# A Whitney Extension Problem for Manifolds

Kevin O’Neill

###### Abstract

The purpose of this paper is to address a manifold-based version of Whitney’s
extension problem: Given a compact set $E\subset\mathbb{R}^{n}$, how can we
tell if there exists a $d$-dimensional, $C^{m}$-smooth manifold
$\mathcal{M}\supset E$? We provide an answer for compact manifolds with
boundary in terms of a Glaeser refinement much like that used in the solution
of the classical Whitney extension problem and a topological condition. This
condition is the existence of a continuous selection for Grassmannian-valued
functions, meant to reflect the collection of possible tangent spaces. We
demonstrate the necessity of this condition in general and its non-redundancy
in an example, while also showing it need not be checked when $d=1$.

##  1 Introduction

Let $C^{m}(\mathbb{R}^{n})$ denote the set of $m$-times continuously
differentiable functions from $\mathbb{R}^{n}$ to $\mathbb{R}$.

The Whitney extension problem —first posed by Whitney in 1934 [22] —asks the
following:

######  Problem 1.1 (Whitney Extension Problem).

Given integers $m,n\geq 1$, a compact subset $E\subset\mathbb{R}^{n}$, and a
function $f:E\to\mathbb{R}$, how can we tell if there exists $F\in
C^{m}(\mathbb{R}^{n})$ such that $F(x)=f(x)$ for all $x\in E$?

If such an $F$ exists, we call it an extension of $f$.

The case of $n=1$ was solved by Whitney [22], before Glaeser solved the $m=1$
case [15]. Fefferman [7] (see also [6],[5]) solved the general case following
notable progress by Brudnyi-Shvartsman (see [3], for instance). We provide a
summary of Fefferman’s solution here before providing specifics in Section 2.

Let $\mathcal{P}^{m}(\mathbb{R}^{n})$ denote the vector space of real-valued
polynomials in $n$ variables and of degree at most $m$. For each $x\in E$,
define

$H(x)=\\{P\in\mathcal{P}^{m}(\mathbb{R}^{n}):P(x)=f(x)\\}$ (1)

If $f$ has an extension $F$, then clearly $J_{x}F\in H(x)$ for all $x\in E$,
where $J_{x}F$ is the $m$-th degree Taylor polynomial of $F$ at $x$. (We omit
the dependence on $m$ in our notation since $m$ will be fixed.)

For each $x\in E$, determine a new set of polynomials $\tilde{H}(x)$
consisting of the elements of $H(x)$ which are compatible with elements of
$H(y)$ according to Taylor’s theorem for all $y$ near $x$. The collection
$(\tilde{H}(x))_{x\in E}$ is called the Glaeser refinement of $(H(x))_{x\in
E}$. A key property of Glaeser refinement is that it is computable using only
linear algebra and limits. (See Section 2, also [9].) The Glaeser refinement
is repeated a finite number of times, terminating in a set $H_{*}(x)$ for each
$x\in E$. The main result of [7] is that $f$ has an extension if and only if
$H_{*}(x)$ is nonempty for all $x\in E$. We will make use of a vector-valued
version of this theorem established in[13].

The purpose of this paper is to address the following variant of the Whitney
extension problem, on the topic of manifolds:

######  Problem 1.2.

Let $d,m,n\geq 1$ be integers satisfying $d<n$. Given a compact subset
$E\subset\mathbb{R}^{n}$, how can we tell if there exists a $d$-dimensional,
$C^{m}$-smooth manifold $\mathcal{M}$ such that $\mathcal{M}\supset E$?

Problem 1.2 appears to have first been formally stated in the collection of
Open Problems connected with the Second Workshop on Whitney Problems in 2009
[1], although it was left open to determine an exact formulation of the
problem. A similar version was also stated in [9].

Related questions have been answered in prior work, largely in discrete
settings. In [14], an algorithm is developed to find a manifold which nearly
optimally interpolates a finite set of points. [12] also constructs an
interpolating manifold under additional assumptions with an improvement in
runtime. For work on an intrinsic version of the problem in which one attempts
to construct a Riemannian manifold matching a collection of distances, see
[10] and [11].

For positive integers $d_{1},d_{2}\geq 1$, let
$\mathcal{P}^{m}(\mathbb{R}^{d_{1}},\mathbb{R}^{d_{2}})$ denote the vector
space of $\mathbb{R}^{d_{2}}$-valued polynomials in $d_{1}$ variables and of
degree at most $m$ and $C^{m}(\mathbb{R}^{d_{1}},\mathbb{R}^{d_{2}})$ denote
the set of $m$-times continuously differentiable functions from
$\mathbb{R}^{d_{1}}$ to $\mathbb{R}^{d_{2}}$. To determine the existence of a
$d$-dimensional, $C^{m}$-smooth manifold containing a set $E$, we assign to
each $x\in E$ a subset $H(x)\subset
O(n)\times\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$. As a
$d$-dimensional manifold in $\mathbb{R}^{n}$ is locally the graph of a
$C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ function in some coordinate system,
we use the orthogonal group, $O(n)$ to determine an orthonormal coordinate
system for $\mathbb{R}^{n}$. The element $(Q,P)\in
O(n)\times\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ is meant to
represent a possible Taylor polynomial $P$ of such a function in the
coordinate system determined by $Q$. An advantage of this approach is that it
does not require one to compute any change of coordinates formulas for jets,
simply acknowledge that they exist (see Lemma 4.2).

The sets $H(x)$ are constructed as follows: For each $Q\in O(n)$, determine a
coordinate system for $\mathbb{R}^{n}$. Then view $x$ as plot of a point
$(y,f(y))$ in that coordinate system. Following (1), include in $H(x)$ all
pairs $(Q,P)$ where $P\in\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
satisfies $P(y)=f(y)$. Do this for all $Q\in O(n)$.

We define a notion of Glaeser refinement for this collection $(H(x))_{x\in E}$
and show in Lemma 3.2 that this refinement terminates in a finite number of
steps, leaving us with a set $H^{*}(x)$ for each $x\in E$. Our main result,
Theorem 3.4, states that there is a $d$-dimensional, $C^{m}$-smooth, compact
manifold with boundary $\mathcal{M}\supset E$ if and only if $H^{*}(x)$ is
nonempty for all $x\in E$ and a certain topological condition is satisfied. We
will refer to such an $\mathcal{M}$ as a manifold extension of $E$, specifying
the particular qualities of $\mathcal{M}$ (existence of boundary, etc.) when
required.

We now take a moment to motivate this topological condition. Consider the
subset of $\mathbb{R}^{4}$

$E_{0}=\\{(x_{1},x_{2},0,0):x_{1}^{2}+x_{2}^{2}\leq 1\\}\cup\\{(\cos t,\sin
t,s\cos t,s\sin t):0\leq t\leq 2\pi,0\leq s\leq 1\\}.$

We ask whether there exists a 3-dimensional, $C^{1}$ manifold containing
$E_{0}$. To begin, let us determine, for each point $p\in E_{0}$, whether
there exists $\delta>0$ and a 3-dimensional, $C^{1}$ manifold
$\mathcal{M}_{p}$ such that

$\mathcal{M}_{p}\cap B(p,\delta)\supset E_{0}\cap B(p,\delta),$

where $B(x,r)$ refers to the open ball of radius $r>0$ centered at
$x\in\mathbb{R}^{n}$. We will also take note of some constraints on
$T_{p}\mathcal{M}_{p}$, where $T_{x}\mathcal{M}$ refers to the tangent space
of the manifold $\mathcal{M}$ at the point $x\in\mathcal{M}$.

If $p$ is of the form $(\cos t_{0},\sin t_{0},s_{0}\cos t_{0},s_{0}\sin
t_{0})$ for fixed $0\leq t_{0}\leq 2\pi$ and $0<s_{0}\leq 1$, then such an
$\mathcal{M}_{p}$ exists, with

$\mathcal{M}_{p}=\\{(r\cos t,r\sin t,s\cos t,s\sin t):t_{0}-\delta\leq t\leq
t_{0}+\delta,0<s_{0}-\delta\leq s\leq s_{0}+\delta,1-\delta<r<1+\delta\\}$

serving as one possible example.

For any $p$ of the form $(\cos t_{0},\sin t_{0},0,0)$ we have the manifold

$\mathcal{M}_{p}=\\{(r\cos t,r\sin t,s\cos t,s\sin t):t_{0}-\delta\leq t\leq
t_{0}+\delta,-\delta<s<\delta,1-\delta<r<1+\delta\\}.$

The tangent space $T_{p}\mathcal{M}_{p}$ is uniquely determined since

$p+c(1,0,0,0)\in E_{0}$

for arbitrarily small values of $c$ and $E\cap B(p,\delta)$ contains a two-
dimensional subset of $\\{(x,y,0,0):x,y\in\mathbb{R}\\}$ for all $\delta>0$.

Thus,

$T_{p}\mathcal{M}_{p}=p+Span\\{(1,0,0,0),(0,1,0,0),(0,0,\cos t_{0},\sin
t_{0})\\}$

for any choice of $\mathcal{M}_{p}$.

The remaining points of $E_{0}$ are of the form $p=(x_{1},x_{2},0,0)$ for
$x_{1}^{2}+x_{2}^{2}<1$. For these points, we may choose

$p+Span\\{(1,0,0,0),(0,1,0,0),(0,0,\cos\theta,\sin\theta)\\}$ (2)

to be both $\mathcal{M}_{p}$ and $T_{p}\mathcal{M}_{p}$, where
$\theta\in[0,2\pi]$ is allowed to be arbitrary. Furthermore, whatever the
choice of $\mathcal{M}_{p}$, $T_{p}\mathcal{M}_{p}$ must be of the form (2)
since

$p+c(1,0,0,0),p+c(0,1,0,0)\in E_{0}$

for arbitrarily small values of $c$ and a vector space is uniquely determined
by an orthonormal basis.

We have shown two things thus far, that for each $p\in E_{0}$, the
intersection of $E_{0}$ with a neighborhood of $p$ lies in a 3-dimensional,
$C^{1}$ manifold, and that in some cases the tangent space of such a manifold
must satisfy certain constraints.

If there exists a single 3-dimensional, $C^{1}$ manifold $\mathcal{M}\supset
E_{0}$, then its tangent space must vary continuously. In this example, this
means, for each $p=(x_{1},x_{2},0,0)$ satisfying $x_{1}^{2}+x_{2}^{2}\leq 1$,
we must choose $\theta(p)\in[0,2\pi]$ such that

$(\cos\theta(p),\sin\theta(p))=(x_{1},x_{2})\text{ for }x_{1}^{2}+x_{2}^{2}=1$

and

$\theta:\\{(x_{1},x_{2}:x_{1}^{2}+x_{2}^{2}\leq 1\\}\to S^{1}\text{ is
continuous}.$

However, this implies no such $\mathcal{M}$ exists as there is no continuous
extension of the identity on $S^{1}\subset\mathbb{R}^{2}$ to a map from the
unit disc to $S^{1}$.

Let $Gr(n,d)$ denote the Grassmannian of $d$-dimensional planes through the
origin in $\mathbb{R}^{n}$. This example may be extended to the case of
fitting $d$-dimensional manifolds in $\mathbb{R}^{n}$ whenever $3\leq d<n$ by
forcing the tangent spaces at $(\cos t,\sin t,0,\ldots,0)$ to be
$\mathbb{R}^{2}+V(t)$, where $V(t)$ is a nontrivial loop in $Gr(n-2,d-2)$. A
similar argument applies as such Grassmannians are not simply connected.

The above example is in contrast to the usual notion of extension (or
interpolation) of smooth functions in Euclidean space, where local extensions
(or interpolants) may be pasted together via a partition of unity to form a
global extension (or interpolant) without worry of topological obstacle.

We are now ready to somewhat formally state the topological condition in
Theorem 3.4. At each $x\in E$, let $\hat{H}(x)\subset Gr(n,d)$ such that
$p+\hat{H}(x)$ is the set of possible tangent planes determined by the
collections of coordinate frames and Taylor polynomials found in $H^{*}(x)$.
The topological condition is that there exists a continuous $f:E\to Gr(n,d)$
such that $f(x)\in\hat{H}(x)$ for all $x\in E$.

The manifold with boundary guaranteed by our main result need not be
connected. In the construction, we first paste together of the graphs of
$C^{m}$ functions on open sets, then obtain a compact manifold with boundary
by intersecting the result with a smooth hypersurface. To determine the
existence of a connected manifold will likely require stronger analysis of
global, topological issues.

A theme in the solution to the Whitney extension problem is that the answer
(yes or no extension) is computable with only linear algebra and the ability
to take a limit. The present form of Glaeser refinement and the computation of
$\hat{H}(x)$ have this same property, though we do not provide a complete
solution regarding the topological problem.

In Section 2, we provide background on some tools necessary for our analysis
as well as the classical Whitney extension problem for
$C^{m}(\mathbb{R}^{n},\mathbb{R}^{d})$, including a formal definition of the
aforementioned Glaeser refinement and the solution given in Theorem 2.5. The
interested reader is encouraged to consult [9] and [8] for more on the topic.

Section 3 begins with a formal discussion of our version of Glaeser refinement
for Problem 1.2, including a proof that it terminates after a controlled
number of iterations. It ends with a statement of our main theorem.

In Section 4, we prove a few technical results that will be useful later on.
As a corollary, we deduce that in order to determine the existence of a
manifold extension, we need only consider a finite set of $Q\in O(n)$,
reducing our computations to, essentially, a finite number of the usual
Glaeser refinements (see Theorem 4.6).

The proof of Theorem 3.4 begins in Section 5, where we use elements of
$H_{*}(x)$ to construct local interpolants (or local manifolds) by direction
application of Theorem 2.5. These interpolants are carefully chosen to be
compatible using our topological condition before being pasted together into a
manifold in Section 6. To smooth out the boundary, we intersect the manifold
with a generic $C^{\infty}$ surface. The pasting process is adapted from a
similar method in [14]. This will complete the proof of Theorem 3.4.

Next, we demonstrate in Section 7 that for $d=1$, one need not check the
topological condition to determine the existence of a manifold extension. A
key tool here is a selection theorem of Michael [19] which provides sufficient
conditions for one to be able to continuously choose a function value from a
collection of subsets when the codomain provides topological obstacles. While
the $d=1$ case may not require this machinery, the approach will shed light on
why the topological condition need be checked in higher dimensions.

As one last remark, we note that in the solution to the classical Whitney
extension problem, a finiteness principle solving a discrete version of the
problem was first proven, then used to address the continuous case. Here, we
jump ahead to the continuous version of the problem since we are able to use
the solution to the classical Whitney extension problem as a black box. It may
be interesting to see what discrete versions of our results may hold.

##  2 Background Material

###  2.1 Basic Notation

We use $C$ and $c$ to denote arbitrary large and small constants,
respectively. A constant $C=C(S)$ depends only on the parameters $S$.
$C_{1},c_{1},\ldots$ will denote recurring constants.

Let $N,D\in\mathbb{N}$. If $F:\mathbb{R}^{N}\to\mathbb{R}^{D}$ is
differentiable, we write its gradient, $\nabla
F:\mathbb{R}^{N}\to\mathbb{R}^{D\times N}$ as the matrix-valued function
satisfying

$F(y)=F(x)+\nabla F(x)(y-x)+o(|y-x|),$

where $(x-y)$ is viewed as a column vector in $\mathbb{R}^{N}$ in the above
matrix-vector multiplication. $\nabla^{k}F$ is the order $k+1$ tensor
consisting of the $k$-th order derivatives of $F$. If $F$ is $C^{m}$ for all
$m\geq 0$, then we say $F$ is $C^{\infty}$.

For a particular $k$-th order derivative, we use the notation
$\partial^{\alpha}F$, where $\alpha=(\alpha_{1},\ldots,\alpha_{N})$ is a
multi-index such that $\alpha_{1}+\ldots+\alpha_{N}=k$. Note that
$\partial^{\alpha}F$ is an $\mathbb{R}^{D}$-valued function.

