# A Stefan-Sussmann theorem for normal distributions on manifolds with
boundary

David Perrella  David Pfefferlé  Luchezar Stoyanov  The University of Western
Australia, 35 Stirling Highway, Crawley WA 6009, Australia

##  I Abstract

An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is
proven for normal distributions. These distributions contain vectors
transverse to the boundary along its entirety. Plain integral manifolds are
not enough to “integrate” a normal distribution; the next best “integrals” are
so-called neat integral manifolds with boundary. The conditions on the
distribution for this integrability is expressed in terms of adapted collars
and integrability of a pulled-back distribution on the interior and on the
boundary.

##  II Introduction

The foundations of integrable distribution theory on manifolds can be
summarised by the following four programs.

  1. 1.

Define integrability of a distribution.

  2. 2.

Ask what “differential” conditions are equivalent to integrability.

  3. 3.

Determine the “regularity” of integral manifolds.

  4. 4.

Look at a notion of “maximal” for integral manifolds and show that there is a
correspondence with foliations.

In the case of constant rank, the Frobenius Theorem addresses program two and
is the main component of the foundations (although, as noted by Lavau [1], the
precise historical list of contributors to the foundations of this theory do
include authors before Frobenius). In the case of non-constant rank, it was
the independent works of Stefan [2] and Sussmann [3] which gave rise to the
completion of the foundations; culminating in the Stefan-Sussmann Theorem
(which again addresses program two). What appears absent from the literature
is an analogous theory on manifolds with boundary, where the integral
manifolds become, in general, manifolds with boundary.

To address this, the present paper takes on the analogous first and second
programs for _normal distributions_ on manifolds with boundary. Normal
distributions are those which contain vectors transverse to the boundary along
its entirety. Our analogue of integrability is what we call _neat
integrability_ and the differential conditions constitute our main result,
namely a Stefan-Sussmann theorem for normal distributions.

In a future paper, we will use the results obtained here to complete the
remaining two analogous programs in the foundations. This will provide means
of guaranteeing examples of _foliations by manifolds with boundary_ on
manifolds with boundary. A general notion of these foliations were considered
by Noakes [4] where he generalised Bott’s Theorem [5] as well as demonstrate
how these foliations may be used to guarantee (ordinary) foliations.

This paper is structured as follows. Distributions and the Stefan-Sussmann
Theorem are first reviewed in section III. We then discuss normal
distributions and state our main result in section IV. Lastly, we give a proof
of this result in section V.

##  III Review of distributions and the Stefan-Sussmann Theorem

The definitions of the standard terms we use from smooth manifold theory
coincide with that of Lee’s book [6] and of Rudolf and Schmidt’s book [7]. In
particular, a submanifold with or without boundary in general is not assumed
to be embedded, only immersed. We compile the relevant theory of distributions
from Rudolf and Schmidt’s book [7] using our own terminology. Let $M$ be a
smooth manifold.

For an arbitrary subset $D\subset TM$, a _$\bm{D}$ -section_ is a vector field
$X$ on $M$ such that for all $p\in M$, $X|_{p}\in D$. We also denote
$D_{p}=D\cap T_{p}M$ for $p\in M$. With this, $D$ is called a _distribution on
$\bm{M}$_ if for all $p\in M$, $D_{p}$ is a vector subspace of $T_{p}M$ and
for any $v\in D_{p}$, there exists a $D$-section $X$ with $X|_{p}=v$. The
_rank of $\bm{D}$_ is the map $r:M\to\mathbb{N}_{0}$ given by $r(p)=\dim
D_{p}$. For a smooth map $f:S\to M$ between manifolds, we set
$f^{*}D=(Tf)^{-1}(D)$ and call $f^{*}D$ the _pullback of $\bm{D}$ on to
$\bm{S}$_ (where $Tf$ denotes the tangent map).

For the following, let $D$ be a distribution on $M$ of rank $r$. An _integral
manifold_ of $D$ is a connected submanifold $I$ of $M$ such that, for $p\in
I$, $Ti|_{p}(T_{p}I)=D_{p}$ ($i:I\subset M$). We say that $D$ is _integrable_
if through every point of $M$, there passes an integral manifold of $D$.

We now introduce the definitions needed to state the Stefan-Sussmann Theorem.
To this end, a _local $D$-section_ is a local section $X:U\to TM$ (where $U$
is an open submaifold of $M$) with $X|_{p}\in D$ for $p\in U$. A distribution
$D$ on $M$ is said to be:

  1. 1.

_involutive_ if its $D$-sections form a Lie subalgebra of the space of vector
fields on $M$,

  2. 2.

_homogeneous_ (after Stefan [2]) if for all local $D$-sections $X$, for any
point $(p,t)\in\mathcal{D}$ in the (maximal) domain of the flow
$\psi^{X}:\mathcal{D}\to M$, one has (after identifying tangent spaces)
$T\psi^{X}_{t}|_{p}(D_{p})=D_{\psi^{X}(p,t)}$.

The final definition pertains to certain nice charts for $D$. Let $p\in M$ and
set $n=\dim M$. A chart $(U,\varphi)$ on $M$ is said to be _adapted to
$\bm{D}$ at $\bm{p}$_ if the following holds (where
$(\partial_{1},...,\partial_{n})$ denotes the local frame of $TM$ induced by
$\varphi$).

  1. 1.

$\varphi(p)=0$ and $\varphi(U)=(-\epsilon,\epsilon)^{n}$ for some
$\epsilon>0$,

  2. 2.

$\partial_{1},...,\partial_{r(p)}$ are local $D$-sections,

  3. 3.

for all $c\in(-\epsilon,\epsilon)^{n-r(p)}$, denoting
$U_{c}=\varphi^{-1}\left((-\epsilon,\epsilon)^{r(p)}\times\\{c\\}\right)$ if
$r(p)<n$ and $U_{c}=U$ otherwise, the rank $r$ of $D$ is constant along
$U_{c}$.

We now state the Stefan-Sussmann Theorem (as formulated in Rudolf and
Schmidt’s book [7]).

######  Theorem III.1 (Stefan-Sussmann Theorem).

Let $M$ be a manifold (without boundary) and $D$ a distribution on $M$. The
following are equivalent.

  1. 1.

$D$ is integrable.

  2. 2.

$D$ is involutive and has constant rank along the integral curves of local
$D$-sections.

  3. 3.

$D$ is homogenous.

  4. 4.

For every $p\in M$, there exists a chart adapted to $D$ at $p$.

The purpose of this paper is to prove an analogous result for what we are
calling _normal distributions on manifolds with boundary_. We now discuss this
notion and state our main result.

##  IV Normal distributions and the main result

The definition of distribution is precisely as before except on a manifold
with boundary (including the notions of rank, pullback and involutivity). For
the remainder of this section, let $D$ be a distribution on a manifold with
boundary $M$.

A connected submanifold $L$ with boundary of $M$ is called an _integral
manifold with boundary of $\bm{D}$_ if for $p\in L$, $Tl|_{p}(T_{p}L)=D_{p}$
($l:L\subset M$). We will say that $D$ is _integrable by boundaries_ if
through every point of $M$, there passes an integral manifold with boundary of
$D$.

The distribution $D$ is said to be _normal_ if for every $p\in\partial M$,
there exists a vector in $D_{p}$ transverse to $\partial M$. To introduce our
analogue of the Stefan-Sussmann Theorem (Theorem III.1), we introduce some
definitions. For the following, let $D$ be a normal distribution on $M$ of
rank $r$.

An integral manifold with boundary $L$ of $D$ is said to be _neat_ if
$\partial L\subset\partial M$. Accordingly, $D$ is said to be _neatly
integrable by boundaries_ if through any point in $M$, there passes a neat
integral manifold of $D$.

