

1. (1) $F_C^\sigma = {\rm Image}\,\,\sigma_1 \cap \cdots \cap {\rm Image}\,\,\sigma_m$,

2. (2) $(\Psi_C^\sigma)_*\omega_\partial = \pi_F^*\omega_F$,

3. (3) The following diagram

   (5.8)

   

   commutes for the map

   \begin{equation*} \Psi_C^\sigma = (\sigma_C \circ \pi_{F_C},(I_1^\sigma, \ldots, I_m)). \end{equation*}

4. (4) The G-action with $G = \mathbb R^m$ has the moment map $\phi_{G,C}^\sigma: \mathcal{N}\text{bhd}_\epsilon^Z(C) \to \mathbb R^m$ given by

   \begin{equation*} \phi_{G,C}^\sigma = (R_1^\sigma, \ldots, R_m^\sigma) \end{equation*}

   for a collection of Poisson-commuting R_i’s satisfying the simultaneous normalization condition

   \begin{equation*} R_i^\sigma|_{H_i} = 0, \quad R_i^\sigma \geq 0 \end{equation*}

   for all i on $\mathcal{N}\text{bhd}^Z(C)$.

5. (5) The map $\mathcal{N}\text{bhd}(\partial M) \to F_C \times {\mathbb C}^m_{{\rm Re} \geq 0}$ is given by the formula

   (5.9)\begin{equation} \widetilde \Psi_C^\sigma(x) = \left(\sigma_C(\pi_{F_C}(x)), R_1^\sigma(x) + \sqrt{-1} I_1^\sigma(x), \ldots, R_m^\sigma(x) + \sqrt{-1}I_m^\sigma(x)\right). \end{equation}

   such that

   (5.10)\begin{equation} (\widetilde \Psi_C^\sigma)_*\omega = \pi_F^*\omega_{F_C} + \sum_{i=1}^m dR_i^\sigma \wedge dI_i^\sigma. \end{equation}

We call these data a σ-splitting data of $\mathcal{N}\text{bhd}(C)$ associated with the choice $\sigma = \{\sigma_1, \cdots, \sigma_m\}$ of sections $\sigma_i: \mathcal N_{H_i} \to H_i$.

We also gather the following consequences of the above discussion separately. The first one, in particular, states that Proposition 1.7 still holds for the Liouville σ-sectors with corners.