On $Gr(n,d)$, introduce the distance function

$\text{dist}(W,W^{\prime})=\|\Pi_{W}-\Pi_{W^{\prime}}\|\text{ for
}W,W^{\prime}\in Gr(n,d),$ (3)

where $\Pi_{W}$ refers to the orthogonal projection of $\mathbb{R}^{n}$ onto
$W$ and $\|\cdot\|$ is any norm on the space of linear operators on
$\mathbb{R}^{n}$.

######  Lemma 2.1.

Let $d,n\in\mathbb{N}$. There exist a constant $C=C(d,n)$ and
$\epsilon_{0}=\epsilon_{0}(d,n)>0$ such that whenever $W,W^{\prime}\in
Gr(n,d)$ satisfy $\text{dist}(W,W^{\prime})\leq\epsilon\leq\epsilon_{0}$,
there exists $Q\in O(n)$ such that $QW^{\prime}=W$ and $\|Q-I_{n}\|\leq
C\epsilon$.

###### Proof.

By a rotation in $\mathbb{R}^{n}$, we may assume without loss of generality
that $W^{\prime}=Span\\{e_{1},\ldots,e_{k}\\}$, where $e_{1},\ldots,e_{n}$ is
the standard basis for $\mathbb{R}^{n}$. Define

$v_{i}=\Pi_{W}(e_{i}),1\leq i\leq d.$

By hypothesis,

$\|v_{i}-e_{i}\|=\|\Pi_{W}(e_{i})-\Pi_{W^{\prime}}(e_{i})\|\leq\text{dist}(W,W^{\prime})\|e_{i}\|\leq\epsilon$
(4)

for $1\leq i\leq d$. We run the Gram-Schmidt process on
$\\{v_{1},\ldots,v_{d}\\}$ to create an orthonormal basis
$\\{w_{1},\ldots,w_{d}\\}$ for $W$.

First, take $u_{1}=v_{1}$. By (4), $\|v_{1}-e_{1}\|\leq\epsilon$. Now suppose
that for $1\leq i\leq k$, there exists $C$ such that

$\|u_{i}-e_{i}\|\leq C\epsilon.$ (5)

Note that (5) implies

$|\|u_{i}\|-1|\leq C\epsilon.$ (6)

The Gram-Schmidt process gives

$u_{k+1}=v_{k+1}-\sum_{i=1}^{k}\frac{\langle v_{k+1},u_{i}\rangle}{\langle
u_{i},u_{i}\rangle}u_{i},$

and ones sees by (4) that

$\displaystyle\|u_{k+1}-e_{k+1}\|$
$\displaystyle\leq\|u_{k+1}-v_{k+1}\|+\|v_{k+1}-e_{k+1}\|$ $\displaystyle\leq
kC\epsilon+C\epsilon\leq C\epsilon.$

Define $w_{i}=\frac{u_{i}}{\|u_{i}\|}$ to create an orthonormal basis and
observe $\|w_{i}-e_{i}\|\leq C\epsilon$ by (5) and (6).

Repeating this process for the orthogonal complements of $W$ and $W^{\prime}$,
we find an orthonormal basis $\\{v_{d+1},\ldots,v_{n}\\}$ for $W^{\perp}$ such
that

$\|v_{i}-e_{i}\|<C\epsilon\text{ for }d+1\leq i\leq n.$ (7)

Form an orthogonal matrix $Q$ with columns $v_{1},\ldots,v_{n}$. Then
$QW^{\prime}=W$ and $\|Q-I_{n}\|\leq C\epsilon$ by (7). ∎

Let $Q\in O(n)$ such that the columns of $Q$ are $v_{1},\ldots,v_{n}$. (Here
and in the sequel, we identify an orthogonal transformation with its matrix
representation with respect to the standard ordered basis of
$\mathbb{R}^{n}$.) Define $\Pi:O(n)\to Gr(n,d)$ by

$\Pi(Q)=Span\\{v_{1},\ldots,v_{d}\\}.$ (8)

The dimension $d$ will be fixed, thus given by context.

The affine Grassmannian $Graff(n,d)$ consists of all $d$-planes in
$\mathbb{R}^{n}$ (containing the origin or not). It has a natural topology
under which i) the map $\mathcal{M}\to Graff(n,d)$ by $x\mapsto
T_{x}\mathcal{M}$ is continuous for any $d$-dimensional $C^{1}$ manifold
$\mathcal{M}\subset\mathbb{R}^{n}$ and ii) the natural projection
$p:Graff(n,d)\to Gr(n,d)$ by

$p(x+W)=W$ (9)

for $x\in\mathbb{R}^{n}$, $W\in Gr(n,d)$ is continuous.

###  2.2 Manifolds and Transversality

Given $x\in\mathbb{R}^{n}$, $\delta>0$, we let $B(x,\delta)$ denote the open
ball of radius $\delta$ with center $x$. If $A\subset\mathbb{R}^{n}$ is
closed, define $\text{dist}(x,A)=\inf_{y\in A}|x-y|$.

We say $\mathcal{M}\subset\mathbb{R}^{n}$ is a $d$-dimensional, $C^{m}$
manifold if for all $x\in\mathcal{M}$, there exists $\delta>0$ such that
$B(x,\delta)\cap\mathcal{M}$ is the graph of a
$C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ function in some coordinate system,
or equivalently, if $B(x,\delta)\cap\mathcal{M}=F(U)$ for some open
$U\subset\mathbb{R}^{d}$ and injective $C^{m}$ function $F:U\to\mathcal{M}$. A
manifold need not be closed as a subset of $\mathbb{R}^{n}$.

We say $\mathcal{M}\subset\mathbb{R}^{n}$ is a $d$-dimensional, $C^{m}$
manifold with boundary if for all $x\in\mathcal{M}$, there exists $\delta>0$
such that $\mathcal{M}=F(U)$ for some injective $C^{m}$ function
$F:U\to\mathcal{M}$ where $U$ is either an open subset of $\mathbb{R}^{d}$ or
a (relatively) open subset of
$\mathbb{R}^{d}_{+}:=\\{(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}:x_{d}\geq 0\\}$.
If $\mathcal{M}$ is a closed subset of $\mathbb{R}^{n}$, we refer to it as a
compact manifold with boundary.

If $\mathcal{M}$ is a $C^{1}$ manifold and $x\in\mathcal{M}$, we denote the
tangent space of $\mathcal{M}$ at $x$ by $T_{x}\mathcal{M}$. If
$\mathcal{M}\subset\mathbb{R}^{N}\times\mathbb{R}^{D}$ is the graph of $F\in
C^{1}(\mathbb{R}^{N},\mathbb{R}^{D})$, then

$T_{(x,F(x))}\mathcal{M}=(x,F(x))+\operatorname{Col}\left(\begin{bmatrix}I_{N}\\\
\nabla F(x)\end{bmatrix}\right),$ (10)

where $I_{N}$ is the $N\times N$ identity matrix and $Col(A)$ refers to the
column space of a matrix $A$.

We say two submanifolds $\mathcal{M}$ and $\mathcal{N}$ of $\mathbb{R}^{n}$
intersect transversely if for all $z\in\mathcal{M}\cap\mathcal{N}$,
$T_{z}\mathcal{M}+T_{z}\mathcal{N}=\mathbb{R}^{n}$, where the addition is the
standard sum of vector subspaces.

The following is a standard result on the transverse intersection of manifolds
(see Theorem 6.30b) in [18] for a slightly more general result, for instance).

######  Theorem 2.2.

Suppose $\mathcal{M}$ and $\mathcal{N}$ are $C^{\infty}$ submanifolds of
$\mathbb{R}^{n}$. If $\mathcal{M}$ and $\mathcal{N}$ intersect transversely,
then $\mathcal{M}\cap\mathcal{N}$ is a $C^{\infty}$ manifold.

The following is a standard application of Thom’s transversality theorem [21]
(see Example 3.1 of [16]):

######  Theorem 2.3.

Let $\mathcal{M}$ and $\mathcal{N}$ be $C^{\infty}$ submanifolds of
$\mathbb{R}^{n}$. Then, for almost every $v\in\mathbb{R}^{n}$, $v+\mathcal{M}$
intersects $\mathcal{N}$ transversely.

###  2.3 Solution to the Classical Whitney Extension Problem

Fix integers $D,m,N\geq 1$. Let
$\mathcal{P}^{m}(\mathbb{R}^{N},\mathbb{R}^{D})=\mathcal{P}^{m}(\mathbb{R}^{N})\oplus\ldots\oplus\mathcal{P}^{m}(\mathbb{R}^{N})$
denote the vector space of $\mathbb{R}^{D}$ valued-valued polynomials in $N$
real variables and of degree at most $m$. If $F\in
C^{m}(\mathbb{R}^{N},\mathbb{R}^{D})$, we denote the $m$-th degree Taylor
polynomial of $F$ by $J_{x}F$. Note that
$J_{x}F\in\mathcal{P}^{m}(\mathbb{R}^{N},\mathbb{R}^{D})$. We will often call
$J_{x}F$ the jet of $F$ at $x$.

Given a compact set $E\subset\mathbb{R}^{N}$, we define a bundle
$\mathcal{H}=(H(x))_{x\in E}$ to be a collection of affine subspaces
$H(x)\subset\mathcal{P}^{m}(\mathbb{R}^{N},\mathbb{R}^{D})$ (called fibers).
We say $\mathcal{H}$ has a section $F\in C^{m}(\mathbb{R}^{N},\mathbb{R}^{D})$
if $J_{x}F\in H(x)$ for all $x\in E$.

If $(H(x))_{x\in E}$ is a bundle, we define a second bundle, called the
Glaeser refinement of $(H(x))_{x\in E}$ and denoted
$\tilde{\mathcal{H}}=(\tilde{H}(x)))_{x\in E}$, as follows:

For all $x_{0}\in E$, $\tilde{H}(x_{0})$ consists of the elements
$P_{0}=(P_{0}^{1},\ldots,P_{0}^{D})\in H(x_{0})$ such that for all
$\epsilon>0$, there exists $\delta>0$ such that for all
$x_{1},\ldots,x_{{\bar{k}}}\in E\cap B(x_{0},\delta)$ there exist
$P_{i}=(P_{i}^{1},\ldots,P_{i}^{D})\in H(x_{i})$ ($1\leq i\leq{\bar{k}})$
satisfying

$|\partial^{\alpha}(P_{i}^{l}-P_{j}^{l})(x_{i})|\leq\epsilon|x_{i}-x_{j}|^{m-|\alpha|}$

for all $0\leq i,j\leq{\bar{k}},1\leq l\leq D$, and $|\alpha|\leq m$. Here,
${\bar{k}}={\bar{k}}(m,N,D)$ is a positive integer. (It is elementary to
verify that each fiber $\tilde{H}(x)$ is an affine space so
$\tilde{\mathcal{H}}$ is indeed a bundle by definition.) Let $(H_{1}(x))_{x\in
E},(H_{2}(x))_{x\in E},\ldots$ denote the iterated Glaeser refinements of
$(H_{0}(x))_{x\in E}=(H(x))_{x\in E}$.

An important property of Glaeser refinement is that it is computable with only
linear algebra and the ability to take a limit. To see this, fix $x_{0}\in
E,P_{0}\in H(x_{0})$ and for $x_{1},\ldots,x_{\bar{k}}\in E$ define

$\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}})=\sum_{\begin{subarray}{c}0\leq
i,j\leq{\bar{k}}\\\ x_{i}\neq x_{j}\end{subarray}}\sum_{|\alpha|\leq
m}\sum_{1\leq l\leq
D}\left[\frac{\partial^{\alpha}(P^{l}_{i}-P^{l}_{j})(x_{i})}{|x_{i}-x_{j}|^{m-|\alpha|}}\right]^{2}.$

For fixed $x_{1},\ldots,x_{\bar{k}}$,
$\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}})$ is a
positive semidefinite quadratic form. Since each $H(x_{i})$ is an affine
space,

$\mathcal{Q}_{\min}(x_{1},\ldots,x_{{\bar{k}}})=\min\\{\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}}):P_{i}\in
H(x_{i}),1\leq i\leq{\bar{k}}\\}$

may be computed with linear algebra. We see that $P_{0}\in\tilde{H}(x_{0})$ if
and only if $\mathcal{Q}_{\min}(x_{1},\ldots,x_{{\bar{k}}})\to 0$ as
$(x_{1},\ldots,x_{{\bar{k}}})\to(x_{0},\ldots,x_{0})$ in $E^{{\bar{k}}}$.

We note that $(\tilde{H}(x))_{x\in E}$ has a section if and only if
$(H(x))_{x\in E}$ has a section. If $(H(x))_{x\in E}$ has a section $F$, then
one may take $P_{i}=J_{x_{i}}F$ in the definition of Glaeser refinement to
show $J_{x}F\in\tilde{H}(x)$ for all $x\in E$. The reverse direction is
trivial since $\tilde{H}(x)\subset H(x)$ for all $x\in E$.

We say $(H(x))_{x\in E}$ is Glaeser-stable if it is equal to its Glaeser
refinement. The following lemma was proven in [13] and adapted from the
corresponding lemma in [7], which in turn was adapted from [15] and [2].

######  Lemma 2.4.

Let $E\subset\mathbb{R}^{N}$ be compact and $(H(x))_{x\in E}$ be a bundle.
Then, there exists $l_{*}=l_{*}(m,N,D)$ such that $H_{k}(x)=H_{l_{*}}(x)$ for
all $k\geq l_{*}$. In particular, $(H_{l_{*}}(x))_{x\in E}$ is Glaeser-stable.

Given $f:E\to\mathbb{R}^{D}$, define

$H(x)=\\{P\in\mathcal{P}^{m}(\mathbb{R}^{N},\mathbb{R}^{D}):P(x)=f(x)\\}.$

The question of whether there exists $F\in
C^{m}(\mathbb{R}^{N},\mathbb{R}^{D})$ such that $F(x)=f(x)$ for all $x\in E$
is equivalent to whether $(H(x))_{x\in E}$ has a section. By Lemma 2.4, this
is the case if and only if $(H_{l_{*}}(x))_{x\in E}$ has a section. This
question is answered by the following theorem from [13].

######  Theorem 2.5.

Let $E\subset\mathbb{R}^{N}$ and $(H(x))_{x\in E}$ be a Glaeser-stable bundle
such that each fiber $H(x)$ is nonempty. Then $(H(x))_{x\in E}$ has a section.

Furthermore, given $x_{0}\in E$ and $P\in H(x_{0})$, one may choose $F$ such
that $J_{x_{0}}=P$.

The $D=1$ case was first proven in [7] and was used directly to establish
Theorem 2.5 via the ‘gradient trick.’

Clearly, if any $H(x)$ is empty, then $(H(x))_{x\in E}$ cannot have a section.
Thus, Theorem 2.5 provides a complete solution to Problem 1.1.

##  3 Glaeser Refinement for Manifolds

Fix integers $d,m,n\geq 1$ satisfying $d<n$ throughout the remainder of the
paper.

Given $Q\in O(n)$, define a coordinate system centered at the origin with axes
$(q_{1},\ldots.q_{n})$, where $q_{1},\ldots,q_{n}$ are the columns of $Q$. We
split the coordinates into the first $d$ (viewed as independent variables) and
the last $n-d$ (viewed as dependent variables), writing, for
$x\in\mathbb{R}^{n}$,

$x_{Q}=(q_{1}\cdot x,\ldots,q_{d}\cdot x)$

and

$x^{\perp}_{Q}=(q_{d+1}\cdot x,\ldots,q_{n}\cdot x).$

That is,

$\begin{bmatrix}x_{Q}\\\ x_{Q}^{\perp}\end{bmatrix}=Q^{T}x,$ (11)

where $x_{Q}\in\mathbb{R}^{d},x_{Q}^{\perp}\in\mathbb{R}^{n-d}$, and $A^{T}$
refers to the transpose of the matrix $A$.