We now move to analogues of the hypotheses within the Stefan-Sussmann theorem.
For the remaining of this section, we assume that $\partial M\neq\emptyset$.
The analogues we present will be in terms of some collar; without mentioning
any general vector field. This simplification is made by the hypotheses
expressing integrability of some constructed distributions on the manifolds
$\operatorname{int}M$ and $\partial M$. One can express our main result purely
in terms of vector fields by applying the (standard) Stefan-Sussmann Theorem
(Theorem III.1) and the Boundary Flowout Theorem (Theorem V.2).

Recall, a _collar_ $\Sigma:\partial M\times[0,1)\to M$ is an embedding onto a
neighbourhood of $\partial M$ in $M$ such that $\Sigma(p,0)=p$ for all
$p\in\partial M$. We will call a collar $\Sigma:\partial M\times[0,1)\to M$ a
_$\bm{D}$ -precollar_ if, for all $(p,t)\in\partial M\times[0,1)$,

$r(\Sigma(p,t))=r(p)\text{ and }\Sigma(p,\cdot)^{\prime}(t)\in
D_{\Sigma(p,t)}.$

The normal distribution $D$ will be called _precollared_ if a $D$-precollar
exists. A collar $\Sigma:\partial M\times[0,1)\to M$ will called a _$D$
-collar_ if, denoting $\jmath:\partial M\subset M$, for all $(p,t)\in\partial
M\times[0,1)$,

$T\Sigma(\cdot,t)|_{p}((\jmath^{*}D)_{p})+T\Sigma(p,\cdot)|_{t}(T_{t}[0,1))=D_{\Sigma(p,t)}.$

Accordingly, the normal distribution $D$ will be called _collared_ if a
$D$-collar exists.

Our analogue of adapted charts is closest to the original. However, to ensure
these charts account for the distribution being normal, we must use the local
vector field generated by the last coordinate of the chart; since this is the
only local vector field which is transverse to the boundary. We have arranged
this by using the last few coordinates of the chart as opposed to the first
few.

Let $p\in M$ and set $n=\dim M$. Then, a chart $(U,\varphi)$ is said to be
_neatly adapted to $\bm{D}$ at $\bm{p}$_ if the following holds.

  1. 1.

$\varphi(p)=0$ and $\varphi(U)=(-\epsilon,\epsilon)^{n}$ if
$p\in\operatorname{int}M$ and
$\varphi(U)=(-\epsilon,\epsilon)^{n}\cap\mathbb{H}^{n}$ otherwise,

  2. 2.

$\partial_{n-r(p)+1},...,\partial_{n}$ are local $D$-sections,

  3. 3.

For all $c\in(-\epsilon,\epsilon)^{n-r(p)}$, denoting
$U_{c}=\varphi^{-1}\left(\\{c\\}\times(-\epsilon,\epsilon)^{r(p)}\right)$ if
$r(p)<n$ and $U_{c}=U$ otherwise, the rank $r$ of $D$ is constant along
$U_{c}$.

With these definitions, we may present our main result.

######  Theorem IV.1.

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $D$ be a
normal distribution of rank $r$ on $M$. Denote by
$\imath:\operatorname{int}M\subset M$ and $\jmath:\partial M\subset M$ the
inclusions of the interior and the boundary into $M$. Then, the pullback
$\imath^{*}D$ is a distribution of rank $r|_{\operatorname{int}M}$ on
$\operatorname{int}M$ and $\jmath^{*}D$ is a distribution of rank
$r|_{\partial M}-1$ on $\partial M$. Moreover, the following are equivalent.

  1. S1.

$D$ is neatly integrable by boundaries.

  2. S2.

$\imath^{*}D$ is integrable and $D$ is precollared.

  3. S3.

$\imath^{*}D$ and $\jmath^{*}D$ are integrable and $D$ is collared.

  4. S4.

For every $p\in M$, there exists a chart neatly adapted to $D$ at $p$.

Using Theorem IV.1 and Proposition V.6 (found in the next section), a
Frobenius theorem may be formulated as follows.

######  Corollary IV.1.

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $D$ be a
normal distribution of constant rank. Then the following are equivalent.

  1. F1.

$D$ is neatly integrable by boundaries.

  2. F2.

$D$ is involutive.

  3. F3.

For every $p\in M$, there exists a chart neatly adapted to $D$ at $p$.

We will now prove Theorem IV.1.

##  V Proof of main result

Our proof will establish five implications directly. Although this is not the
optimal amount (four is optimal), three of our chosen implications are
trivial, making for only two substantial implications. We will now establish
some preliminary observations of distributions which we used throughout in
proving these implications.

###  V.1 Preliminary observations of distributions and of normal distributions

We first state two additional results found in Rudolf and Schmidt’s book [7]
for later use. The first is on the “regularity” of integral manifolds of
distributions. They are in fact _weakly embedded_. We introduce some notation
to express this and related notions to simplify subsequent proofs.

Let $A,B,M,N$ be manifolds with boundary and $f:M\to N$ be a map between
manifolds with boundary.

  1. 1.

If $A\subset M$, the restriction is denoted $f|_{A}:A\to N$ as usual.

  2. 2.

If $f(M)\subset B$, we will write the induced map as $f|^{B}:M\to B$. We call
$f|^{B}$ the _co-restriction of $\bm{f}$ to $B$_ if $B\subset N$ and call it
the _co-extension of $\bm{f}$ to $\bm{B}$_ if $B\supset Y$. If $f(A)\subset
B$, we define $f|_{A}^{B}=(f|_{A})^{B}$ for succinctness.

  3. 3.

If $A$ is an submanifold with boundary in $M$ and $f$ is an injective
immersion and $B$ is the unique submanifold with boundary of $N$ such that
$f|_{A}^{B}$ is a diffeomorphism, we denote $f(A)=B$.

  4. 4.

If $M$ is a manifold and $A$ is a submanifold of $M$, it is said that
_$\bm{A}$ is weakly embedded in $\bm{M}$_ if for any smooth map $F:S\to M$
between manifolds such that $F(S)\subset A$, we have that the co-restriction
$F|^{A}$ is also smooth.

  5. 5.

We will denote by $\text{Pts}(A)$ the set of points of $A$ when it is
necessary for distinction.

The following is Proposition 3.5.15 in Rudolf and Schmidt’s book [7].

######  Proposition V.1.

Let $M$ be a manifold and $D$ an integrable distribution on $M$. Then, all
integral manifolds of $D$ are weakly embedded in $M$.

Another important result found in Rudolf and Schmidt’s book [7] is on the
existence of maximal integral manifolds. An integral manifold $I$ of a
distribution $D$ on a manifold $M$ is _maximal_ if, for any integral manifold
$H$ of $D$ with a point in common with $I$, $H$ is an open submanifold of $I$.
The following is Theorem 3.5.17 in Rudolf and Schmidt’s book [7] adapted to
our context.

######  Theorem V.1.

Let $M$ be a manifold and $D$ an integrable distribution on $M$. Then, through
every point of $M$, there passes a unique maximal integral manifold of $D$.

This gives rise to a convenient containment property of integral curves.

######  Proposition V.2.

Let $M$ be a manifold and $D$ an integrable distribution on $M$. Let $X$ be a
$D$-section, $\gamma:(-\epsilon,\epsilon)\to M$ be an integral curve of $X$
and $I$ be a maximal integral manifold of $D$ with $\gamma\cap
I\neq\emptyset$. Then, $\gamma\subset I$.

###### Proof.

First let $H$ be a maximal integral manifold of $D$. We see that $X$ is
tangent to $H$. Hence, there is a $h$-related vector field $\tilde{X}$ to $X$
where $h:H\subset M$ is the inclusion. Then consider $\gamma^{-1}(H)$. Let
$t\in\gamma^{-1}(H)$. Then considering the integral curves of $\tilde{X}$, by
uniqueness, there is a neighbourhood $(t-\delta,t+\delta)$ of $t$ for some
$\delta>0$ with $(t-\delta,t+\delta)\subset\gamma^{-1}(I)$. Hence,
$\gamma^{-1}(I)$ is open in the interval $(-\epsilon,\epsilon)$.

By assumption $\gamma^{-1}(I)\neq\emptyset$. Hence, by the above,
$\gamma^{-1}(I)$ is open and nonempty in $(-\epsilon,\epsilon)$. Now consider
the compliment
$(-\epsilon,\epsilon)\backslash\gamma^{-1}(I)=\gamma^{-1}(M\backslash I)$. By
Theorem V.1, replacing each $p\in M$ with the unique maximal integral manifold
$L$ of $D$ containing $p$, we obtain a partition $\mathcal{C}$ of $M$ by the
maximal integral manifolds of $D$. Hence, $M\backslash
I=\cup_{H\in\mathcal{C}\backslash\\{I\\}}H$ so that,

$(-\epsilon,\epsilon)\backslash\gamma^{-1}(I)=\gamma^{-1}(M\backslash
I)=\gamma^{-1}(\cup_{H\in\mathcal{C}\backslash\\{I\\}}H)=\cup_{H\in\mathcal{C}\backslash\\{I\\}}\gamma^{-1}(H).$

Hence, using the above again, $\gamma^{-1}(M\backslash I)$ is open in
$(-\epsilon,\epsilon)$. Since $(-\epsilon,\epsilon)$ is connected, it follows
that $\gamma^{-1}(I)=(-\epsilon,\epsilon)$. That is, $\gamma\subset I$. ∎

We now make some observations which focus on normal distributions. These will
enable the use of collars with normal distributions as well as prove the
opening statement of Theorem IV.1. We first state two results found in Lee’s
book for later use.

Let $M$ be a manifold with boundary. Let $v\in TM$ be a vector on the boundary
$\partial M$. If $v$ is transverse to $\partial M$, following Lee [6], $v$ is
said to be:

  1. 1.

_inward-pointing_ if there exists a curve $\gamma:[0,\epsilon)\to M$ for some
$\epsilon>0$ such that $\gamma^{\prime}(0)=v$;

  2. 2.

_outward-pointing_ if there exists a curve $\gamma:(-\epsilon,0]\to M$ for
some $\epsilon>0$ such that $\gamma^{\prime}(0)=v$.

We will make use of the following, which is Proposition 5.41 from Lee’s book
[6] and a useful corollary.

######  Proposition V.3.

For $p\in\partial M$, $T_{p}M$ is partitioned by the set of tangent vectors,
the set of inward-pointing vectors and the set of outward-pointing vectors.

######  Corollary V.1.

A vector $v\in TM$ on the boundary is tangent to $\partial M$ if and only if
there exists a curve $\gamma:(-\epsilon,\epsilon)\to M$ for some $\epsilon>0$
such that $\gamma^{\prime}(0)=v$.

For collar constructions, we will frequently use the Boundary Flowout Theorem
(Theorem 9.24 in Lee’s book [6]).

######  Theorem V.2 (Boundary Flowout Theorem).

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $N$ an
inward-pointing vector field on $M$. Then, there exists a smooth
$\delta:\partial M\to\mathbb{R}^{+}$ and embedding $\Phi:U\to M$ where $U$ is
an open submanifold with boundary in $\partial M\times[0,\infty)$ given by,

$U=\\{(p,t)\in\partial M\times[0,\infty):t<\delta(p)\\},$

such that $\Phi(p,\cdot):[0,\delta(p))\to M$ is an integral curve of $N$
starting at $p$ for $p\in\partial M$.

For the remaining of this section, let $M$ be a manifold with boundary and $D$
be a normal distribution on $M$ with rank $r$.

We first note that boundaries are essential for integrability of normal
distributions by boundaries.

######  Proposition V.4.

If $K$ is an integral manifold with boundary of the normal distribution $D$,
then $\operatorname{int}K\cap\partial M=\emptyset$.

###### Proof.

Assume there is a $p\in\operatorname{int}K\cap\partial M$. Taking a vector
$N\in D_{p}$ transverse to $\partial M$, we get an $i$-related vector $n$ on
$\operatorname{int}K$ ($i:\operatorname{int}K\subset M$) and hence a curve
$\gamma:(-\epsilon,\epsilon)\to\operatorname{int}K$ with
$\gamma^{\prime}(0)=n$. However, then $\gamma|^{M}:(-\epsilon,\epsilon)\to M$
is with $(\gamma|^{M})^{\prime}(0)=N$, contradicting $N$ being transverse to
$\partial M$ from Proposition V.3. ∎

######  Proposition V.5.

There exists an inward-pointing vector field $N$ on $M$ which is also a
$D$-section.

###### Proof.

For each $p\in\partial M$, we may choose an open neighbourhood $U_{p}$ in $M$
containing $p$ and a $D$-section $N_{p}$ such that $N_{p}$ is inward-pointing
on $U_{p}\cap\partial M$. Then, for the cover
$\mathcal{U}=\\{U_{p}:p\in\partial M\\}\cup\\{\operatorname{int}M\\}$ of $M$,
take a partition of unity $\mathcal{P}=\\{\rho_{p}:p\in\partial
M\\}\cup\\{\rho\\}$ on $M$ subordinated to $\mathcal{U}$. Then, the vector
field $N=\sum_{p\in\partial M}\rho_{p}N_{p}$ on $M$ is inward-pointing and is
a $D$-section. ∎

The Boundary Flowout Theorem (Theorem V.2) enables the use of collars with $D$
as follows.

######  Proposition V.6.

There is a collar $\Sigma:\partial M\times[0,1)\to M$ whose diffeomorphism co-
restriction $\sigma=\Sigma|^{V}$ (where $V=\Sigma(\partial M\times[0,1))$)
satisfies the following. There is an inward-pointing $i^{*}D$-section
$\tilde{N}$ ($i:V\subset M$) such that for each $p\in\partial V=\partial M$,
the map $\sigma(p,\cdot):[0,1)\to V$ is an integral curve starting at $p$.

###### Proof.

Given such an $N$ from Proposition V.5 and the resulting embedding $\Phi:U\to
M$ from Theorem V.2, as seen in the proof of the Collar neighbourhood Theorem
(Theorem 9.25 in Lee’s book [6]) we get an embedding $\Sigma:\partial
M\times[0,1)\to M$ given by $\Sigma(p,t)=\Phi(p,\delta(p)t)$. The map
$\sigma=\Sigma|^{V}$ trivially inherits the desired properties from $\Phi$. ∎

We will also require some simple pull-back properties for distributions.

######  Proposition V.7.

Let $M$ be a manifold with boundary and $D$ a distribution on $M$. Let $S$ be
a manifold with boundary of codimension $1$ and $F:S\to M$ an injective
immersion. Let $\omega$ be a $1$-form on $M$ such that $F^{*}\omega=0$ and $X$
be a $D$-section with $\omega(X)=1$. Then, $F^{*}D$ is a distribution on $S$
of rank $r\circ F-1$ where $r$ is the rank of $D$.

###### Proof.

Let $p\in S$ and denote $q=F(p)$. From $\dim{S}=\dim{M}-1$,
$\omega|_{q}(X|_{q})=1$, and injectivity of $TF|_{p}:T_{p}S\to T_{q}M$, we
have that $\mathop{\text{im}}{TF|_{p}}=\ker\omega|_{q}$. Now, consider the
linear map $A_{q}:T_{q}M\to T_{q}M$ given by
$A_{q}(u)=u-\omega|_{q}(u)X|_{q}$. We claim that
$TF|_{p}((F^{*}D)_{p})=A_{q}(D_{q})$.