The original vector $x$ may be recovered from $x_{Q}$ and $x^{\perp}_{Q}$ by
the formula

$x=\begin{bmatrix}q_{1}&\ldots&q_{d}\end{bmatrix}x_{Q}+\begin{bmatrix}q_{d+1}&\ldots&q_{n}\end{bmatrix}x^{\perp}_{Q}=Q\begin{bmatrix}x_{Q}\\\
x_{Q}^{\perp}\end{bmatrix},$

where $x_{Q}$ and $x^{\perp}_{Q}$ are interpreted as column vectors for matrix
multiplication. We call $x_{Q}$ and $x_{Q}^{\perp}$ the $Q$-coordinates of $x$
and the resulting coordinate system the $Q$-coordinate system. The definition
of the $Q$-coordinate system depends on $d$, though we hide this dependence in
our notation.

Given $Q\in O(n)$ and $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$, let
$G_{Q}(F)$ be the graph of $F$ in the $Q$-coordinate system, that is,

$G_{Q}(F):=\\{Q[(v,F(v))]:v\in\mathbb{R}^{n-d}\\}.$

Note that $G_{Q}(F)$ is a $C^{m}$ manifold defined so that $x\in G_{Q}(F)$ if
and only if $F(x_{Q})=x_{Q}^{\perp})$.

At each point $x\in E$, let

$H(x)\subset O(n)\times\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ (12)

such that

$H^{Q}(x):=\\{P:(Q,P)\in H(x)\\}$ (13)

is an affine space for fixed $Q\in O(n)$ and all $x\in E$. In this case, we
say $(H(x))_{x\in E}$ is an M-bundle and each $H(x)$ is a fiber of the
M-bundle at $x$. We say an M-bundle $(H(x))_{x\in E}$ is nontrivial if $H(x)$
is nonempty for all $x\in E$. We will occasionally refer to a pair $(Q,P)$ as
a jet.

Intuitively, the idea is that given a manifold $\mathcal{M}\ni x$ and $Q\in
O(n)$ such that $\mathcal{M}$ is locally the graph of a $C^{m}$ function $F$
in $Q$ coordinates near $x$, $(Q,J_{x_{Q}}F)$ should be considered ‘a’ jet of
$\mathcal{M}$ at $x$. The graph $G_{Q}(J_{x_{Q}}F)$ well-approximates the
graph $G_{Q}(F)$ near $x$.

A natural choice of M-bundle is

$H_{0}(x)=\\{(Q,P)\in O(n)\times
P^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d}):P(x_{Q})=x_{Q}^{\perp}\\}.$ (14)

The purpose of an M-bundle is to keep track of all such possible
approximations. We cannot fix a single $Q$ since it is possible that a given
tangent space may not be viewable as a function in $Q$ coordinates for some
given $Q$.

However, we should acknowledge that by including all $Q\in O(n)$ as
possibilities for coordinate systems, we are including a lot of redundancies.
First, a portion of a manifold may be viewed as the graph of a function with
many different choices for independent and dependent variables. For instance,
if $\mathcal{M}=\\{(x,y,z):x=y=z\\}\subset\mathbb{R}^{3}$, then $\mathcal{M}$
may be viewed as a function of $x$, writing $\mathcal{M}=G_{Q_{1}}(F)$, where

$Q_{1}=I_{3},F(t)=(t,t),$

or as a function of $y$ by $\mathcal{M}=G_{Q_{2}}(F)$, or a function of $z$ by
$\mathcal{M}=G_{Q_{3}}(F)$, where

$Q_{2}=\begin{bmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{bmatrix},Q_{3}=\begin{bmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{bmatrix}.$

Second, there are natural symmetries due to changes of coordinates within the
independent variables. For instance, in the case where $d=2$ and $n=3$,
$G_{Q_{4}}(y_{1}+y_{2}^{2})=G_{Q_{5}}$, where

$Q_{4}=I_{3},Q_{5}=\begin{bmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{bmatrix}.$

Similar redundancies can occur by elements of $O(n)$ which fix the last $n-d$
coordinates, rather than the first $d$.

One could avoid the redundant information found in an M-bundle by forming
equivalence classes in
$O(n)\times\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ (see Lemma 4.2
and Corollary 4.3 as a starting point). However, the calculations involved may
be particularly difficult (see the proof of Lemma 4.2), so it appears easier
to keep the extra information and develop tools for translating from one
coordinate system to another, as we will do. Of particular relevance to this
issue will be the notion of a consistent M-bundle, defined later in this
section.

######  Definition 3.1.

Given an M-bundle $(H(x))_{x\in E}$, we define its Glaeser refinement
$(\tilde{H}(x))_{x\in E}$ as follows.

Following (13), write $H(x)=\bigcup_{Q\in O(n)}H^{Q}(x)$ for $x\in E$. We say
$P_{0}\in\widetilde{H^{Q}}(x_{0})$ if and only if

  1. 1.

$P_{0}\in H^{Q}(x_{0})$ and

  2. 2.

For all $\epsilon>0$, there exists $\delta>0$ such that if
$x_{1},\ldots,x_{{\bar{k}}}\in E\cap B(x_{0},\delta)$, there exist $P_{i}\in
H^{Q}(x_{i})$ such that

$|\partial^{\alpha}(P_{i}-P_{j})((x_{i})_{Q})|\leq\epsilon|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}\text{for
all }|\alpha|\leq m,0\leq i,j,\leq{\bar{k}}.$ (15)

Lastly, let

$\tilde{H}(x)=\bigcup_{Q\in O(n)}\widetilde{H^{Q}}(x).$

Observe that $\widetilde{H^{Q}}(x)=\tilde{H}^{Q}(x)$, where
$(\tilde{H}(x))_{x\in E}$ is the Glaeser refinement of $(H(x))_{x\in E}$ and

$\tilde{H}^{Q}(x)=\\{P:(Q,P)\in\tilde{H}(x)\\}$

as in (13); we will use both notations interchangeably, referring to
$(\tilde{H}^{Q}(x))_{x\in E}$ as the Glaeser refinement of $(H^{Q}(x))_{x\in
E}$. We say an M-bundle is Glaeser stable if it is its own Glaeser refinement.

The definition of Glaeser refinement for $H^{Q}(x)$ is nearly equivalent to
the definition of Glaeser refinement given in Section 2 for the usual type of
bundle. However, for bundles we consider $x_{i}\in\mathbb{R}^{d}$ and
$B(x_{0},\delta)$ is a $d$-dimensional ball. For M-bundles,
$x_{i}\in\mathbb{R}^{n}$ and $B(x_{0},\delta)$ is an $n$-dimensional ball. The
compatibility of the $P_{i}$ may be tested through consideration of the
$(x_{i})_{Q}\in\mathbb{R}^{d}$, though the original $x_{i}\in\mathbb{R}^{n}$
are still relevant.

To see this difference expressed in an example, consider the Dirichlet
function

$f(x)=\begin{cases}1&x\in\mathbb{Q}\\\ 0&x\notin\mathbb{Q}\end{cases},$

which is known to be discontinuous. In particular, there is no $C^{1}$
extension of $f:[0,1]\to\mathbb{R}$ to the entire real line. Defining
$H(x)=\\{f(x)+m(t-x):m\in\mathbb{R}\\}$ for $x\in[0,1]$, one may readily
verify that the Glaeser refinement of the bundle $(H(x))_{x\in E}$ has empty
fibers; for each $x\in[0,1]$ there exist arbitrarily close $t$ to $x$ such
that $|f(t)-f(x)|=1$. Here, $f(x)+m(t-x)$ is viewed as a polynomial in $t$.

Now consider the graph of the Dirichlet function: the set
$E=\\{(x,1):x\in\mathbb{Q}\cap[0,1]\\}\cup\\{(x,0):x\in[0,1]\setminus\mathbb{Q}\\}$.
We define an M-bundle as in (14) and observe that

$H^{I_{2}}(x)=\begin{cases}\\{1+m(t-x):m\in\mathbb{R}\\}&x\in\mathbb{Q}\\\
\\{0+m(t-x):m\in\mathbb{R}\\}&x\notin\mathbb{Q}\end{cases}$

so that $P(x)=y$ for all $P\in H(z)$, where $z=(x,y)\in E$, so that $z\in
G_{I_{2}}(P)$. (See also (14).)

This M-bundle remains nontrivial after Glaeser refinement because the points
$(x,0)$ and $(x^{\prime},1)$ are far apart in $\mathbb{R}^{2}$, even when $x$
is near $x^{\prime}$. For Glaeser refinement, $H^{I_{2}}(x)$ need only be
checked against $H^{I_{2}}(x^{\prime})$ if both $x$ and $x^{\prime}$ are
rational, or if both $x$ and $x^{\prime}$ are irrational. We note there is a
$C^{1}$ manifold containing $E$, for instance,
$\\{(x,0):x\in(-.5,1.5)\\}\cup\\{(x,0):x\in(-.5,1.5)\\}$.

######  Lemma 3.1.

If $(H(x))_{x\in E}$ is an M-bundle, then so is $(\tilde{H}(x))_{x\in E}$.

###### Proof.

The part of this statement requiring proof is that $\tilde{H}^{Q}(x)$ is an
affine space.

Let $x_{0}\in E,Q\in O(n)$ and suppose
$P_{0},P_{0}^{\prime}\in\tilde{H}^{Q}(x_{0})$. We want to show $\lambda
P_{0}+(1-\lambda)P_{0}^{\prime}\in\tilde{H}^{Q}(x_{0})$ for
$\lambda\in\mathbb{R}$.

Let $\epsilon>0$. There exists $\delta>0$ such that for
$x_{1},\ldots,x_{\bar{k}}\in E\cap B(x_{0},\delta)$ there exist
$P_{i},P_{i}^{\prime}\in H^{Q}(x_{i})$ ($1\leq i\leq{\bar{k}}$) satisfying

$|\partial^{\alpha}(P_{i}-P_{j})((x_{i})_{Q})|\leq\epsilon/R|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}\text{for
all }|\alpha|\leq m,0\leq i,j,\leq{\bar{k}}$

and

$|\partial^{\alpha}(P^{\prime}_{i}-P^{\prime}_{j})((x_{i})_{Q})|\leq\epsilon/R|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}\text{for
all }|\alpha|\leq m,0\leq i,j,\leq{\bar{k}},$

where $R=R(\lambda)$ is to be determined.

Define $P_{i}^{\prime\prime}=\lambda P_{i}+(1-\lambda)P_{i}^{\prime}$ for
$0\leq i\leq{\bar{k}}$. Then,

$\displaystyle|\partial^{\alpha}(P^{\prime\prime}_{i}-P^{\prime\prime}_{j})((x_{i})_{Q})|$
$\displaystyle\leq|\lambda|\cdot|\partial^{\alpha}(P_{i}-P_{j})((x_{i})_{Q})|+|1-\lambda|\cdot|\partial^{\alpha}(P^{\prime}_{i}-P^{\prime}_{j})((x_{i})_{Q})|$
$\displaystyle\leq(|\lambda|+|1-\lambda|)\epsilon/R|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}$
$\displaystyle=\epsilon|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}$

for all $|\alpha|\leq m,0\leq i,j,\leq{\bar{k}}$ by setting
$R=(|\lambda|+|1-\lambda|)$. Thus,
$P_{0}^{\prime\prime}\in\tilde{H}^{Q}(x_{0})$. ∎

To address the computability of Glaeser refinement for M-bundles, fix
$x_{0}\in E,Q\in O(n),P_{0}\in H^{Q}(x_{0})$ and for
$x_{1},\ldots,x_{\bar{k}}\in E$ define

$\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}})=\sum_{\begin{subarray}{c}0\leq
i,j\leq{\bar{k}}\\\ x_{i}\neq x_{j}\end{subarray}}\sum_{|\alpha|\leq
m}\sum_{1\leq l\leq
n-d}\left[\frac{\partial^{\alpha}(P^{l}_{i}-P^{l}_{j})((x_{i})_{Q})}{|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}}\right]^{2}.$

As before, for fixed $x_{0},\ldots,x_{\bar{k}}$,
$\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}})$ is a
positive semidefinite quadratic form in $P_{1},\ldots,P_{{\bar{k}}}$. Since
each $H^{Q}(x_{i})$ is an affine space,

$\mathcal{Q}_{\min}(x_{1},\ldots,x_{{\bar{k}}})=\min\\{\mathcal{Q}(P_{1},\ldots,P_{{\bar{k}}};x_{1},\ldots,x_{{\bar{k}}}):P_{i}\in
H^{Q}(x_{i}),1\leq i\leq{\bar{k}}\\}$

may be computed with linear algebra. We see that
$P_{0}\in\tilde{H}^{Q}(x_{0})$ if and only if
$\mathcal{Q}_{\min}(x_{1},\ldots,x_{{\bar{k}}})\to 0$ as
$(x_{1},\ldots,x_{{\bar{k}}})\to(x_{0},\ldots,x_{0})$ in $E^{{\bar{k}}}$. As
in the case of bundles, the Glaeser refinement of M-bundles may also be
computed using linear algebra and a limit. While this is done here
independently for an infinite number of $Q\in O(n)$, Theorem 4.6 will show
this may be reduced to a finite number.

######  Lemma 3.2 (Termination Lemma).

Let $E\subset\mathbb{R}^{n}$ be compact and $(H_{0}(x))_{x\in E}$ be an
M-bundle with iterated Glaeser refinements $(H_{k}(x))_{x\in E}$, $k\geq 1$.
There exists $l^{*}=l(n,m,d)$ such that $H_{k}(x)=H_{l^{*}}(x)$ for all $k\geq
l^{*}$ and $x\in E$.

###### Proof.

Since each $H_{k+1}^{Q}(x)$ depends only on $H^{Q}_{k}(y)$ for $y$ near $x$
(and not the whole of $H_{k}(y)$), it suffices to show that
$H^{Q}_{k}(x)=H^{Q}_{l^{*}}(x)$ for all $k\geq l^{*},x\in E$, and $Q\in O(n)$.

We observe that Glaeser refinement has the following three properties:

  1. 1.

If $(H^{Q}(x))_{x\in E}$ and $(\hat{H}^{Q}(x))_{x\in E}$ agree in a
neighborhood of a point $x_{0}\in E$, then so do their Glaeser refinements.

  2. 2.

Each $H^{Q}(x)$ is an affine space, thus has a well-defined dimension. We have

$\dim\tilde{H}^{Q}(x)\leq\liminf_{y\to x,y\in E}\dim H^{Q}(y).$

  3. 3.

If $H^{Q}(x)=\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ for all $x\in
E$, then the Glaeser refinement of $(H^{Q}(x))_{x\in E}$ is itself.

Having proven termination of Glaeser refinement for general M-bundles, for the
remainder of the paper fix $(H_{0}(x))_{x\in E}$ as in (14), using
$(H_{1}(x))_{x\in E},(H_{2}(x))_{x\in E}$, and so on to denote its Glaeser
refinements.

From these three properties, one can arrive at the desired conclusion using a
standard argument (see the proof of Lemma 4.1 in [9]).

∎

As mentioned in the Introduction, we may naively hope to prove that there
exists a manifold extension if and only if the Glaeser refinement terminates
in a nontrivial M-bundle, but this statement is false. We now introduce the
ideas needed to formalize the topological condition.

Given a compact set $E\subset\mathbb{R}^{n}$, and $K(x)\subset Gr(n,d)$ for
$x\in E$, we say $(K(x))_{x\in E}$ is a Gr-bundle. A section of such a Gr-
bundle is a continuous function $f:E\to Gr(n,d)$ such that $f(x)\in K(x)$ for
all $x\in E$. Here, continuity is determined by the topology on $Gr(n,d)$
induced by the metric in (3).

Let $Q\in O(n)$ and $A$ be an $(n-d)\times d$ matrix. Define

$Q\oplus A:=\operatorname{Col}\left(Q\begin{bmatrix}I_{d}\\\
A\end{bmatrix}\right)\in Gr(n,d).$

Note that for $x\in\mathbb{R}^{n}$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$, $Q\oplus\nabla F(x_{Q})$ is parallel
to the tangent plane of $G_{Q}(F)$ at $x$; in particular, by (10)
$T_{x}G_{Q}(F)=x+Q\oplus\nabla F(x_{Q})$. That is,

$p(T_{x}G_{Q}(F))=Q\oplus\nabla F(x_{Q}),$

where $p$ is as in (9).