Indeed, one readily sees that $(F^{*}D)_{p}=(TF|_{p})^{-1}(D_{q})$ and
$\ker\omega|_{q}=\mathop{\text{im}}{A_{q}}$ so that,

$TF|_{p}((F^{*}D)_{p})=TF|_{p}((TF|_{p})^{-1}(D_{q}))=\mathop{\text{im}}{TF|_{p}}\cap
D_{q}=\ker\omega|_{q}\cap D_{q}=\mathop{\text{im}}{A_{q}}\cap D_{q}.$

We now consider $\mathop{\text{im}}{A_{q}}\cap D_{q}$. For any $u,v\in T_{q}M$
such that $v=A_{q}(u)$, we have $v=u-\omega|_{q}(u)X|_{q}$. Hence, since
$X|_{q}\in D_{q}$ and $D_{q}$ is a vector subspace of $T_{q}M$, we have $u\in
D_{q}$ if and only if $v\in D_{q}$. This shows that
$\mathop{\text{im}}{A_{q}}\cap D_{q}=A_{q}(D_{q})$. Hence, in total,

$TF|_{p}((F^{*}D)_{p})=\mathop{\text{im}}{A_{q}}\cap D_{q}=A_{q}(D_{q}),$

as claimed.

To compute the dimension of $(F^{*}D)_{p}$, we consider the linear map
$B_{q}=A_{q}|_{D_{q}}$. Since $\dim{\ker{B_{q}}}=1$, the vector subspace
$TF|_{p}((F^{*}D)_{p})=A_{q}(D_{q})$ has dimension
$\dim\text{im}B_{q}=r(q)-1$. Hence, because $TF|_{p}$ is a linear injection,
$(F^{*}D)_{p}$ is a vector subspace of $T_{p}S$ of dimension $r(q)-1=(r\circ
F-1)(p)$.

Now, let $v\in(F^{*}D)_{p}$. Then, $TF(v)\in D$ and $v\in T_{p}S$ so that
$w=TF|_{p}(v)\in D_{q}$. Hence, there is a $D$-section $W$ with $W|_{q}=w$.
Consider the vector field $A(W)=W-\omega(W)X$. For $p^{\prime}\in S$, we have
that,

$A(W)|_{F(p^{\prime})}=A_{F(p^{\prime})}(W|_{F(p^{\prime})})\in
A_{F(p^{\prime})}(D_{F(p^{\prime})})=TF|_{p^{\prime}}((F^{*}D)_{p^{\prime}}),$

so that $W|_{F(p^{\prime})}=TF|_{p^{\prime}}(u)$ for some
$u\in(F^{*}D)_{p^{\prime}}\subset T_{p^{\prime}}S$. Hence, since $F$ is an
immersion, there exists a unique map $Y:S\to TS$ with $Y|_{p^{\prime}}\in
T_{p^{\prime}}S$ for $p^{\prime}\in S$ such that $A(W)\circ F=TF\circ Y$.
Then, since $F(S)$ is an submanifold with boundary in $M$, using a trivial
extension of Proposition 8.23 to the case of $M$ with boundary from Lee’s book
[6] (on restriction of vector fields to submanifolds with boundary), we obtain
that $Y$ is a vector field on $S$. It is then clear that $Y$ is a
$F^{*}D$-section with $Y|_{p}=v$. ∎

######  Remark V.1.

Let $M$ be a manifold with boundary and $D$ be a subset of $TM$. If $D$ is a
distribution of rank $r$ on $M$, then for any open submanifold $U$ with
boundary, denoting $\iota:U\subset M$, the subset $i*D$ is a distribution of
rank $r|_{U}$ on $U$. Using smooth bump functions, the converse can also be
established. Namely, if every point $p\in M$ has a neighbourhood $U$ for which
$i^{*}D$ is a distribution on $U$ (where $\iota:U\subset M$), then $D$ is a
distribution on $M$.

With these pull-back properties we may in particular prove the opening
statement of Theorem IV.1.

######  Lemma V.1.

The pullback $\jmath^{*}D$ ($\jmath:\partial M\subset M$) is a distribution on
$\partial M$ of rank $r|_{\partial M}-1$.

###### Proof.

Take a collar $\Sigma$ as in Proposition V.6 and adopt the notation from
there. We have that $i^{*}D$ is a distribution of rank $r|_{V}$ whereby
$\tilde{N}$ is a $i^{*}D$-section (see Remark V.1). With the diffeomorphism
$\sigma$, we have a smooth function
$f=(\pi_{2}\circ\sigma^{-1})|^{\mathbb{R}}$ where $\pi_{2}:\partial
M\times[0,1)\to[0,1)$ is the projection onto $[0,1)$.

Then, note that $\partial V=\partial M$ so that $\partial M$ is an embedded
submanifold of $V$ where the inclusion $\jmath^{\prime}:\partial M\subset V$
satisfies $i\circ\jmath^{\prime}=\jmath$. Setting $\omega=df$, $\omega$ is a
1-form on $V$ we have $\jmath^{\prime*}\omega=0$ and $\omega(\tilde{N})=1$.
Hence, by Proposition V.7,
$\jmath^{*}D=(i\circ\jmath^{\prime})^{*}D=\jmath^{\prime*}(i^{*}D)$ is a
distribution on $\partial M$ of rank $r|_{V}|_{\partial M}-1=r|_{\partial
M}-1$. ∎

Lemma V.1, combined with Remark V.1, allows us to record the following; which
is the opening statement of Theorem IV.1.

######  Corollary V.2.

Let $M$ be a manifold with boundary and $D$ be a normal distribution of rank
$r$ on $M$. Then, the pullback $\imath^{*}D$
($\imath:\operatorname{int}M\subset M$) is a distribution of rank
$r|_{\operatorname{int}M}$ on $\operatorname{int}M$ and $\jmath^{*}D$
($\jmath:\partial M\subset M$) is a distribution of rank $r|_{\partial M}-1$
on $\partial M$.

###  V.2 Implication S1 $\Rightarrow$ S2

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $D$ be a
normal distribution of rank $r$ on $M$. Assume that $D$ is neatly integrable.
Let $\imath:\operatorname{int}M\subset M$ and $\jmath:\partial M\subset M$
denote the inclusions.

######  Lemma V.2.

The pullback distribution $\imath^{*}D$ is integrable.

###### Proof.

Let $p\in\operatorname{int}M$. Take a neat integral manifold $L$ of $D$ with
$p\in L$. Then, we have $p\in\operatorname{int}L$. Since $L$ is connected,
$\operatorname{int}L$ is connected also. Since
$\operatorname{int}L\subset\operatorname{int}M$, we have that
$\operatorname{int}L$ is immersed in $\operatorname{int}M$. Clearly,
$Ti|_{q}(T_{q}\operatorname{int}L)=(\imath^{*}D)_{q}$ for
$q\in\operatorname{int}L$ (where
$i:\operatorname{int}L\subset\operatorname{int}M$). ∎

######  Lemma V.3.

Let $L$ be a neat integral manifold with boundary of $D$. Then,
$Tj|_{p}(T_{p}\partial L)=(\jmath^{*}D)_{p}$ for $p\in\partial L$ where
$j:\partial L\subset\partial M$.