Given an M-bundle $\mathcal{H}=(H(x))_{x\in E}$, define

$\hat{H}(x)=\\{W\in Gr(n,d):\text{ there exists }(Q,P)\in H(x)\text{ such that
}Q\oplus\nabla P(x_{Q})=W\\}$

and define a Gr-bundle $\hat{\mathcal{H}}=(\hat{H}(x))_{x\in E}$. This Gr-
bundle is meant to represent the collection of possible tangent spaces
determined by the jets in the M-bundle $\mathcal{H}$. Note that the $\oplus$
operation used to define $\hat{\mathcal{H}}$ is computable with just linear
algebra.

######  Lemma 3.3.

Let $E\subset\mathbb{R}^{n}$ be compact and suppose $E\subset\mathcal{M}$,
where $\mathcal{M}$ is either a $C^{m}$, $d$-dimensional manifold or a
$C^{m}$, $d$-dimensional compact manifold with boundary. Then,

  1. 1.

The M-bundle $\mathcal{H}_{*}=(H_{l^{*}}(x))_{x\in E}$ is nontrivial and

  2. 2.

The Gr-bundle $\widehat{\mathcal{H}_{*}}$ has a section.

Furthermore, if $x\in E$ and $Q\in O(n)$ such that

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta)$

for some $\delta>0$, then $(Q,J_{x}F)\in H_{l^{*}}(x)$.

###### Proof.

Given $x_{0}\in E$, choose $Q$, $\delta_{0}>0$, and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that

$\mathcal{M}\cap B(x_{0},\delta_{0})=G_{Q}(F)\cap B(x_{0},\delta_{0}),$

that is,

$y_{Q}^{\perp}=F(y_{Q})$

for all $y\in\mathcal{M}\cap B(x_{0},\delta_{0})$.

Fix $\epsilon>0$. By Taylor’s theorem applied to $F$, there exists $\delta>0$
such that $s,t\in B((x_{0})_{Q},\delta)$ implies

$|\partial^{\alpha}(J_{s}F-J_{t}F)(s)|\leq\epsilon|s-t|^{m-|\alpha|}\text{for
all }|\alpha|\leq m,0\leq i,j,\leq{\bar{k}}.$ (16)

Given $x_{1},...,x_{{\bar{k}}}\in E\cap B(x_{0},\delta)$, let

$P_{i}=J_{(x_{i})_{Q}}(F((x_{i})_{Q}),$ (17)

for $0\leq i\leq{\bar{k}}$, noting that
$P_{i}((x_{i})_{Q})=(x_{i})_{Q}^{\perp}$, thus $(Q,P_{i})\in H_{0}(x_{i})$.
Thus, as $(x_{1})_{Q},...,(x_{{\bar{k}}})_{Q}\in E\cap B((x_{0})_{Q},\delta)$,
$(Q,P_{0})\in H_{1}(x_{0})$ by definition of Glaeser refinement by (16).

By similar argument (noting that $\mathcal{M}$ is also locally the graph of a
$C^{m}$ function in $Q$ coordinates at each $y\in E\cap B(x_{0},\delta_{0})$),
we see that $(Q,P_{y})\in H_{1}(y)$ for all $y\in E\cap B(x_{0},\delta_{0})$,
where

$P_{y}=J_{y_{Q}}F(y_{Q}).$ (18)

By the above reasoning with $H_{k}$ and $H_{k+1}$ in place of $H_{0}$ and
$H_{1}$, respectively, $(Q,P_{y})\in H_{k}(y)$ for all $k\geq 0$. In
particular, $(Q,P_{0})\in H_{l^{*}}(x_{0})$ and since $x_{0}\in E$ was
arbitrary, the M-bundle $(H_{l^{*}}(x))_{x\in E}$ is nontrivial. This
establishes the first point.

Given $x\in\mathcal{M}$, choose $Q\in O(n)$ such that $\mathcal{M}\cap
B(x,\delta)$ is the graph of a $C^{m}$ function
$F:\mathbb{R}^{d}\to\mathbb{R}^{n-d}$ in $Q$ coordinates for some $\delta>0$.

As before, let $x\in E$, choose $Q$, $\delta>0$, and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta),$

By the prior argument, $(Q,J_{x_{Q}}F)\in H_{l^{*}}(x)$. By (10),

$T_{x}\mathcal{M}=x+Q\oplus\nabla F(x_{Q})=x+Q\oplus\nabla J_{x_{Q}}F(x_{Q}).$

thus, the element of $Gr(n,d)$ parallel to $T_{x}\mathcal{M}$ lies in
$\widehat{H_{l^{*}}}(x)$. Since $T_{x}\mathcal{M}$ varies continuously in $x$,
and the natural projection $p$ in (9) is continuous,

$p(T_{x}\mathcal{M})=Q\oplus\nabla J_{x_{Q}}F(x_{Q})\in\hat{H}(x)$

is a continuous section of $(\widehat{H_{l^{*}}}(x))_{x\in E}$.

∎

At this point, we have shown that if $E$ has a manifold extension, then
repeated Glaeser refinements of $(H_{0}(x))_{x\in E}$ terminate in a
nontrivial Glaeser stable M-bundle $(H_{l^{*}}(x))_{x\in E}$. To complete our
answer to Problem 1.2, we show that any such M-bundle implies a manifold
extension of $E$.

The proof leads to a slightly stronger result; to state it, we require some
more terminology.

An M-bundle is proper if $P(x_{Q})=x_{Q}^{\perp}$ for all $(Q,P)\in H(x)$ and
all $x\in E$, that is, if $H(x)\subset H_{0}(x)$ for all $x\in E$. If an
M-bundle is proper, so are all its Glaeser refinements.

Let $(H(x))_{x\in E}$ be a proper M-bundle. Let $x\in\mathbb{R}^{n}$, $P\in
H^{Q}(x)$, and choose $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that
$J_{x_{Q}}F=P$. If $R\in O(n)$ such that $G_{Q}(F)$ is locally the graph of a
$C^{m}$ function $F^{\prime}$ in $R$ coordinates in a neighborhood of $x$, we
define the realization of the jet $(Q,P)$ at $x$ in the $R$ coordinate system
as

$J_{x,R}(Q,P)=(R,J_{x_{R}}F^{\prime})\in
O(n)\times\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d}).$

This object is shown to be well-defined (independent of choice of $F$) in
Lemma 4.2.

We say an M-bundle $(H(x))_{x\in E}$ is consistent if, whenever $x\in E$,
$(Q,P)\in H(x)$, and $G_{Q}(P)$ is locally the graph of a $C^{m}$ function in
$R$ coordinates in a neighborhood of $x$ for $R\in O(n)$, then
$J_{x,R}(Q,P)\in H(x)$. We will show that $(H_{l^{*}}(x))_{x\in E}$ is
consistent in Lemma 4.5.

######  Remark 3.2.

Observe that if $\mathcal{M}$ is locally the graph of a $C^{m}$ function $F$
in $Q$ coordinates and $F^{\prime}$ in $R$ coordinates, then

$p(T_{x}\mathcal{M})=Q\oplus\nabla F(x_{Q})$

and

$p(T_{x}\mathcal{M})=R\oplus\nabla F^{\prime}(x_{R}).$

Thus, when $J_{x,R}(Q,P)=(R,P^{\prime})$, we have

$Q\oplus\nabla P(x_{Q})=R\oplus\nabla P^{\prime}(x_{R}).$

The following theorem is the main result of this paper.

######  Theorem 3.4.

Fix positive integers $m$ and $d\leq n$ and let $E\subset\mathbb{R}^{n}$. Let
$\mathcal{H}=(H(x))_{x\in E}$ be a nontrivial, Glaeser-stable, proper,
consistent M-bundle.

If the Gr-bundle $\widehat{\mathcal{H}}=(\hat{H}(x))_{x\in E}$ has a section,
then there exists a $C^{m}$, $d$-dimensional, compact manifold with boundary
$\mathcal{M}$ such that $E\subset\mathcal{M}\subset\mathbb{R}^{n}$.

Furthermore, if for $x_{0}\in E$, $W_{0}\in\hat{H}(x_{0})$ and
$\widehat{\mathcal{H}}$ has a section $f$ such that $f(x_{0})=W_{0}$, then for
all $(Q,P)\in H(x)$ satisfying $Q\oplus\nabla P((x_{0})_{Q})=W_{0}$, one may
take $\mathcal{M}$ such that

$\mathcal{M}\cap B(x_{0},\delta)=G_{Q}(F)\cap B(x_{0},\delta)$

for some $\delta>0$ and $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
satisfying $J_{(x_{0})_{Q}}F=P$.

We can do better in the case $m=1$.

By a section of an M-bundle $(H(x))_{x\in E}$, we mean a manifold
$\mathcal{M}\supset E$ such that the following holds. Let $x\in E$ and choose
$Q\in O(n)$ such that

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta)$

for some $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ and $\delta>0$. Then,
$(Q,J_{x_{Q}}F)\in H(x)$.

The above definition suggests we only consider proper M-bundles, so that the
section contains the set $E$. It also suggests M-bundles should be consistent
since any small neighborhood of a manifold may be viewed as the graph of a
function in many different coordinate systems; the definition of a section
requires that the resulting jet lie in $H(x)$ for any such coordinate system.

######  Theorem 3.5.

Let $m=1$. Fix positive integers $d\leq n$ and let $E\subset\mathbb{R}^{n}$.
Let $\mathcal{H}=(H(x))_{x\in E}$ be a nontrivial, Glaeser-stable, proper,
consistent M-bundle.

If the Gr-bundle $\widehat{\mathcal{H}}=(\hat{H}(x))_{x\in E}$ has a section,
then $\mathcal{H}$ has a section. Furthermore, if $\Pi\in\hat{H}(x)$ and
$\widehat{\mathcal{H}}$ has a section $f$ such that $f(x)=\Pi$, then for all
$(Q,P)\in H(x)$ satisfying $Q\oplus\nabla P(x_{Q})=\Pi$, one may take the
section of $\mathcal{H}$ to be a compact manifold with boundary $\mathcal{M}$
such that

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta)$

for some $\delta>0$ and $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
satisfying $J_{x_{Q}}F=P$.

We conjecture Theorem 3.5 holds for $m\geq 2$; however, this does not appear
to follow from our current pasting process in Section 6.

##  4 Some Technical Lemmas

######  Lemma 4.1.

Let $Q,R\in O(n)$, $x\in\mathbb{R}^{n}$, and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$. Then, there exist $\delta>0$ and
$F^{\prime}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that

$G_{Q}(F)\cap B(x,\delta)=G_{R}(F^{\prime})\cap B(x,\delta)$

if and only if the top $d\times d$ minor of

$A:=R^{T}Q\begin{bmatrix}I_{d}\\\ \nabla F(x_{Q})\end{bmatrix}$

is invertible.

In either case, we say $G_{Q}(F)$ and $R$ are compatible near $x$.

###### Proof.

The tangent plane $T_{x}[G_{Q}(F)]$ is parallel to the plane spanned by the
columns of $Q\begin{bmatrix}I_{d}\\\ \nabla F(x_{Q})\end{bmatrix}$. We ask if
there exists an $(n-d)\times d$ matrix $B$ such that

$R\begin{bmatrix}I_{d}\\\ B\end{bmatrix}$

has the same column space as $Q\begin{bmatrix}I_{d}\\\ \nabla
F(x_{Q})\end{bmatrix}$, so that $T_{x}[G_{Q}(F)]$ may be viewed as an affine
function in $R$ coordinates. Equivalently, we ask that

$\operatorname{Col}(A)=\operatorname{Col}\left(\begin{bmatrix}I_{d}\\\
B\end{bmatrix}\right).$

By basic properties of Gaussian elimination, this is true if and only if the
top $d\times d$ minor of $A$ is invertible. ∎

######  Lemma 4.2.

Let $x\in\mathbb{R}^{n},(Q,P)\in H_{0}(x)$. If $G_{Q}(P)$ and $R\in O(n)$ are
compatible near $x$, then $J_{x,R}(Q,P)$ is well-defined.

Furthermore,

$J_{x_{R}}F^{\prime}(x_{R})=x_{R}^{\perp};$ (19)

thus, $(R,J_{x_{R}}F^{\prime})=J_{x,R}(Q,P)\in H_{0}(x)$.

###### Proof.

Let $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $J_{x_{Q}}F=P$.
All $y\in G_{Q}(F)$ satisfy

$y_{Q}^{\perp}=F(y_{Q}).$ (20)

Since $G_{Q}(P)$ and $R$ are compatible near $x$, so are $G_{Q}(F)$ and $R$,
as this property depends only on $\nabla F$ by Lemma 4.1. Thus, there exist
$\delta>0$ and $F^{\prime}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such
that $y\in G_{Q}(F)\cap B(x,\delta)$ satisfy

$y_{R}^{\perp}=F^{\prime}(y_{R}).$ (21)

In particular,

$x_{R}^{\perp}=F^{\prime}(x_{R})$

since $x\in G_{R}(F^{\prime})$. This establishes (19).

By (11),

$\begin{bmatrix}y_{Q}\\\
y_{Q}^{\perp}\end{bmatrix}=Q^{T}R\begin{bmatrix}y_{R}\\\
y_{R}^{\perp}\end{bmatrix}:=\begin{bmatrix}A_{1}&A_{2}\\\
A_{3}&A_{4}\end{bmatrix}\begin{bmatrix}y_{R}\\\ y_{R}^{\perp}\end{bmatrix},$

thus (20) becomes

$A_{3}y_{R}+A_{4}y_{R}^{\perp}=F(A_{1}y_{R}+A_{2}y_{R}^{\perp}).$ (22)

Let $S(k)$ represent the following statement:

$\nabla^{k}F^{\prime}(y_{R})=f(Q,R,\nabla
F(A_{1}y_{R}+A_{2}y_{R}^{\perp}),\ldots,\nabla^{k}F(A_{1}y_{R}+A_{2}y_{R}^{\perp}))$

for $y_{R}$ in a neighborhood of $x_{R}$, where $f$ is infinitely
differentiable.

Our goal is to prove $S(k)$ for $1\leq k\leq m$. We do so by induction on $k$.

Let $k=1$. Taking the derivative of (22) with respect to $y_{R}$, one has

$A_{3}+A_{4}\frac{\partial y_{R}^{\perp}}{\partial y_{R}}=\nabla
F(A_{1}y_{R}+A_{2}y_{R}^{\perp})(A_{1}+A_{2}\frac{\partial
y_{R}^{\perp}}{\partial y_{R}}).$

The above is a linear equation in $\nabla F^{\prime}=\frac{\partial
y_{R}^{\perp}}{\partial y_{R}}$, the solution having $C^{\infty}$ dependence
on the parameters in the equation: $A_{i}$ (meaning $Q$ and $R$) and $\nabla
F(A_{1}y_{R}+A_{2}y_{R}^{\perp})$.

Suppose $S(k)$ holds. By a simple application of the multivariate Chain Rule,
one sees that the $(k+1)$-st order derivatives of $F^{\prime}$ depend only on
$Q,R$, $\nabla
F(A_{1}y_{R}+A_{2}y_{R}^{\perp}),\ldots,\nabla^{k+1}F(A_{1}y_{R}+A_{2}y_{R}^{\perp})$,
and $\nabla F^{\prime}(A_{1}y_{R}+A_{2}y_{R}^{\perp})$, which in turn depends
only on $Q,R$, and $\nabla F(A_{1}y_{R}+A_{2}y_{R}^{\perp})$ by the base case
$k=1$. Thus, we have established $S(k+1)$ and by induction, $S(k)$ holds for
all $k$ up to $m$.

In particular, $\nabla F^{\prime}(x_{R}),\ldots,\nabla^{m}F^{\prime}(x_{R})$
depend only on $Q,R$, and $\nabla F(x_{Q}),\ldots,\nabla^{m}F(x_{Q})$ (which
in turn depend only on $P$), proving our claim.

∎

While we will not make direct use of the following corollary, we include it to
shed light on the situation and establish a logical basis for showing that “is
the realization of said jet in some coordinate system” is an equivalence
relation.

######  Corollary 4.3.

Fix
$x\in\mathbb{R}^{n},P\in\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d}),Q\in
O(n)$ such that $P(x_{Q})=x_{Q}^{\perp}$. If $G_{Q}(P)$ and $R\in O(n)$ are
compatible near $x$, then $J_{x,Q}(J_{x,R}(Q,P))=(Q,P)$.

###### Proof.

By Lemma 4.2, to compute $J_{x,R}(Q,P)$ we may choose $\delta>0$ and
$F^{\prime}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that
$G_{R}(F^{\prime})\cap B(x,\delta)=G_{Q}(P)\cap B(x,\delta)$ so
$J_{x,R}(Q,P)=(R,J_{x_{R}}F^{\prime})$. (Here, we choose $P$ as a $C^{m}$
function $F$ such that $J_{x_{Q}}F=P$.)

To compute $J_{x,Q}(R,J_{x_{R}}F^{\prime})$ we may find $\delta^{\prime}>0$
and $F^{\prime\prime}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that
$G_{R}(F^{\prime})\cap B(x,\delta^{\prime})=G_{Q}(F^{\prime\prime})\cap
B(x,\delta^{\prime})$. However, we already know from our prior computation
that taking $F^{\prime\prime}=P$ and $\delta^{\prime}=\delta$ works. We
observe $J_{x_{Q}}P=P$ to complete the proof. ∎

######  Lemma 4.4 (Local Interpolants).

Let $(H(x))_{x\in E}$ be a proper, Glaeser-stable M-bundle. Let $Q\in O(n)$
and $x_{0}\in E$.

If $P\in H^{Q}(x_{0})$, then there exists $\delta>0$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q}(F)\supset E\cap
B(x_{0},\delta)$. Furthermore, in this scenario, one may take
$J_{(x_{0})_{Q}}F=P$ and $J_{x_{Q}}F\in H^{Q}_{l^{*}}(x)$ for all $x\in E\cap
B(x_{0},\delta)$.

Conversely, if there exists $\delta>0$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q}(F)\supset E\cap
B(x_{0},\delta)$, then $P\in H_{l^{*}}^{Q}(x_{0})$.

###### Proof.

Suppose $P\in H^{Q}(x_{0})$. Letting $\epsilon=1$, there exists $\eta>0$ such
that whenever $x_{1},\ldots,x_{\bar{k}}\in E\cap B(x_{0},\eta)$, there exist
$P_{1},\ldots,P_{\bar{k}}\in H^{Q}(x_{i})$ satisfying

$|\partial^{\alpha}(P_{i}-P_{j})((x_{i})_{Q})|\leq|(x_{i})_{Q}-(x_{j})_{Q}|^{m-|\alpha|}\text{
for all }|\alpha|\leq m,0\leq i,j\leq{\bar{k}}.$ (23)

Let $E^{\prime}=\\{x_{Q}:x\in E\cap B(x_{0},\eta)\\}$. We have shown the
bundle $(H^{\prime}(x_{Q}))_{x_{Q}\in E^{\prime}}$, defined by

$H^{\prime}(x_{Q})=\\{P\in\mathcal{P}(\mathbb{R}^{d},\mathbb{R}^{n-d}):P\in
H^{Q}(x)\\},$

is nontrivial. Note this bundle is well-defined in the sense that if
$x,x^{\prime}\in E\cap B(x_{0},\eta)$, then $x_{Q}=x^{\prime}_{Q}$ implies
$x=x^{\prime}$. Suppose for the sake of contradiction that there exist
$x,x^{\prime}\in E\cap B(x_{0},\eta)$ such that $x\neq x^{\prime}$, yet
$x_{Q}=(x^{\prime})_{Q}$. However, if $P\in H^{Q}(x)$ and $P^{\prime}\in
H^{Q}(x^{\prime})$, then $P(x_{Q})=x_{Q}^{\perp}$ and
$P^{\prime}(x^{\prime}_{Q})=(x^{\prime})_{Q}^{\perp}$, contradicting (23) in
the case $\alpha=0,P_{i}=P,P_{j}=P^{\prime}$.

Our goal is to show that the Glaeser stability of $(H(x))_{x\in E}$ in the
M-bundle sense implies the Glaeser stability of $(H^{\prime}(x))_{x\in
E^{\prime}}$ in the usual sense.

Let $(y_{0})_{Q}\in E^{\prime},P_{0}\in H^{\prime}(y_{0})$ and $\epsilon>0$.
Choose $(y_{1})_{Q},\ldots,(y_{\bar{k}})_{Q}\in E^{\prime}\cap
B((y_{0})_{Q},\delta)$ for $\delta>0$ to be determined. By Glaeser stability
of $(H(x))_{x\in E}$ (in the M-bundle sense), there exists
$0<\delta^{\prime}<\eta$ such that if $y_{1},\ldots,y_{\bar{k}}\in E\cap
B(y_{0},\delta^{\prime})$, there exist
$P_{1},\ldots,P_{\bar{k}}\in\mathcal{P}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such
that

$P_{i}\in H^{Q}(y_{i})\text{ for all }0\leq i\leq{\bar{k}}$

and

$|\partial^{\alpha}(P_{i}-P_{j})((y_{i})_{Q})|\leq\epsilon|(y_{i})_{Q}-(y_{j})_{Q}|^{m-|\alpha|}\text{
for all }|\alpha|\leq m,0\leq i,j\leq{\bar{k}}.$

Letting $\alpha=0,\epsilon=1,$ and $j=0$, we have

$|(y_{i})_{Q}^{\perp}-P_{0}((y_{0})_{Q})|\leq|(y_{i})_{Q}-(y_{0})_{Q}|.$ (24)

Since $P_{0}$ is differentiable, we have

$|P_{0}((y_{i})_{Q})-(y_{0})_{Q}^{\perp}|=|P_{0}((y_{i})_{Q})-P_{0}((y_{0})_{Q})|\leq
C|(y_{i})_{Q}-(y_{0})_{Q}|$ (25)

for $C$ depending only on $P_{0}$ and $\delta^{\prime}$.

Thus, by the triangle inequality, (24), and (25),

$|(y_{i})_{Q}^{\perp}-(y_{0})_{Q}^{\perp}|\leq|(y_{i})_{Q}^{\perp}-P_{0}((y_{i})_{Q})|+|P_{0}((y_{i})_{Q})-(y_{0})_{Q}^{\perp}|\leq
C^{\prime}|(y_{i})_{Q}-(y_{0})_{Q}|$ (26)

and by (26),

$|y_{i}-y_{0}|\leq|(y_{i})_{Q}^{\perp}-(y_{0})_{Q}^{\perp}|+|(y_{i})_{Q}-(y_{0})_{Q}|\leq
C^{\prime\prime}|(y_{i})_{Q}-(y_{0})_{Q}|$

where $C^{\prime\prime}$ depends only on $P_{0}$ and $\delta^{\prime}$.

By taking $\delta=\delta^{\prime}/C^{\prime\prime}$, we obtain that for any
$y_{1},\ldots,y_{\bar{k}}\in E^{\prime}\cap B((y_{0})_{Q},\delta)$, there
exist
$P_{1},\ldots,P_{\bar{k}}\in\mathcal{P}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such
that

$P_{i}\in H^{\prime}(y_{i})\text{ for all }0\leq i\leq{\bar{k}}$

and

$|\partial^{\alpha}(P_{i}-P_{j})((y_{i})_{Q})|\leq\epsilon|(y_{i})_{Q}-(y_{j})_{Q}|^{m-|\alpha|}\text{
for all }|\alpha|\leq m,0\leq i,j\leq{\bar{k}}.$

Thus, $(H^{\prime}(y))_{y\in E^{\prime}}$ is Glaeser stable (in the usual
sense).

Furthermore, since $(H(x))_{x\in E}$ is proper,

$P(x_{Q})=x_{Q}^{\perp}\text{ for all }x_{Q}\in E^{\prime}.$

By Theorem 2.5, there exists $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
such that $F(x_{Q})=x_{Q}^{\perp}$ for all $x_{Q}\in E^{\prime}$; thus,
$G_{Q}(F)\supset E\cap B(x_{0},\delta)$. Furthermore, we may take $F$ such
that $J_{(x_{0})_{Q}}F=P$.

For the reverse direction, suppose there exists $\delta>0$ and $F\in C^{m}$
such that $G_{Q}(F)\supset E\cap B(x_{0},\delta)$. For all $x\in E\cap
B_{\delta}(x_{0})$, $(Q,J_{x_{Q}}F)\in H_{0}(x)$ since
$F(x_{Q})=x_{Q}^{\perp}$. Since $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$,
the jets $J_{x_{Q}}F$ are Taylor compatible and the $(Q,J_{x_{Q}}F)$ survive
each round of Glaeser refinement, remaining in $H_{l^{*}}(x_{Q})$, as in the
proof of Lemma 3.3. In particular, $(Q,J_{(x_{0})_{Q}}F)\in H_{l^{*}}(x_{0})$.

∎

######  Lemma 4.5.

Let $E\subset\mathbb{R}^{n}$ be compact and $(H_{l^{*}}(x))_{x\in E}$ be as
previously defined. Let $x\in E$ and $(Q,P)\in H_{l^{*}}(x)$. If $G_{Q}(P)$
and $R\in O(n)$ are compatible near $x$, then $J_{x,R}(Q,P)\in H_{l^{*}}(x)$.
That is, $(H_{l^{*}}(x))_{x\in E}$ is consistent.

###### Proof.

The M-bundle $(H_{l_{*}}(x))_{x\in E}$ is proper by definition and Glaeser-
stable by Lemma 3.2. Thus, by Lemma 4.4, there exists $\delta>0$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q}(F)\supset E\cap
B(x_{0},\delta)$. Since $G_{Q}(P)$ and $R$ are compatible near $x$, so are
$G_{Q}(F)$ and $R$, as compatibility depends only on $\nabla F$ by Lemma 4.1.
Thus, there exist $\delta^{\prime}>0$ and $F^{\prime}\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q}(F)\cap
B(x,\delta^{\prime})=G_{R}(F^{\prime})\cap B(x,\delta^{\prime})$. By Lemma 4.4
(this time in the opposite direction),
$J_{x,R}(Q,P)=(R,J_{(x_{R})}F^{\prime})\in H_{l^{*}}(x)$. ∎

For $S\subset O(n)$, define

$H_{0}^{S}(x)=\\{(Q,P)\in S\times
P^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d}):P(x_{Q})=x_{Q}^{\perp}\\}$

and denote the fibers of its iterated Glaeser refinements by

$H_{k}^{S}(x)=\bigcup_{Q\in S}H_{k}^{Q}(x).$

The following theorem reduces the refinements of $(H^{Q}(x))_{x\in E}$
required in Theorem 3.4 from those involving all $Q\in O(n)$ to a finite
number of $Q$’s.

######  Theorem 4.6.

Fix integers $m,n,d$. There exists a finite $S_{0}\subset O(n)$ such that the
following holds.

Let $E\subset\mathbb{R}^{n}$. Then then there exists a $C^{m}$,
$d$-dimensional, compact manifold with boundary $\mathcal{M}$ such that
$E\subset\mathcal{M}\subset\mathbb{R}^{n}$ if and only if

  1. 1.

The M-bundle $\mathcal{H}^{S_{0}}_{*}=(H^{S_{0}}_{l^{*}}(x))_{x\in E}$ is
nontrivial and

  2. 2.

The Gr-bundle
$\widehat{\mathcal{H}^{S_{0}}_{*}}=(\hat{H}^{S_{0}}_{l^{*}}(x))_{x\in E}$ has
a continuous section.

Furthermore, if for $x\in E$, $\Pi\in\hat{H}^{S_{0}}_{l^{*}}(x)$ and
$\widehat{(\mathcal{H}^{S_{0}})^{*}}$ has a section $f$ such that $f(x)=\Pi$,
then for all $(Q,P)\in H^{S_{0}}_{l^{*}}(x)$ satisfying $Q\oplus\nabla
P(x_{Q})=\Pi$, one may take $\mathcal{M}$ to satisfy

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta)$

for some $\delta>0$ and $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
satisfying $J_{x_{Q}}F=P$.

###### Proof.

Suppose both 1) and 2) above hold.

Since $H_{l^{*}}^{S_{0}}(x)\subset H_{l^{*}}(x)$ for all $x\in E$, the bundle
$(H_{l^{*}}(x))_{x\in E}$ is nontrivial. It is by definition proper, by Lemma
4.5 consistent, and by Lemma 3.2 Glaeser-stable. Furthermore, since
$\widehat{H^{S_{0}}_{l^{*}}}(x)\subset\widehat{H_{l^{*}}}(x)$ for all $x\in
E$, the bundle $(\widehat{H_{l^{*}}}(x))_{x\in E}$ has a continuous section.
By Theorem 3.4, there exists a $d$-dimensional, $C^{m}$ manifold
$\mathcal{M}\supset E$.

Now suppose there exists a $d$-dimensional, $C^{m}$ compact manifold with
boundary $\mathcal{M}\supset E$. We may mostly repeat the argument in the
proof of Lemma 3.3; we need only show there exists a finite set $S_{0}\subset
O(n)$, chosen independently of $E$ and $\mathcal{M}$, satisfying the following
property. For all $x\in\mathcal{M}$, there exists $Q\in S_{0}$ such that
$T_{x}\mathcal{M}$ is the graph of a function in $Q$ coordinates.

Choose $S_{0}$ to be the set of $n\times n$ permutation matrices, that is,
those matrices which have a single 1 per row and a single 1 per column with
the remainder of the entries being 0. We claim $S_{0}$ satisfies the desired
properties.

Since $\mathcal{M}$ is locally the graph of a $C^{m}$ function in
$Q_{0}$-coordinates, where the first $d$ columns of $Q_{0}$ span a $d$-plane
parallel to $T_{x}\mathcal{M}$, by Lemma 4.1 we want to find $Q\in S_{0}$ such
that the top $d\times d$ minor of $Q^{T}A$ is invertible, where

$A=Q_{0}\begin{bmatrix}I_{d}\\\ \nabla F(x_{Q})\end{bmatrix}.$

The matrix $A$ is of rank $d$; therefore, there exist a subset
$\\{a_{i_{1}},\ldots,a_{i_{d}}\\}$ of its rows ($i_{1},\ldots,i_{d}$ all
distinct) such that the $d\times d$ matrix with rows
$a_{i_{1}},\ldots,a_{i_{d}}$ is invertible. The result follows by choosing $Q$
to have a 1 in the $i_{1}$-position of its first column, the $i_{2}$-position
of its second column, up to putting a 1 in the $i_{d}$ position of the $d$-th
column and filling out the rest of $Q$’s columns in any way such that it
becomes a permutation matrix.

∎

In the above proof, we chose a particular choice of $S_{0}$ for which the
conclusion of Theorem 4.6 was particularly easy to verify. There exist smaller
sets $S_{0}$ with the same property; for instance, one could remove
permutation matrices from the $S_{0}$ used above which share the same first
$d$ columns with another remaining element. However, we provide no attempt to
determine the smallest such $S_{0}$.

##  5 Constructing Local Interpolants

We now begin the proof of Theorem 3.4. Suppose

  1. 1.

The M-bundle $\mathcal{H}=(H(x))_{x\in E}$ is nontrivial, Glaeser-stable,
proper, and consistent;

  2. 2.

The Gr-bundle $(\hat{H}(x))_{x\in E}$ has a continuous section $f:E\to
Gr(n,d)$;

  3. 3.

For some fixed $x_{0}\in E$, $f(x_{0})=W_{0}\in\hat{H}(x_{0})$. Moreover,
$Q_{0}\oplus\nabla P_{0}(x_{0})=\Pi_{0}$, where $(Q_{0},P_{0})\in H(x_{0})$.