###### Proof.

Let $p\in\partial L$. Then $\partial L\neq\emptyset$ and denoting $l:L\subset
M$ and $g:\partial L\subset L$, we get,

$\begin{split}T\jmath|_{p}(Tj|_{p}(T_{p}\partial L))&=T(\jmath\circ
j)|_{p}(T_{p}\partial L)\\\ &=T(l\circ g)|_{p}(T_{p}\partial L)\\\
&=Tl|_{p}(Tg|_{p}(T_{p}\partial L))\\\ &\subset Tl|_{p}(T_{p}L)\\\
&=D_{p}.\end{split}$

Hence, $Tj|_{p}(T_{p}\partial
L)\subset(T\jmath|_{p})^{-1}(D_{p})=(\jmath^{*}D)_{p}$. Since
$\dim{T_{p}L}=\dim{D_{p}}$, we have by Proposition V.1 that
$\dim{Tj|_{p}(T_{p}\partial L)}=\dim{(\jmath^{*}D)_{p}}$. Hence, we must have
that $Tj|_{p}(T_{p}\partial L)=(\jmath^{*}D)_{p}$. ∎

######  Lemma V.4.

The pullback $\jmath^{*}D$ is integrable.

###### Proof.

Let $p\in\partial M$. Again, take a neat integral manifold with boundary $L$
of $D$ with $p\in L$. Then $p\in\partial L$ from Proposition V.4. Let $J$ be
the connected component of $\partial L$ containing $p$ (an open submanifold of
$\partial L$). By Lemma V.3, $J$ is an integral manifold of $\jmath^{*}D$. ∎

######  Lemma V.5.

For every $p\in\partial M$, there exists a neat integral manifold $L$ of $D$
with boundary such that $p\in L$ and $\operatorname{int}L$ is a maximal
integral manifold of $\imath^{*}D$.

###### Proof.

Let $p\in\partial M$. By assumption, there exists a neat integral manifold $K$
of $D$ with boundary such that $p\in K$. Then, $\operatorname{int}K$ is an
integral manifold of $\imath^{*}D$. Then, from Theorem V.1, there exists a
maximal integral manifold $I$ of $\imath^{*}D$ whereby $\operatorname{int}K$
is an open submanifold of $I$. Since also $\operatorname{int}K$ is an open
submanifold of $K$ and $I\cap K=\text{Pts}(\operatorname{int}K)$, we have a
manifold with boundary $L$ with $\text{Pts}(L)=I\cup K$ (so $p\in L$) such
that $I$ is an open submanifold of $L$ and $K$ is an open submanifold with
boundary in $L$. It follows easily that $L$ is an integral manifold with
boundary of $D$. Moreover, since again $\operatorname{int}K$ is an open
submanifold of $I$, we obtain that $\operatorname{int}L=I$ and $\partial
L=\partial K\subset\partial M$ (so $L$ is neat). ∎

For the following, fix a collar $\Sigma:\partial M\times[0,1)\to M$ as in
Proposition V.6 and inherit the surrounding notation from there.

######  Lemma V.6.

Let $p\in\partial M$. Then, there exists a neat integral manifold $L$
containing $p$ and a collar $\rho$ on $L$ such that $l\circ\rho=\Sigma\circ
k$.

###### Proof.

Fix an integral manifold $L$ of $D$ with boundary such that $p\in L$ as in
Lemma V.5. Denote by $k:\partial L\times[0,1)\subset\partial M\times[0,1)$ and
$l:L\subset M$ the inclusions.

Consider the diffeomorphism $\sigma=\Sigma|^{V}:\partial M\times[0,1)\to V$,
the inclusion $h_{1}:V\cap L\subset V$ and the $h_{1}$-related vector field
$\tilde{n}$ to $\tilde{N}$. We see that $\tilde{n}$ on $V\cap L$ is inward-
pointing. Applying the Boundary Flowout Theorem (Theorem V.2) to $\tilde{n}$,
write $U_{1}$ for the obtained neighbourhood of $\partial(V\cap
L)\times\\{0\\}=\partial L\times\\{0\\}$ and the obtained embedding
$\tilde{\rho}_{1}:U_{1}\to V\cap L$. By existence and uniqueness of integral
curves, we have that $h_{1}\circ\tilde{\rho}_{1}=\sigma\circ k|_{U_{1}}$. With
the embedding $\rho_{1}=\tilde{\rho}_{1}|^{L}$, we have
$l\circ\rho_{1}=\Sigma\circ k|_{U_{1}}$.

Now, the open submanifold $V\cap\operatorname{int}L$ of $L$ is such that, the
connected components of $V\cap L$ are maximal integral manifolds of the
distribution $i^{*}D$ on the manifold $V\cap\operatorname{int}M$ where
$i:V\cap\operatorname{int}M\subset M$. Let $\tilde{N}^{\prime}$ be the
$i^{\prime}$-related vector field to $\tilde{N}$ where
$i^{\prime}:V\cap\operatorname{int}M\subset V$. Then, $\tilde{N}^{\prime}$ is
a $i^{*}D$-section. Hence, from Proposition V.2, the integral curves of
$\tilde{N}^{\prime}$ which intersect $V\cap\operatorname{int}L$ are contained
in $V\cap\operatorname{int}L$. Hence, the equation $l\circ\rho_{1}=\Sigma\circ
k|_{U_{1}}$ implies that, with $U_{2}=\partial L\times(0,1)$, there exists a
map $\tilde{\rho}_{2}:U_{2}\to V\cap\operatorname{int}L$ such that
$h_{2}\circ\tilde{\rho}_{2}=\tilde{\Sigma}\circ k_{2}$ where
$\tilde{\Sigma}=\Sigma|_{\partial M\times(0,1)}^{\operatorname{int}M}$ is an
embedding, $k_{2}:\partial L\times(0,1)\subset\partial M\times(0,1)$,
$h_{2}:V\cap\operatorname{int}L\subset\operatorname{int}M$.

Since $V\cap\operatorname{int}L$ is an open submanifold of the integral
manifold $\operatorname{int}L$ of $\imath^{*}D$, we have by Proposition V.1,
that $V\cap\operatorname{int}L$ is weakly embedded in $\operatorname{int}M$.
Hence, $\tilde{\rho}_{2}$ is smooth. Differentiating the equation
$h_{2}\circ\tilde{\rho}_{2}=\tilde{\Sigma}\circ k_{2}$, we obtain by the
Inverse Function Theorem that $\tilde{\rho}_{2}$ is a local diffeomorphism and
hence an embedding (because of injectivity). Hence the map
$\rho_{2}=\tilde{\rho_{2}}|^{L}$ is also an embedding and satisfies
$l\circ\rho_{2}=\Sigma\circ k|_{U_{2}}$.

We also see that $\rho_{1}$ and $\rho_{2}$ agree on $U_{1}\cap U_{2}$ so there
exists a (unique) smooth map $\rho:\partial L\times[0,1)\to L$ with
$\rho_{|U_{1}}=\rho_{1}$ and $\rho_{|U_{2}}=\rho_{2}$. It follows that $\rho$
satisfies $l\circ\rho=\Sigma\circ k$ and that $\rho$ is an embedding. ∎

######  Lemma V.7.

The collar $\Sigma$ is a $D$-collar.

###### Proof.

Let $(p,t)\in\partial M\times[0,1)$. Then, take a neat integral manifold with
boundary $L$ with $p\in L$ as in Lemma V.6 and inherit the notation from
there. Since $\rho$ is an embedding, the relation $l\circ\rho=\Sigma\circ k$
gives us,

$\begin{split}D_{\Sigma(p,t)}&=D_{\rho(p,t)}\\\
&=Tl|_{\rho(p,t)}(T_{\rho(p,t)}L)\\\
&=Tl|_{\rho(p,t)}(T\rho|_{(p,t)}(T_{(p,t)}(\partial L\times[0,1))))\\\
&=T\Sigma|_{(p,t)}(Tk|_{(p,t)}(T_{(p,t)}(\partial L\times[0,1)))).\end{split}$

Now, considering the projections $\pi_{1},\pi_{2}$ of $\partial M\times[0,1)$
(onto the indicated factors) and likewise $\tau_{1},\tau_{2}$ on $\partial
L\times[0,1)$, with $j:\partial L\subset\partial M$, we obtain the following
linear isomorphisms and relations.