We define choices $\delta_{x}\in(0,\infty),R_{x}\in O(n)$, and $F_{x}\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ for all $x\in E$ as follows.

Since $f$ is continuous on the compact set $E$, it is uniformly continuous.
Choose $0<\delta_{f}<1$ such that if $y,y^{\prime}\in E$ and
$|y-y^{\prime}|<2\delta_{f}$, then $\text{dist}(f(y),f(y^{\prime}))<c_{0}/2$,
where $c_{0}>0$ is a small constant to be chosen later.

Since $\mathcal{H}$ is proper, Glaeser-stable, and non-empty, by Lemma 4.4,
there exists $0<\delta<\delta_{f}/5$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q_{0}}(F)\supset E\cap
B(x_{0},4\delta)$. Furthermore, $J_{(x_{0})_{Q_{0}}}F=P_{0}$ and

$J_{x_{Q_{0}}}F\in H^{Q_{0}}(x)\text{ for all }x\in E\cap B(x_{0},4\delta).$
(27)

Let $R\in O(n)$ such that $W_{0}=\Pi R$, where $\Pi$ is as in (8). Then,
$\Pi_{0}$ and $R$ are compatible near $x_{0}$ since $W_{0}=G_{R}(h)$, where
$h:\mathbb{R}^{d}\to\mathbb{R}^{n-d}$ is the zero function. Thus, taking
$\delta$ smaller if needed, there exists $F^{\prime}\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that

$G_{R}(F^{\prime})\cap B(x_{0},4\delta)=G_{Q_{0}}(F)\cap B(x_{0},4\delta).$

Since $G_{R}(F^{\prime})$ and $G_{Q_{0}}(F)$ have the same tangent space at
$x_{0}$,

$\nabla F^{\prime}((x_{0})_{R})=0.$ (28)

Again taking $\delta$ smaller if necessary, we may, by the uniform continuity
of the tangent space of $G_{R}(F^{\prime})$ on closed balls, assume that

$\text{dist}(R\oplus\nabla F^{\prime}(x_{R}),R\oplus\nabla
F^{\prime}(y_{R}))<c_{0}/2$ (29)

for all $x,y\in B(x_{0},\delta)$. In particular,

$\text{dist}(R\oplus\nabla F^{\prime}(x_{R}),W_{0})<c_{0}/2$ (30)

for all $x\in B(x_{0},\delta)$.

Let $\delta_{x_{0}}=\delta,F_{x_{0}}=F^{\prime}$, and $R_{x_{0}}=R$, as above.

If $x\in B(x_{0},2\delta_{x_{0}})\setminus\\{x_{0}\\}$, choose $Q\in O(n)$
such that $\Pi Q=R_{x_{0}}\oplus\nabla F_{0}^{\prime}(x_{R_{x_{0}}})$ and $F$
such that

$G_{R_{x_{0}}}(F_{0}^{\prime})\cap B(x,20\delta)=G_{Q}(F)\cap
B(x_{0},20\delta)$

for some $0<\delta<\delta_{x_{0}}/10$. In particular, $\nabla F(x_{Q})=0$. By
the consistency of $(H(x))_{x\in E}$ and (27), $J_{y_{Q}}F\in H^{Q}(y)$ for
all $y\in B(x,20\delta)$.

For such $x$, define $\delta_{x}=\delta,R_{x}=Q$, and $F_{x}=F$.

If $x\notin B(x_{0},2\delta_{0})$, choose $(Q,P)\in H(x)$ such that
$Q\oplus\nabla P(x_{Q})=f(x)$. By Lemma 4.4, there exists
$0<\delta<\delta_{x_{0}}/20$ and $F\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
such that $G_{Q}(F)\supset E\cap B(x,\delta)$. Observe that by choice of
$\delta$, $B(x,20\delta)$ and $B(x_{0},\delta_{x_{0}})$ are disjoint.

As before, let $R\in O(n)$ such that

$f(x)=\Pi R.$ (31)

Then, $f(x)$ and $R$ are compatible near $x$, so taking $\delta$ smaller if
needed, there exists $F^{\prime}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
such that

$G_{R}(F^{\prime})\cap B(x,20\delta)=G_{Q}(F)\cap B(x,20\delta).$

Observe that, as before, (28) holds by (31).

For such $x$, define $\delta_{x}=\delta,R_{x}=R$, and $F_{x}=F^{\prime}$.

By making $\delta_{x}$ smaller if necessary, we have, for all $x\in E$,

  1. 1.

$G_{R_{x}}(F_{x})\cap B(x,20\delta_{x})\supset E\cap B(x,20\delta_{x})$ (32)

  2. 2.

The bound

$|\nabla F_{x}(y_{R_{x}})|\leq c_{0}$ (33)

holds for all $y_{R}\in B(x_{R},20\delta_{x})\subset\mathbb{R}^{d}$, hence for
all $y\in B(x,20\delta_{x})\subset\mathbb{R}^{n}$. This is allowed because
$\nabla F_{x}$ is continuous and $\nabla F_{x}(x_{R_{x}})=0$.

  3. 3.

If $x,x^{\prime}\in E$ and $|x-x^{\prime}|<\delta_{f}$, then

$\text{dist}(\Pi R_{x},R_{x^{\prime}})<c_{0}$ (34)

  4. 4.

If $x\in B(x_{0},\delta_{x_{0}})$, then

$G_{R_{x}}(F_{x})\cap B(x,20\delta_{x})=G_{R_{x_{0}}}(F_{x_{0}})\cap
B(x,20\delta_{x}).$ (35)

  5. 5.

If $x\notin B(x_{0},\delta_{x_{0}})$, then

$B(x,\delta_{x})\cap B(x_{0},\delta_{x})=\emptyset.$ (36)

The third property above requires some justification. If $x=x_{0}$ and
$x^{\prime}\notin B(x_{0},\delta_{x_{0}})$ or $x,x^{\prime}\notin
B(x_{0},\delta_{x_{0}})$, then $\Pi R_{x}=f(x)$ and $\Pi
R_{x^{\prime}}=f(x^{\prime})$, so (34) follows from the definition of
$\delta_{f}$. If $x=x_{0}$ and $x^{\prime}\in B(x_{0},\delta_{x_{0}})$ or
$x,x^{\prime}\in B(x_{0},\delta_{x_{0}})$, then it follows from (30) and (29),
respectively.

The last case is if $x\in B(x_{0},\delta_{x_{0}})$ and $x^{\prime}\notin
B(x_{0},\delta_{x_{0}})$. Here,

$|x^{\prime}-x_{0}|\leq|x^{\prime}-x|+|x-x_{0}|\leq\delta_{f}+\delta_{f}=2\delta_{f};$

thus, by the definition of $\delta_{f}$,

$\text{dist}(\Pi R_{x^{\prime}},W_{0})<c_{0}/2.$

By (30),

$\text{dist}(\Pi R_{x},W_{0})<c_{0}/2.$

Thus, by the triangle inequality,

$\text{dist}(\Pi R_{x^{\prime}},\Pi R_{x})<c_{0},$

establishing (34).

We add one more condition to the $F_{i}$. Let

$E_{i}:=\\{(z)_{Q_{i}}:z\in E\cap
B(z_{i},3\delta_{0})\\}\subset\mathbb{R}^{d}.$ (37)

We require that $F_{i}$ be $C^{\infty}$ at $x$ whenever
$d(x,E_{i})>\delta_{0}$. This is attainable by a standard argument using a
smooth partition of unity to paste together the original $F_{i}$ with a
$C^{\infty}$ approximating function.

Since $E$ is compact, there exist $x_{1},\ldots,x_{M}\in E$ such that

$E\subset\bigcup_{1\leq i\leq M}B(x_{i},\delta_{x_{i}}).$ (38)

Let $\delta_{0}=\min_{1\leq i\leq M}\delta_{x_{i}}>0$. Observe that

$\delta_{0}<\delta_{x_{0}}/10<\delta_{f}/50;$ (39)

else $E\subset B(x_{0},\delta_{x_{0}})$ and we are done. Since
$\\{B(x,\delta_{0}):x\in E\\}$ is an open cover of $E$, we again apply
compactness to determine $y_{1},\ldots,y_{M^{\prime}}\in E$ such that

$E\subset\bigcup_{1\leq j\leq M^{\prime}}B(y_{j},\delta_{0}/3).$

Next, we use the classical Vitali covering lemma (found in [20], for example),
stated below:

######  Lemma 5.1.

Let $B(x_{1},r_{1}),\ldots,B(x_{J},r_{J})$ be balls in a metric space. Then
there exist indices $j_{1},\ldots j_{m}$ such that

$B(x_{1},r_{1})\cup\ldots\cup B(x_{J},r_{J})\subset
B(x_{j_{1}},3r_{j_{1}})\cup\ldots\cup B(x_{j_{m}},3r_{j_{m}})$

and $B(x_{j_{1}},r_{j_{1}}),\ldots,B(x_{j_{m}},r_{j_{m}})$ are disjoint.

Applying the Vitali covering lemma to $\\{B(y_{j},\delta_{0}/3):1\leq j\leq
M^{\prime}\\}$, there exist $z_{1},\ldots,z_{N}\in E$ such that

$E\subset\bigcup_{1\leq j\leq N}B(z_{j},\delta_{0})$

where the $B(z_{j},\delta_{0}/3)$ are disjoint. Here,
$\\{z_{1},\ldots,z_{N}\\}\subset\\{y_{1},\ldots,y_{M^{\prime}}\\}$ and the
$z_{j}$ are distinct.

From this, we may deduce the existence of a constant $C=C(n)$ such that for
all $1\leq j\leq N$,

$|\\{k:B(z_{j},10\delta_{0})\cap B(z_{k},10\delta_{0})\neq\emptyset\\}|\leq
C.$

We summarize the results of this section, replacing $F_{j}(\cdot)$ by
$F_{j}(\cdot-(z_{j})_{Q_{j}})-(z_{j})_{Q_{j}}^{\perp}$.

There exist $\delta_{0}>0$, $z_{1},\ldots,z_{N}\in E$, $Q_{1},\ldots,Q_{N}\in
O(n)$, and $F_{1},\ldots,F_{N}\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{n-d})$ such
that for all $1\leq i,j\leq N$,

  1. 1.

$E\subset\bigcup_{1\leq i\leq N}B(z_{i},\delta_{0})$ (40)

  2. 2.

$F_{i}(0)=0;$ (41)

  3. 3.

$z_{i}+G_{Q_{i}}(F_{i})\supset E\cap B(z_{i},20\delta_{0});$ (42)

  4. 4.

$|\nabla F_{i}(w)|\leq c_{0}$ (43)

for $|w|\leq 8\delta_{0}$;

  5. 5.

$\text{dist}(\Pi Q_{i},\Pi Q_{j})<c_{0}$ (44)

whenever $B(z_{i},10\delta_{0})\cap B(z_{j},10\delta_{0})\neq\emptyset$ by
(34) and (39).

  6. 6.

$|\\{k:B(z_{i},10\delta_{0})\cap B(z_{k},10\delta_{0})\neq\emptyset\\}|\leq
C.$ (45)

  7. 7.

As follows from (35), if $z_{i}\in B(x_{0},5\delta_{0})$, then

$[z_{i}+G_{Q_{i}}(F_{i})]\cap B(z_{i},\delta_{0})=G_{R_{x_{0}}}(F_{x_{0}})\cap
B(x_{0},\delta_{0}).$ (46)

where $x_{0},R_{x_{0}},F_{x_{0}}$ are as at the beginning of the section.

  8. 8.

$F_{i}\text{ is }C^{\infty}\text{ at distance greater than }\delta_{0}\text{
from }E_{i},$ (47)

where $E_{i}$ is as in (37).

##  6 Pasting Local Interpolants Together

In this section, we write elements $z\in\mathbb{R}^{n}$ as
$(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{n-d}$. Let
$\Pi_{d}:\mathbb{R}^{n}\to\mathbb{R}^{d}$ send $z\in\mathbb{R}^{n}$ to its
first $d$ coordinates, or equivalently, $x$. We write
$B_{d}(p,r),B_{n-d}(p,r)$, and $B_{n}(p,r)$ to write the open ball of radius
$r$ with center $p$ in $\mathbb{R}^{d},\mathbb{R}^{n-d}$, and
$\mathbb{R}^{n}$, respectively. Similarly, $B_{d}$, $B_{n-d}$, and $B_{n}$
will denote the unit balls in dimensions $d,n-d$, and $n$, respectively.

For $1\leq i\leq N$, let $o_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the rigid
motion defined by

$o_{i}(z)=Q_{i}z+z_{i}.$

Let $\mathbf{cyl}=\delta_{0}(B_{d}\times B_{n-d})$, and for $r>0$ let
$\mathbf{cyl}^{r}=r\delta_{0}(B_{d}\times B_{n-d})$. Define
$\phi:\mathbb{R}^{n}\to\mathbb{R}$ by letting

$\phi(z)=|y|^{2}$

for $z=(x,y)\in\mathbf{cyl}^{2}$, and $\phi(z)=0$ otherwise. Let
$\mathbf{cyl}_{i}=o_{i}(\mathbf{cyl})$ and likewise
$\mathbf{cyl}^{r}_{i}=o_{i}(\mathbf{cyl}^{r})$ for $r>0$.

Observe that

$B_{n}(z_{i},r\delta_{0})\subset\mathbf{cyl}_{i}^{r}\subset
B_{n}(z_{i},2r\delta_{0})$ (48)

for all $r>0$.

In the language of this section, (41), (42), and (48) imply

$z_{i}+G_{Q_{i}}(F_{i})\supset E\cap\mathbf{cyl}_{i}^{10}.$ (49)

######  Lemma 6.1.

For $1\leq i\leq N$ and let
$S_{i}=\\{\mathbf{cyl}^{5}_{i_{1}},\ldots,\mathbf{cyl}^{5}_{i_{|S_{i}|}}\\}$
be the set of cylinders $\mathbf{cyl}_{i}^{5}$ which intersect
$\mathbf{cyl}^{5}_{i}$.

Then, there exist rigid motions $U_{i_{1}},\ldots,U_{i_{|S_{i}|}}$ satisfying
$U_{i_{j}}(z_{i_{j}})=z_{i_{j}}$ and translations
$T_{i_{1}},\ldots,T_{i_{|S_{i}|}}$ such that for $1\leq j\leq|S_{i}|$

  1. 1.

$T_{i_{j}}U_{i_{j}}\mathbf{cyl}^{5}_{i_{j}}\text{ is the translation of
}\mathbf{cyl}^{5}_{i}\text{ by a vector contained in }\mathbb{R}^{d};$ (50)

  2. 2.

$|(I_{n}-U_{i_{j}})v|<C(n,d)c_{0}|v-z_{i_{j}}|;$ (51)

and

  3. 3.

$|T_{i_{j}}(0)|<c_{0}\delta_{0}.$ (52)

A collection of cylinders satisfying the conclusion of the above lemma, as
well as a couple of additional properties, would be considered a cylinder
packet in [14]. We omit these additional properties since they are not
necessary for our purposes.

###### Proof.

Without loss of generality, let $Q_{i}=I_{n}$ and $z_{i}=0$.

Let $\Pi_{i_{j}}$ denote the orthogonal projection onto $\Pi Q_{i_{j}}$.