$\displaystyle\left(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)}\right):T_{(p,t)}(\partial
M\times[0,1))\to T_{p}\partial M\times T_{t}[0,1)$
$\displaystyle\left(T\tau_{1}|_{(p,t)},T\tau_{2}|_{(p,t)}\right):T_{(p,t)}(\partial
L\times[0,1))\to T_{p}\partial L\times T_{t}[0,1)$ $\displaystyle\pi_{1}\circ
k=j\circ\tau_{1}$ $\displaystyle\pi_{2}\circ k=\tau_{2}.$

With this, applying Lemma V.3 and a similar computation to the above, we get,

$\displaystyle T\pi_{1}|_{(p,t)}(Tk|_{(p,t)}(T_{(p,t)}(\partial
L\times[0,1))))=Tj|_{p}(T_{p}\partial L)=(\jmath^{*}D)_{p}$ $\displaystyle
T\pi_{2}|_{(p,t)}(Tk|_{(p,t)}(T_{(p,t)}(\partial L\times[0,1))))=T_{t}[0,1).$

Using again the fact that $\left(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)}\right)$
is a linear isomorphism, we see that for any subset $A\subset
T_{(p,t)}(\partial M\times[0,1))$,

$A=T\text{Id}(\cdot,t)|_{p}(T\pi_{1}|_{(p,t)}(A))+T\text{Id}(p,\cdot)|_{t}(T\pi_{2}|_{(p,t)}(A)).$

Then, from the above computations, it easily follows that,

$T\Sigma(\cdot,t)|_{p}((\jmath^{*}D)_{p})+T\Sigma(p,\cdot)|_{t}(T_{t}[0,1))=D_{\Sigma(p,t)}.$

∎

We summarise our observations.

######  Corollary V.3.

Let $M$ be a manifold with boundary and $D$ be a normal distribution of rank
$r$ on $M$. Then we have the implication S1 $\Rightarrow$ S3 in Theorem IV.1.

###  V.3 Implication S2 $\Rightarrow$ S1

For this implication, we require analysing the product $A\times[0,1)$ where
$A$ is a manifold. The domain of a collar is what this analysis will be
applied to.

####  V.3.1 Product analysis

Let $A$ be a smooth manifold and $D$ be a distribution of rank $r$ on $A\times
B$, where either $B=[0,1)$ or $B=(0,1)$. Set $M=A\times B$ and $n=\dim M$.
Denote by $\pi_{1}:M\to A$ and $\pi_{2}:M\to B$ the projections.

The following argument uses many elements from the proof of Lemma 19.5 in
Lee’s book [6].

######  Lemma V.8.

Assume $B=[0,1)$. Let $p\in A$ such that $D$ has constant rank along the curve
$\gamma_{p}=\text{\emph{Id}}(p,\cdot):[0,1)\to M$. Then, setting $R=r(p,0)$,
there exists a neighbourhood $U$ of $(p,0)$ in $M$ and local 1-forms
$\omega_{R+1},...,\omega_{n}:U\to T^{*}M$ and an $\epsilon>0$ such that
$\gamma_{p}([0,\epsilon))\subset U$ for which
$D_{\gamma_{p}(t)}=\bigcap_{i=1}^{n}\ker\omega_{R+i}|_{\gamma_{p}(t)}$ for
$t\in[0,\epsilon)$.

###### Proof.

Let $v_{1},...,v_{R}$ be a basis for $D_{(p,0)}$. Then, extend each $v_{i}$ to
a $D$-section $X_{i}$. Then, the subset $V\subset M$ for which
$X_{1},...,X_{R}$ are linearly independent is an open neighbourhood of
$(p,0)$. So, $X_{1}|_{V},...,X_{R}|_{V}$ are linearly independent local vector
fields on $M$. Hence, in a neighbourhood $U$ of $(p,0)$, there exist local
vector fields $Y_{R+1},...,Y_{n}:U\to TM$ such that
$(X_{1}|_{U},...,X_{R}|_{U},Y_{R+1},...,Y_{n})$ forms a local frame for $TM$
on $U$. Then, consider the local co-frame
$(\omega_{1},...\omega_{R},\omega_{R+1},...,\omega_{n})$ of $T^{*}M$ on $U$
dual to $(X_{1}|_{U},...,X_{R}|_{U},Y_{R+1},...,Y_{n})$.

For $q\in U$, we get,

$\bigcap_{i=1}^{n}\ker\omega_{R+i}|_{q}=\text{span}(X_{1}|_{q},...,X_{R}|_{q})\subset
D_{q}.$

Since $U$ is a neighbourhood of $(p,0)$, there is an $\epsilon>0$ with
$\gamma_{p}([0,\epsilon))\subset U$. For $t\in[0,\epsilon)$, we hence get,
$\bigcap_{i=1}^{n}\ker\omega_{R+i}|_{\gamma_{p}(t)}\subset D_{\gamma_{p}(t)}$.
Moreover, $\bigcap_{i=1}^{n}\ker\omega_{R+i}|_{\gamma_{p}(t)}$ and
$D_{\gamma_{p}(t)}$ both have dimension $R$. Hence, we must have that
$D_{\gamma_{p}(t)}=\bigcap_{i=1}^{n}\ker\omega_{R+i}|_{\gamma_{p}(t)}$. ∎

Hence, we get the following uniqueness result.

######  Corollary V.4.

Assume $B=[0,1)$. Assume that $E$ is a distribution on $M$ such that
$D_{(p,t)}=E_{(p,t)}$ for $(p,t)\in\operatorname{int}M$ and for each for $p\in
A$, $D$ and $E$ have constant rank along the curves
$\gamma_{p}=\text{\emph{Id}}(p,\cdot):[0,1)\to M$ for $p\in A$. Then, $D=E$.

For what follows, denote $\iota_{t}=\text{Id}(\cdot,t):A\to M$ for $t\in B$
and recall that,

$(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)}):T_{(p,t)}(A\times B)\to T_{p}A\times
T_{t}B,$

is a linear isomorphism.

######  Lemma V.9.

Assume $B=[0,1)$. Let $D_{1}$ be a distribution on $A$ of rank $r_{1}$. Then,
$E=\pi_{1}^{*}D_{1}$ is a distribution of rank $r_{1}\circ\pi_{1}+1$ on $M$.
If $D_{1}$ is integrable, then $E$ is neatly integrable by boundaries.

###### Proof.

For $(p,t)\in M$, we have $E_{(p,t)}=(T\pi_{1}|_{(p,t)})^{-1}((D_{1})_{p})$ so
that, since $(D_{1})_{p}$ is a vector subspace of $T_{p}A$, $E_{(p,t)}$ is a
vector subspace of $T_{(p,t)}M$. Since $(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)})$
is a linear isormorphism, we have that
$T\pi_{1}|_{(p,t)}(E_{(p,t)})=(D_{1})_{p}$ and
$T\pi_{2}|_{(p,t)}(E_{(p,t)})=T_{t}[0,1)$, so
$(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)})(E_{(p,t)})=(D_{1})_{p}\times
T_{t}[0,1)$ and hence $E_{(p,t)}$ has dimension
$\dim{(D_{1})_{p}}+1=(r_{1}\circ\pi_{1}+1)(p)$.

Now, let $u\in E_{(p,t)}$. Since $u\in T_{(p,t)}M$, setting
$v=T\pi_{1}|_{(p,t)}(u)$, we have, for some $\alpha\in\mathbb{R}$,

$u=T\iota_{t}|_{p}(v)+\alpha\frac{\partial}{\partial
t}|_{(p,t)}=T\pi_{1}|_{(p,t)}(u)+\alpha\frac{\partial}{\partial t}|_{(p,t)},$

since $(T\pi_{1}|_{(p,t)},T\pi_{2}|_{(p,t)})$ is a linear isomorphism,
$T_{t}[0,1)$ is 1-dimensional and
$T\pi_{2}|_{(p,t)}\left(\frac{\partial}{\partial t}|_{(p,t)}\right)=0$.

Moreover, $v=T\pi_{1}|_{(p,t)}(u)\in(D_{1})_{p}$. Hence, there is a
$D_{1}$-section $W$ with $W|_{p}=v$. Then, the map $X:M\to TM$ defined by
$X(p,t)=T\iota_{t}|_{p}(W_{p})$, is a vector field on $M$. Then, consider the
vector field $Y=X+\alpha\frac{\partial}{\partial t}$.

Clearly $Y|_{(p,t)}=u$ and for $(p^{\prime},t^{\prime})\in M$,
$T\pi_{1}|_{(p^{\prime},t^{\prime})}(X|_{(p^{\prime},t^{\prime})})=V_{p^{\prime}}\in(D_{1})_{p^{\prime}}$
and hence
$X|_{(p^{\prime},t^{\prime})}\in(\pi_{1}^{*}D_{1})_{(p^{\prime},t^{\prime})}$.
So, $Y$ is an $E$-section with $Y|_{(p,t)}=u$.

Now assume that $D_{1}$ is integrable. Then, let $(p,t)\in M$. Then, take a
integral manifold $I$ of $D_{1}$ containing $p$. Then, $I\times[0,1)$ is a
connected submanifold of $M$ with boundary whereby
$\partial(I\times[0,1))\subset\partial M$.

Now, let $(p^{\prime},t^{\prime})\in I\times[0,1)$. Denoting by
$\tau_{1}:I\times[0,1)\to I$ the projection onto the first factor, we have
$\iota\circ\tau_{1}=\pi_{1}\circ\iota^{\prime}$ where $\iota:I\subset A$ and
$\iota^{\prime}:I\times[0,1)\subset M$ denote inclusions. Combined with the
fact that $I$ is an integral manifold of $D_{1}$, we get,

$T\pi_{1}|_{(p^{\prime},t^{\prime})}(T\iota^{\prime}|_{p^{\prime},t^{\prime})}(T_{(p^{\prime},t^{\prime})}(I\times[0,1))))=T\iota|_{p^{\prime}}(T_{p^{\prime}}I)=(D_{1})_{p^{\prime}}.$

Thus,
$T\iota^{\prime}(T_{(p^{\prime},t^{\prime})}(I\times[0,1)))\subset(T\pi_{1}|_{(p,t)})^{-1}((D_{1})_{p^{\prime}})=E_{(p^{\prime},t^{\prime})}$.
Moreover, since $(T\iota^{\prime}|_{p^{\prime},t^{\prime})}$ is an immersion,

$\dim{T\iota^{\prime}|_{p^{\prime},t^{\prime})}(T_{(p^{\prime},t^{\prime})}(I\times[0,1))}=\dim{T_{(p^{\prime},t^{\prime})}(I\times[0,1)}=\dim{I}+1=\dim{(D_{1})_{p^{\prime}}}+1=\dim
E_{(p^{\prime},t^{\prime})}.$

Hence, we must have
$T\iota^{\prime}(T_{(p^{\prime},t^{\prime})}(I\times[0,1)))=(T\pi_{1}|_{(p,t)})^{-1}((D_{1})_{p^{\prime}})=E_{(p^{\prime},t^{\prime})}$.
Hence, $I\times[0,1)$ is a neat integral manifold with boundary containing
$(p,t)$. ∎

######  Lemma V.10.

Assume $B=(0,1)$. Assume $D$ is integrable and that $\frac{\partial}{\partial
t}$ is a $D$-section. Then, there exists an integrable distribution $D_{1}$ on
$A$ such that $D=\pi_{1}^{*}D_{1}$.

###### Proof.

Consider, for example, the embedding $\iota_{1/2}=\text{Id}(\cdot,1/2):A\to M$
and the pull-back $D_{1}=\iota_{1/2}^{*}D$. Indeed, with
$f=\pi_{2}|^{\mathbb{R}}$, we get the 1-form $\omega=df$ on $M$. Moreover, we
have that $\iota_{1/2}^{*}\omega=0$ and $\omega\left(\frac{\partial}{\partial
t}\right)=1$. Hence, by Proposition V.7, we have that $D_{1}$ is a
distribution on $A$ of rank $r\circ\iota_{1/2}-1$.

Now, let $I$ be a maximal integral manifold of $D$. Then, let $p\in\pi_{1}(I)$
so that $(p,t)\in I$ for some $t\in(0,1)$. Then,
$\gamma_{p}=\text{Id}(p,\cdot)$ is an integral curve of
$\frac{\partial}{\partial t}$ whereby $\gamma_{p}\cap I\neq\emptyset$. Hence,
by Proposition V.2, $\gamma_{p}\subset I$. Hence, $\\{p\\}\times(0,1)\subset
I$. Hence, $\text{Pts}(I)=\pi_{1}(I)\times(0,1)$. Since $\pi_{2}|_{I}$ is a
submersion, by the Preimage Theorem, there exists a smooth structure on
$\pi_{1}(I)$, which we denote by $I_{1}$, such that $I_{1}$ is a connected
submanifold of $A$ whereby $I=I_{1}\times(0,1)$. From this, it follows that
$D_{1}$ is integrable. ∎

######  Lemma V.11.

Assume $B=[0,1)$. Assume that $\imath^{*}D$ is integrable (where
$\imath:\operatorname{int}M\subset M$ is the inclusion),
$\frac{\partial}{\partial t}$ is a $D$-section and that $D$ is of constant
rank along the curves $\gamma_{p}=\text{Id}(p,\cdot):[0,1)\to N\times[0,1)$
for $p\in N$. Then, $D$ is neatly integrable by boundaries.

###### Proof.

First, since $\imath^{*}D$ is integrable and $\frac{\partial}{\partial t}$ is
a $D$-section, it follows from Lemma V.10 that there exists an integrable
distribution $D_{1}$ on $N$ such that
$\imath^{*}D=\tilde{\pi}_{1}^{*}D_{1}=(\pi_{1}\circ\imath)^{*}D_{1}$, where
$\tilde{\pi}_{1}:\operatorname{int}M\to A\times(0,1)$ is projection onto the
first factor.

Then, consider $E=\pi_{1}^{*}(D_{1})$. We have from Lemma V.9 that $E$ is a
neatly integrable distribution of rank $r_{1}\circ\pi_{1}+1$ (where $r_{1}$ is
the rank of $D_{1}$). Hence, $E$ is of constant rank along the curves
$\gamma_{p}=\text{Id}(p,\cdot):[0,1)\to M$. Moreover, we see that
$\imath^{*}E=\imath^{*}(\pi_{1}^{*}D_{1})=(\pi_{1}\circ\imath)^{*}D_{1}=\imath^{*}D$
and hence $D_{(p,t)}=E_{(p,t)}$ for all $(p,t)\in\operatorname{int}M$.

Hence, by Corollary V.4, we must have that $D=E$ and hence, $D$ is neatly
integrable by boundaries. ∎

With this, we derive implication $S2\Rightarrow S1$ in Theorem IV.1.

####  V.3.2 Application of the product analysis

######  Lemma V.12.

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $D$ be a
normal distribution of rank $r$ on $M$. Let $\imath:\operatorname{int}M\subset
M$ and $\jmath:\partial M\subset M$ denote the inclusions. Then we have the
implication $S2\Rightarrow S1$ in Theorem IV.1.