Fix $1\leq i\leq N$ and $1\leq i_{j}\leq|S_{i}|$. Since
$\mathbf{cyl}^{5}_{i_{j}}$ intersects $\mathbf{cyl}^{5}_{i}$, by (48),
$B_{n}(z_{i},10\delta_{0})$ intersects $B_{n}(z_{i_{j}},10\delta_{0})$. Thus,
by (44), $\|\Pi_{i_{j}}-\Pi_{d}\|\leq c_{0}$ By Lemma 2.1 and taking $c_{0}$
sufficiently small, there exists $R_{i_{j}}\in O(n)$ such that
$R_{i_{j}}Q_{i}\mathbb{R}^{d}=\mathbb{R}^{d}$ and $\|R_{i_{j}}\|\leq
C(n,d)c_{0}$. We obtain (51) by letting

$U_{i_{j}}(z)=R_{i_{j}}(z-z_{i_{j}})+z_{i_{j}}.$

Defining

$T_{i_{j}}(z)=z-(z_{i_{j}})_{I_{n}}^{\perp}=z-y_{i_{j}}$

gives us (50), as

$\displaystyle T_{i_{j}}U_{i_{j}}\mathbf{cyl}_{i_{j}}^{3}$
$\displaystyle=T_{i_{j}}U_{i_{j}}(Q_{i_{j}}\mathbf{cyl}_{i}^{3}+z_{i_{j}})$
$\displaystyle=T_{i_{j}}(\mathbf{cyl}_{i}^{3})+z_{i_{j}}$
$\displaystyle=\mathbf{cyl}_{i}^{3}+x_{i_{j}}.$

Lastly, observe that since $|z_{i_{j}}-z_{i}|<20\delta_{0}$, by (42),
$z_{i_{j}}\in G_{I_{n}}(F_{i})$. By (43),

$|y_{i_{j}}|=|F_{i}(x_{i_{j}})|\leq\|\nabla F_{i}\|\cdot|x_{i_{j}}|\leq
6c_{0}\delta_{0},$

establishing (52). ∎

Given a $C^{k}$ manifold $\mathcal{M}$, one may define a function
$D:\mathbb{R}^{n}\to\mathbb{R}$ by

$D(x)=d(x,\mathcal{M})^{2},$

where $d(x,A)=\inf_{y\in A}d(x,y)$ for any subset $A\subset\mathbb{R}^{n}$.

The following theorem is adapted from Theorem 13 and Observation 3 in [14]. It
says, roughly, that given a $C^{k}$ function $F:\mathbb{R}^{n}\to\mathbb{R}$
satisfying similar properties as $D$ above, there exists a $C^{k-2}$ manifold
$\mathcal{N}$ such that $F$ approximates the squared distance function from
$\mathcal{N}$. We will also be interested in quantitative bounds on
$\mathcal{N}$, in particular with regards to the reach.

######  Theorem 6.2.

Suppose $F:B_{n}(0,1)\to\mathbb{R}$ is a $C^{k}$ function satisfying

  1. 1.

$|\partial^{\alpha}_{x,y}F(x,y)|\leq C_{0}$

for $(x,y)\in B_{n}(0,1)$ and $|\alpha|\leq k$.

  2. 2.

For $(x,y)\in\mathbb{R}^{d}\times\mathbb{R}^{n-d}$, we have

$c_{1}[|y|^{2}+\rho^{2}]\leq F(x,y)+\rho^{2}\leq C_{1}[|y|^{2}+\rho^{2}],$

where $0<\rho<c$ and $c$ is a small enough constant depending only on
$C_{0},c_{1},C_{1},k,n$.

Then, there exist constants $c_{2}$ and $C$, depending only on
$C_{0},c_{1},C_{1},k,n$ such that the following hold:

  1. 1.

For $z\in B_{n}(0,c_{2})$, let $N(z)$ be the subspace spanned by the
eigenvectors of the Hessian $\partial^{2}F(z)$ corresponding to the largest
$n-d$ eigenvalues and let $\Pi_{hi}(z):\mathbb{R}^{n}\to N(z)$ be the
orthogonal projection from $\mathbb{R}^{n}$ onto $N(z)$. Then,

$|\partial^{\alpha}\Pi_{hi}(z)|\leq C$

for $z\in B_{n}(0,c_{2})$ and $|\alpha|\leq k-2$. In particular, $N(z)$ is a
$C^{k-2}$ function of $z$.

  2. 2.

There exists a map $\Psi:B_{d}(0,c_{2})\to B_{n-d}(0,c_{2})$ such that

$\\{z\in B_{d}(0,c_{2})\times B_{n-d}(0,c_{2})|\Pi_{hi}(z)\nabla
F(z)=0\\}=\\{(x,\Psi(x))|x\in B_{d}(0,c_{2})\\}.$

Furthermore, $\Psi$ satisfies the quantitative bounds

$|\Psi(0)|\leq C\rho;|\partial^{\alpha}\Psi|\leq C^{|\alpha|}$ (53)

on $B_{d}(0,c_{2})$ and for $1\leq|\alpha|\leq k-2$.

In particular, $\\{z\in B_{d}(0,c_{2})\times
B_{n-d}(0,c_{2})|\Pi_{hi}(z)\nabla F(z)=0\\}$ is a $C^{k-2}$ graph.

  3. 3.

Any $z\in B_{n}(0,c_{2})$ may be expressed uniquely in the form
$(x,\Psi(x))+v$ where $x\in B_{d}(0,c_{2})$ and
$v\in\Pi_{hi}(x,\Psi(x))\mathbb{R}^{n}\cap B_{n}(0,c_{2})$. Furthermore, $x$
and $v$ are both $C^{k-2}$ functions of $z\in B_{n}(0,c_{2})$ with derivatives
up to order $k-2$ bounded in absolute value by a constant $C$.

Let $\theta\in C^{\infty}(\mathbb{R}^{d},[0,1])$ be a bump function strictly
positive on $B_{d}(0,1)$, equal to 0 on the complement of $B_{d}(0,1)$, and
satisfying the bounds

$|\partial^{\alpha}\theta(x)|\leq C$ (54)

for $x\in\mathbb{R}^{d},|\alpha|\leq k$.

Define $G:\bigcup_{i}\mathbf{cyl}^{5}_{i}\to\mathbb{R}$ by

$G(z)=\frac{\sum_{\mathbf{cyl}_{i}\ni
z}\phi(o_{i}^{-1}(z))\theta(\Pi_{d}(o_{i}^{-1}(z))/(6\delta_{0}))}{\sum_{\mathbf{cyl}_{i}\ni
z}\theta(\Pi_{d}(o_{i}^{-1}(z))/(6\delta_{0}))}$ (55)

Let

$\mathcal{A}_{i}:=o_{i}(\delta_{0}B_{d})\subset\mathbf{cyl}_{i};$
$\mathcal{A}^{r}_{i}:=o_{i}(r\delta_{0}B_{d})\subset\mathbf{cyl}^{r}_{i}$

for $r>0$;

$\mathcal{A}^{r}:=\bigcup_{i}\mathcal{A}^{r}_{i};$

and, given $R>0$,

$\mathcal{A}^{r}_{R}:=\\{z\in\mathbb{R}^{n}:\inf_{\bar{z}\in\mathcal{A}^{r}}|z-\bar{z}|<R\\}.$

The following is based on Lemma 16 in [14].

######  Lemma 6.3.

Let $G$ be as in (55). Then, for all $1\leq i\leq N$ and
$w\in\mathcal{A}^{4}_{i}$, the function $G_{w}:B_{n}\to\mathbb{R}$ by

$G_{w}(z)=\frac{G(w+\delta_{0}Q_{i}(z))}{\delta_{0}^{2}}$ (56)

satisfies

  1. 1.

$|\partial^{\alpha}_{x,y}G_{w}(x,y)|\leq C$

for $(x,y)\in B_{n}$, $|\alpha|\leq k$, and a constant $C<\infty$.

  2. 2.

For $(x,y)\in B_{n}$, we have

$c_{1}[|y|^{2}+c_{0}^{2}]\leq G_{w}(x,y)+c_{0}^{2}\leq
C_{1}[|y|^{2}+c_{0}^{2}],$ (57)

where $c_{0}$ is independent of any of $C_{0},c_{1},C_{1},k,d,n$.

###### Proof.

By standard application of the chain rule, it suffices to show

$|\partial^{\alpha}G(z)|\leq C\delta_{0}^{2-|\alpha|}$ (58)

for all $|\alpha|\leq m+2$ and $z\in B(w,\delta_{0})$.

By definition,

$|\partial^{\alpha}\phi(o_{i}^{-1}(z))|\leq\delta_{0}^{2-|\alpha|}$

whenever $\theta(\Pi_{d}(o_{i}^{-1}(z^{\prime}))/(6\delta_{0}))$ is not
uniformly zero for $z^{\prime}$ in a neighborhood of $z$ (that is, whenever
$\phi(o_{i}^{-1}(z))$ is relevant for computations of $G$ and
$\partial^{\alpha}G$).

Define, for $z\in\cup_{i}\mathbf{cyl}^{5}_{i},$

$\theta_{i}(z)=\frac{\theta(\Pi_{d}(o_{i}^{-1}(z))/(6\delta_{0}))}{\sum_{\mathbf{cyl}_{j}\ni
z}\theta(\Pi_{d}(o_{j}^{-1}(z))/(6\delta_{0}))}$

so that

$G(z)=\sum_{\mathbf{cyl}_{i}}\phi(o_{i}^{-1}(z))\theta_{i}(z).$ (59)

By repeated applications of the chain rule,

$|\partial^{\alpha}\theta_{i}(z)|\leq C\delta_{0}^{-|\alpha|};$

thus, by multiple applications of the product rule to (59) and (45), we obtain
(58).

To prove the second claim, let $(x,y)\in B_{n}(0,1)$ and
$z^{\prime}=w+\delta_{0}Q_{i}(x,y)\in\mathbf{cyl}_{i}^{5}$. Observe that
$|y|=d(z^{\prime},5\mathcal{A}_{i})/(2\delta_{0})$

Observe that, for $z\in\mathbf{cyl}_{i}^{5}$, $G(z)$ is a convex combination
of
$d(z,5\mathcal{A}_{i})^{2},d(z,5\mathcal{A}_{i_{1}})^{2},\ldots,d(z,5\mathcal{A}_{i_{|S_{i}|}})^{2}$,
where
$S_{i}=\\{\mathbf{cyl}^{5}_{i_{1}},\ldots,\mathbf{cyl}^{5}_{i_{|S_{i}|}}\\}$
is the set of cylinders which intersect $\mathbf{cyl}^{5}_{i}$ and $i_{1}=i$.

It follows from Lemma 6.1 that for each $1\leq j\leq|S_{i}|$, there is a rigid
motion of $\mathbf{cyl}^{5}_{i_{j}}$ such that the image of
$5\mathcal{A}_{i_{j}}$ is contained in $o_{i}(\mathbb{R}^{d})$; furthermore,
this rigid motion moves no point of $\mathbf{cyl}^{5}_{i_{j}}$ more than
$Cc_{0}\delta_{0}$. Thus,
$|d(z,5\mathcal{A}_{i_{j}})-d(z,5\mathcal{A}_{i})|\leq Cc_{0}\delta_{0}$ for
all $1\leq j\leq|S_{i}|$.

We now see that, for some $\lambda_{1},\ldots,\lambda_{|S_{i}|}\in[0,1]$
satisfying $\sum_{j}\lambda_{j}=1$,

$\displaystyle G(z^{\prime})$
$\displaystyle=\sum_{j}\lambda_{j}d(z^{\prime},5\mathcal{A}_{i_{j}})^{2}$
$\displaystyle\leq\sum_{j}\lambda_{j}[d(z^{\prime},5\mathcal{A}_{i})+Cc_{0}\delta_{0}]^{2}$
$\displaystyle\leq[2\delta_{0}|y|+Cc_{0}\delta_{0}]^{2}.$

Thus,

$\displaystyle G_{w}(x,y)$ $\displaystyle\leq[|y|+Cc_{0}]^{2}\leq
4|y|^{2}+Cc_{0}^{2}$ $\displaystyle G_{w}(x,y)+c_{0}^{2}$ $\displaystyle\leq
C(|y|^{2}+c_{0}^{2}),$

establishing the right-hand-side of (57).

Since $z^{\prime}\in\mathbf{cyl}^{5}_{i}$ and $|S_{i}|\leq C$ by (45),
$\lambda_{1}=\theta_{1}(z^{\prime})\geq\sup_{x^{\prime}\in
B_{d}(0,9/10)}\theta(x^{\prime})/C>0$ and

$G(z^{\prime})\geq\frac{1}{C^{\prime}}[2\delta_{0}|y|]^{2},$

hence

$G_{w}(x,y)\geq\frac{1}{C^{\prime\prime}}|y|^{2}$

and

$G_{w}(x,y)+c_{0}^{2}\geq\frac{1}{C^{\prime\prime}}(|y|^{2}+c_{0}^{2})$

establishing the left-hand-side of (57).

∎

The following lemma is based on Lemma 15 in [14].

######  Lemma 6.4.

Let

$\mathcal{M}_{put}:=\\{z\in\mathcal{A}^{4}_{c_{2}\delta_{0}/2}|\Pi_{hi}(z)\nabla
G(z)=0\\}.$

Then, $\mathcal{M}_{put}$ is a $d$-dimensional, $C^{m+2}$ submanifold of
$\mathbb{R}^{n}$ such that the following holds for some constant $c_{put}$: If
$z\in\mathbf{cyl}_{i}^{4}$ for some $i$ and
$d(z,\mathcal{M}_{put})<c_{2}\delta_{0}/4$, then $z$ has a unique closest
point in $\mathcal{M}_{put}$.

Furthermore, if $z\in\mathcal{A}^{4}_{i}$ for some $1\leq i\leq N$, then
$d(z,\mathcal{M}_{put})<c_{2}\delta_{0}/2$.

###### Proof.

We first show that $\mathcal{M}_{put}$ is indeed a manifold. Given
$z^{\prime}\in\mathcal{M}_{put}$, there exists $1\leq i\leq N$ and
$w\in\mathcal{A}_{i}$ such that $z^{\prime}=w+\delta_{0}Q_{i}(z)$ for some
$z\in B_{n}$. Define $G_{w}$ as in (56). By taking a rigid motion in
$\mathbb{R}^{n}$, we may assume $w=0$ and $Q_{i}=I_{n}$. By Lemma 6.3, $G_{w}$
satisfies the hypotheses of Theorem 6.2 with $\rho=c_{0}$ and $k=m+2$; thus,
by (56), there exists

$\Psi:B_{d}(0,c_{2}\delta_{0})\to B_{n-d}(0,c_{2}\delta_{0})$

such that

$\\{z\in B_{d}(0,c_{2}\delta_{0})\times
B_{n-d}(0,c_{2}\delta_{0})|\Pi_{hi}(z)\nabla G(z)=0\\}=\\{(x,\Psi(x))|x\in
B_{d}(0,c_{2}\delta_{0})\\}$ (60)

and

$|\Psi(0)|\leq Cc_{0}\delta_{0}^{2}\leq
Cc_{0}\delta_{0};|\partial^{\alpha}\Psi|\leq
C^{|\alpha|}\delta_{0}^{2-|\alpha|}$

on $B_{d}(0,c_{2}\delta_{0})$ and for $1\leq|\alpha|\leq m$.

We take $c_{0}$ small enough that $Cc_{0}<c_{2}/4$ so that
$|\Psi(0)|<c_{2}\delta_{0}/4$ and

$d(z,\mathcal{A}^{4})<c_{2}\delta_{0}/4\text{ for all }z\in\mathcal{M}_{put}.$
(61)

Furthermore, any $z\in B_{n}(0,c_{2}\delta_{0}/2)$ may be expressed uniquely
in the form $(x,\Psi(x))+v$ where $x\in B_{d}(0,c_{2}\delta_{0})$ and
$v\in\Pi_{hi}(x,\Psi(x))\mathbb{R}^{n}\cap B_{n}(0,c_{2}\delta_{0})$.
Furthermore, $x$ and $v$ are both $C^{m}$ functions of $z\in
B_{n}(0,c_{2}\delta_{0})$ with derivatives $\partial^{\alpha}$ up to order $m$
bounded in absolute value by a constant $C\delta_{0}^{2-|\alpha|}$.

If $z\in\mathbf{cyl}_{i}^{4}$ and $d(z,\mathcal{M}_{put})<c_{2}\delta_{0}/4$,
by (61), $d(z,\mathcal{A}^{4})<c_{2}\delta_{0}/2$. Thus, $z$ has a unique
closest point in the graph of $\Psi$, and therefore on $\mathcal{M}_{put}$.