###### Proof.

Assume that $\imath^{*}D$ is integrable and $D$ is precollared.

Let $\Sigma:\partial M\times[0,1)\to M$ be a $D$-precollar. Denote by
$\sigma=\Sigma|^{V}$ the co-restriction diffeomorphism onto $V=\Sigma(\partial
M\times[0,1))$. The inclusions we require are denoted by $\iota:V\subset M$,
$\tilde{\imath}:\operatorname{int}V\subset V$, $\tilde{\jmath}:\partial
V\subset V$ and $\imath^{\prime}:\partial M\times(0,1)\subset\partial
M\times[0,1)$. Lastly, we set $\tilde{D}=\tilde{\imath}^{*}D$.

First, notice that since $\tilde{D}=\tilde{\imath}^{*}D$ is a distribution, it
is clear that then $D^{\prime}=\sigma^{*}\tilde{D}$ is a distribution on
$\partial M\times[0,1)$ since $\sigma$ is a diffeomorphism. Moreover, since
$\imath^{*}D$ is integable, it follows easily that
$\tilde{\imath}^{*}\tilde{D}$ is integrable. Since again $\sigma$ is a
diffeomorphism, it is clear then that $\iota^{\prime*}D^{\prime}$ is
integrable.

Now, since $\Sigma$ is a $D$-precollar, the vector field
$\tilde{N}=T\sigma\circ\frac{\partial}{\partial t}\circ\sigma^{-1}$ is a
$\tilde{D}$-section and $\tilde{D}$ has constant rank along the curves
$\sigma(p,\cdot)$ for $p\in\partial V=\partial M$. Hence,
$\frac{\partial}{\partial t}$ is a $D^{\prime}$-section and $D^{\prime}$ has
constant rank along the curves $\gamma_{p}=\text{Id}(p,\cdot)$ for
$p\in\partial M$.

Hence, altogether, by Lemma V.11, $D^{\prime}$ is neatly integrable by
boundaries. Hence, since $\sigma$ is a diffeomorphism, it easily follows that
$\tilde{D}$ is neatly integrable by boundaries. From this, since again
$\imath^{*}D$ is integrable by assumption, we obtain altogether that $D$ is
neatly integrable by boundaries. ∎

###  V.4 Implications S3 $\Rightarrow$ S2, S3 $\Rightarrow$ S4 and S4
$\Rightarrow$ S1

Let $M$ be a manifold with boundary $\partial M\neq\emptyset$ and $D$ be a
normal distribution of rank $r$ on $M$. Let $\imath:\operatorname{int}M\subset
M$ and $\jmath:\partial M\subset M$ denote the inclusions.

######  Remark V.2.

Note that collared immediately implies pre-collared. That is, the desired
implication $S3\Rightarrow S4$ is trivial.

######  Lemma V.13.

Assume that $D$ is collared and $\imath^{*}D$ and $\jmath^{*}D$ are
integrable. For all $p\in M$, then, there exists a chart $(U,\varphi)$ neatly
adapted to $D$ at $p$.

###### Proof.

Let $\Sigma:\partial M\times[0,1)\to M$ be a $D$-collar. Suppose $p\in\partial
M$. Since $\jmath^{*}D$ is integrable, by Theorem III.1, there exists a chart
$(U^{\prime},\tilde{\varphi})$ on $\partial M$ adapted to $\jmath^{*}D$ at
$p$. We may choose $(U^{\prime},\tilde{\varphi})$ so that
$\tilde{\varphi}(U^{\prime})=(-\epsilon,\epsilon)^{n-1}$ for some
$0<\epsilon\leq 1$. Then, $(\varphi^{\prime 1},...,\varphi^{\prime
n-1})=(\tilde{\varphi}^{r(p)},...,\tilde{\varphi}^{n},\tilde{\varphi}^{1},...,\tilde{\varphi}^{r(p)-1})$
defines a new chart $(U^{\prime},\varphi^{\prime})$. Then, with the open set
$V^{\prime}=U^{\prime}\times[0,\epsilon)$, setting $U=\Sigma(V^{\prime})$
gives the chart $(U,\varphi)$ where
$\varphi=(\varphi^{\prime}\times\text{id})\circ\sigma^{\prime-1}$ where we
have set $\sigma^{\prime}=\Sigma|_{V^{\prime}}^{U}$ and
$\text{id}:[0,\epsilon)\to[0,\epsilon)$. Due to the fact that $\Sigma$ is a
$D$-collar, it is easy to see that $(U,\varphi)$ is neatly adapted to $D$.

Suppose instead that $p\in\operatorname{int}M$. Then, from Theorem III.1,
since $\imath^{*}D$ is integrable, there exists a chart $(U,\tilde{\varphi})$
of $\operatorname{int}M$ (and hence $M$) adapted to $\imath^{*}D$. Then,
$(\varphi^{1},...,\varphi^{n})=(\varphi^{\prime r(p)+1},...,\varphi^{\prime
n},\varphi^{\prime 1},...,\varphi^{\prime r(p)})$ defines a new chart
$(U,\varphi)$. It is easy to see that $(U,\varphi)$ is neatly adapted to $D$.
∎

The final implication is essentially the same as the proof presented in
Ruldolf and Schmidt’s book [7] for the regular Stefan-Sussmann Theorem.

######  Lemma V.14.

Assume that at every point $p\in M$, there exists a chart $(U,\varphi)$ neatly
adapted to $D$ at $p$. Then $D$ is neatly integrable by boundaries.

###### Proof.

Set $n=\dim M$. Let $p\in M$. Then, take a chart $(U,\varphi)$ neatly adapted
to $D$ at $p$. The case $r(p)=n$ is trivial. Now suppose $r(p)<n$.

We have for some $\epsilon>0$ that either
$\varphi(U)=(-\epsilon,\epsilon)^{n}$ or
$\varphi(U)=(-\epsilon,\epsilon)^{n}\cap\mathbb{H}^{n}$. Write $S$ for the
structure on $\varphi^{-1}(\\{0\\}\times(-\epsilon,\epsilon)^{r(p)})$ as an
embedded submanifold with boundary in $\mathbb{R}^{n}$ where $\partial
S=S\cap\mathbb{H}^{n}$. Then, $L=(\varphi^{-1})(S)$ is an embedded submanifold
with boundary in $M$ with $\partial L\subset\partial M$. Now, since $D$ is of
constant rank on $L$ and $\partial_{n-r(p)+1},...,\partial_{n}$ are local
$D$-sections, it follows easily that $L$ is an integral manifold with boundary
of $D$. Altogether, $L$ is a neat integral manifold with boundary of $D$ where
$p\in L$. ∎

We summarise the observations of this section.

######  Corollary V.5.

Let $M$ be a manifold with boundary and $D$ be a normal distribution of rank
$r$ on $M$. Then we have the implications $S3\Rightarrow S2$, $S3\Rightarrow
S4$ and $S4\Rightarrow S1$ in Theorem IV.1.

###  V.5 Mutual equivalence of S1, S2, S3 and S4

We need only to verify that we have collected sufficient implications for
mutual equivalence.

###### Proof of Theorem IV.1.

As mentioned, the required opening statement is Corollary V.2. In the
following diagram, we label indicated implication by the result which
established it.

S1S2S3S4V.3V.5V.12V.5V.5

From this it is clear that S1, S2, S3 and S4 are all equivalent. ∎

##  VI Acknowledgements

The first author would like to acknowledge his supervisors and co-authors
David Pfefferlé and Luchezar Stoyanov for the valuable and frequent
discussions which occurred while this paper was being written. This paper was
written while the first author received an Australian Government Research
Training Program Scholarship at The University of Western Australia.

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