Since $\mathcal{A}_{c_{2}\delta_{0}/2}$ is open, there exists $\eta>0$ such
that $B_{n}(z^{\prime},\eta)\subset\mathcal{A}_{c_{2}\delta_{0}/2}$. We have
shown that $\mathcal{M}_{put}$ agrees with the graph of $\Psi$ on
$B_{n}(z^{\prime},\eta)$; thus, $\mathcal{M}_{put}$ is a manifold.

Let $w\in\mathcal{A}^{4}_{i}$. By repeating the application of Theorem 6.2 to
$G_{w}$, we find that $\mathcal{M}_{put}$ is locally of the form

$\\{w+Q_{i}(x,\Psi^{\prime}(x)):x\in B_{d}(0,c_{2}\delta_{0})\\}$

where $|\Psi^{\prime}(0)|<c_{2}\delta_{0}/2$. Thus,
$d(w,\mathcal{M}_{put})<c_{2}\delta_{0}/2$.

∎

Let $\mathcal{M}\subset\mathbb{R}^{n}$ be a $C^{m+2}$, $d$-dimensional
manifold and $D$ be an open set containing $\mathcal{M}$. We say a $C^{m+2}$
map $\pi:D\to\mathcal{M}$ is a disc bundle if for any $z\in\mathcal{M}$,
$\pi(z)=z$ and $D_{z}:=\pi^{-1}(z)$ is isometric to $B_{n-d}(0,r)$, where $r$
is independent of $z$.

We call a $C^{m}$ map $s:\mathcal{M}\to D$ a section (or global section) of
$D$ if for any $z\in\mathcal{M},s(z)\in D_{z}$. We say $s$ is a local section
if it is from $U\to D$, where $U\subset\mathcal{M}$ is an open set.

###### Proof of Theorem 3.4.

We are now ready to complete the proof of Theorem 3.4. Let

$D_{put}=\\{z+rv:-1\leq r\leq 1,z\in\mathcal{M}_{put},v\in
N(z),\|v\|\leq\frac{1}{2}c_{2}\delta_{0}\\},$

where $N(z)$ is the $(n-d)$ dimensional subspace of $\mathbb{R}^{n}$
consisting of vectors orthogonal to $T_{z}\mathcal{M}_{put}$.

Consider the graphs

$G_{i}:=\\{z_{i}+Q_{i}(x,F_{i}(x)):x\in
B_{d}(0,4\delta_{0})\\}\subset\mathbf{cyl}_{i}^{4},1\leq i\leq N.$

By (43) and taking $c_{0}$ sufficiently small,

$d(z,\mathcal{A}^{4}_{i})<c_{0}\delta_{0}<c_{2}\delta_{0}/4\text{ for all
}z\in G_{i},1\leq i\leq N.$ (62)

By Lemma 6.4, each point of $\mathcal{A}^{4}_{i}$ lies within distance
$c_{2}\delta_{0}/4$ of $\mathcal{M}_{put}$. By the triangle inequality and
(62),

$d(z,\mathcal{M}_{put})<c_{2}\delta_{0}/2.$

Thus, each point of $G_{i}$ has a unique closest point on $\mathcal{M}_{put}$,
so $G_{i}$ is the graph of a local section $J_{i}$ of the disc bundle
$D_{put}$ on an open set $U_{i}\subset\mathcal{M}_{put}$ by Lemma 6.4.

We construct a smooth partition of unity on $\mathcal{M}_{put}$ as follows.
Let

$\tilde{\theta}_{j}(z)=\begin{cases}\theta\left(\frac{\Pi_{d}(o_{j}^{-1}z)}{4\delta_{0}}\right)&z\in\mathbf{cyl}^{4}_{j}\\\
0&\text{else},\end{cases}$

where $\theta$ is as in (54). Define

$\theta_{j}(z)=\frac{\tilde{\theta}_{j}(z)}{\sum_{j}\tilde{\theta}_{j}(z)}.$

Define

$J(x):=\sum_{j}\theta_{j}(x)J_{i}(x).$ (63)

Ideally, $J$ would be a global section of $D_{put}$; however, it may not be
well-defined near the boundary of $\mathcal{M}_{put}$ (as all
$\tilde{\theta}_{j}$ may be 0). Instead, we take $J$ as a local section of
$D_{put}$ on $\mathcal{M}_{put}\cap(\cup_{i}\mathbf{cyl}_{i}^{3})$ and observe
that

$G:=\\{x+J(x):x\in\mathcal{M}_{put}\cap(\cup_{i}\mathbf{cyl}_{i}^{3})\\}$

is a $C^{m}$ manifold.

Let $z\in E$ and write $z=x+v$, where $x\in\mathcal{M}_{put}$ and $v\in N(x)$.
If $J_{i}$ is defined on $x$, $z\in G_{i}$, as $G_{i}$ agrees with $E$ on
$B_{n}(z_{i},20\delta_{0})$. Thus, $J_{i}(x)=v$, $J(x)=v$, and $z\in G$.

Similarly, since by (46) each $G_{i}$ must agree with
$G_{R_{x_{0}}}(F_{x_{0}})$ on $B(x_{0},\delta_{0})$, $G$ must as well. Observe
that even if one takes $Q\in O(n)$ other than $R_{x_{0}}$, the jet at $x_{0}$
will just be realized in a different coordinate system.

Let $\mathbf{1}_{i}$ denote the indicator function of $\mathbf{cyl}_{i}^{2}$
and fix $\varphi\in C^{\infty}(\mathbb{R}^{n},\mathbb{R})$ such that
$0\leq\varphi\leq 1$, $\varphi\equiv 1$ on $B(0,\delta_{0}/100)$ and
$\varphi\equiv 0$ on $\mathbb{R}^{n}\setminus B(0,\delta_{0}/50)$.

Define $I=\sum_{i}\mathbf{1}_{i}$. The convolution $I*\varphi$ satisfies the
following properties:

  * •

$I*\varphi\in C^{\infty}$.

  * •

$I*\varphi\equiv 1$ on $\cup_{i}\mathbf{cyl}^{99/50}_{i}$.

  * •

$I*\varphi\equiv 0$ on
$\mathbb{R}^{n}\setminus(\cup_{i}\mathbf{cyl}_{i}^{101/50})$.

Thus, by Sard’s theorem, there exists $0<t<1$ such that
$R:=\varphi^{-1}([t,1])$ satisfies the following properties:

  * •

$\cup_{i}\mathbf{cyl}^{2}_{i}\subset
R\subset\cup_{i}\mathbf{cyl}_{i}^{101/50}$.

  * •

$R$ has $C^{\infty}$ boundary $\partial R$.

To see the first point, observe that $\varphi^{-1}((0,1])$ is an open set
which contains the closure of $\cup_{i}\mathbf{cyl}_{i}^{2}$. By taking $t$
small enough (which we can since Sard’s theorem says the $C^{\infty}$ boundary
holds for almost every $t$), we can ensure this first condition.

We would like to take $\mathcal{M}\cap R$ as the desired $C^{m}$ manifold with
boundary. Clearly, $E\subset\mathcal{M}\cap R$ since
$E\subset\cup_{i}\mathbf{cyl}_{i}\subset R$. However, we still need to ensure
that $\mathcal{M}\cap R$ has a $C^{m}$ boundary. This will hold if for all
$z\in\mathcal{M}\cap\partial R$, $T_{z}\mathcal{M}+T_{z}\partial
R=\mathbb{R}^{n}$.

There is no guarantee that this holds in general. However, we may modify the
constructions of $\mathcal{M}$ and $R$ as follows to ensure that it does.

Observe that $\mathcal{M}_{put}$ is actually a $C^{\infty}$ manifold; thus, by
(47) pasting together local sections via a $C^{\infty}$ partition of unity
creates a global section whose graph is a $C^{\infty}$ manifold at a distance
at least $2\delta_{0}$ from $E$ (in particular at the intersection with
$\partial R$).

By Theorem 2.3, there exists arbitrarily small $v\in\mathbb{R}^{n}$ such that
$v+\partial R$ has transverse intersection with $\mathcal{M}$. Thus, by
Theorem 2.2, for such a $v$, $\mathcal{M}\cap(v+R)$ is the desired $C^{m}$
manifold with boundary, as its boundary $\mathcal{M}\cap(v+\partial R)$ is a
$C^{\infty}$ manifold.

∎

###### Proof of Theorem 3.5.

Recall the end of the proof of Theorem 3.4, when the local sections of
$\mathcal{M}_{put}$ are pasted together ((63)). We are given a finite number
of $F_{i}\in C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$, each such that
$(Q_{i},J_{x_{Q_{i}}}F_{i})\in H(x)$ whenever $x\in E\cap(z_{i},\delta_{0})$.
To prove Theorem 3.5, we simply must show that the pasting process preserves
this property; the case of $Q\neq Q_{i}$ is taken care of by the consistency
of $(H(x))_{x\in E}$.

Let $z_{0}\in E$ and consider the pasting of local sections via the smooth
partition of unity at $z_{0}$. By a rigid motion, we may assume that the
closest point of $\mathcal{M}_{put}$ to $z_{0}$ is the origin $\vec{0}$ in
$\mathbb{R}^{n}$ and that

$T_{\vec{0}}\mathcal{M}_{put}=\mathbb{R}^{d}.$ (64)

The section $J(x)$ may be viewed as the graph of a $C^{1}$ function
$F:\mathbb{R}^{d}\to\mathbb{R}^{n-d}$. Our goal is to compute $\nabla F(0)$
and show the resulting jet lies in $H(z_{0})$.

Let $X(z)$ denote the closest point of $\mathcal{M}_{put}$ and $V(z)=z-X(z)$,
as in Theorem 6.2. Then we may write

$(w,F(w))=X((w,F(w))+V((w,F(w)).$

Choose $w_{i}$ so that

$(w_{i},F_{i}(w_{i}))=X((w,F(w))+V((w_{i},F_{i}(w_{i})).$ (65)

From the pasting process,

$V((w,F(w))=\sum_{i=1}^{I}\Theta_{i}(w_{i})V((w_{i},F_{i}(w_{i})),$ (66)

where $F_{1},\ldots,F_{I}$ are the (possibly relabeled) local interpolants on
the cylinders containing $z_{0}$, $(\Theta_{i})_{i=1}^{I}$ is part of a smooth
partition of unity defined by

$\Theta_{i}(w_{i})=\theta_{i}(X((w_{i},F_{i}(w_{i})))),$

where the $\theta_{i}$ are the smooth partition of unity on
$\mathcal{M}_{put}$ originally used for pasting.

By (64) and the twice differentiablity of $\mathcal{M}_{put}$, we have, for
$z=(x,y)$ near the origin,

$X(z)=(x,0_{n-d})+O(|z|^{2})\text{ and }V(z)=(0_{d},y)+O(|z|^{2}),$ (67)

Observe that applying (67) to (65), we have

$w_{i}=w+O(|w_{i}|^{2}),$

or

$w=w_{i}+O(|w|^{2})$ (68)

by the inverse function theorem. Similarly,

$\Theta_{i}(w_{i})=\Theta_{i}(w)+O(|w|^{2}).$

Thus, by (66), the differentiability of $F$ and the $F_{i}$’s, and (68)

$\displaystyle F(w)$
$\displaystyle=\sum_{i=1}^{I}\Theta_{i}(w)F_{i}(w)+O(|w|^{2})+O\left(\sum_{i=1}^{I}|w_{i}|^{2}\right)$
$\displaystyle=\sum_{i=1}^{I}\Theta_{i}(w)F_{i}(w)+O(|w|^{2}).$

By a standard application of the product rule, and the fact that
$\sum_{i=1}^{I}\nabla\Theta_{i}(0)=0$

$\nabla F(0)=\sum_{i=1}^{I}\Theta_{i}(0)\nabla F_{i}(0).$ (69)

The jet $F(0)+\nabla F(0)\cdot x\in H^{I_{n}}(z_{0})$, as $H^{I_{n}}(z_{0})$
is an affine space, hence closed under convex combinations. ∎

##  7 On Continuous Selections

Given a set $A$, we let $2^{A}$ denote its power set. If $X$ and $Y$ are
topological spaces, we say a map $\varphi:X\to 2^{Y}$ is lower semi-continuous
when for every open $V\subset Y$, the set $\\{x\in X:\varphi(x)\cap
V\neq\emptyset\\}$ is an open subset of $X$. We call a function $f:X\to Y$
satisfying $f(x)\in\varphi(x)$ for all $x\in X$ a selection for $\varphi$ and
$g:A\to Y$ satisfying $f(x)\in\varphi(x)$ for all $x\in A\subset X$ a
selection for $\varphi$ on $A$.

Given a topological space $X$, and a cover $\mathcal{U}$ of $X$, a refinement
of $\mathcal{U}$ is a cover $\mathcal{V}$ such that for all $V\in\mathcal{V}$,
there exists $U\in\mathcal{U}$ such that $V\subset U$. The covering dimension
of $X$, denoted $\dim X$, is the smallest number $n$ such that every finite
open covering $\mathcal{U}$ of $X$ has a finite, open refinement $\mathcal{V}$
such that each $x\in X$ is in at most $n+1$ elements of $\mathcal{V}$. For a
closed subset $A\subset X$, we let the covering dimension of $X\setminus A$ in
$X$, denoted $\dim_{X}(X\setminus A)$, be the smallest $n$ such that $\dim
C\leq n$ for all closed $C\subset X\setminus A$.

We will use the fact that covering dimension is a topological invariant, that
$\dim Y\leq\dim X$ for closed $Y\subset X$, that $[0,1]^{d}$ has covering
dimension $d$ (a result known as Lebesgue’s covering theorem [17]), and the
countable sum theorem (see Theorem 3.6 of [4], for instance).

######  Theorem 7.1 (Countable Sum Theorem).

Suppose $X=\cup_{i=1}^{\infty}X_{i}$, where $X$ is a normal space and each
$X_{i}$ is a closed subspace of $X$ with $\dim X_{i}\leq n$. Then $\dim X\leq
n$.

We say $X$ is $C^{n}$-contractible if every continuous image of $S^{m}$ in $X$
is contractible. In other words, $\pi_{m}(X)=\\{0\\}$ for all $m\leq n$, where
$\pi_{m}(X)$ refers to the $m$-th homotopy group of $X$. Given
$\mathcal{S}\subset 2^{X}$, we say $\mathcal{S}$ is equi-$LC^{n}$ if for every
$y\in\cup_{S\in\mathcal{S}}S$ and open $U\ni y$, there exists open $V\ni y$
such that, for every $S\in\mathcal{S}$, every continuous image of an
$m$-sphere ($m\leq n$) in $S\cap V$ is contractible in $S\cap U$.

The following theorem follows from Theorem 1.2 in [19].

######  Theorem 7.2.

Let $X$ be a paracompact space, $A\subset X$ closed with $\dim_{X}(X\setminus
A)\leq D+1$, $Y$ a complete metric space, $\mathcal{S}\subset 2^{Y}$
equi-$LC^{D}$, and $\varphi:X\to 2^{Y}$ lower semi-continuous.

If every $S\in\mathcal{S}$ is $C^{D}$-contractible, then given any selection
$f:A\to Y$ for $\varphi$ on a closed $A\subset X$, there exists a selection
$F:X\to Y$ for $\varphi$ such that $F(x)=f(x)$ for $x\in A$.

######  Lemma 7.3.

Let $E\subset\mathbb{R}^{n}$ be compact and let $(H(x))_{x\in E}$ be a
nontrivial, Glaeser-stable, proper M-bundle. Then,

$\dim E\leq d.$

###### Proof.

Similar to the start of the proof of Theorem 3.4, we may apply Lemma 4.4 to
obtain, for each $x\in X$, $\delta_{x}>0$, $Q_{x}\in O(n)$, $F_{x}\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that

$G_{Q_{x}}(F_{x})\cap B_{n}(x,2\delta_{x})\supset E\cap B_{n}(x,2\delta_{x}).$

By compactness of $E$, there exist $x_{1},\ldots,x_{n}\in E$ such that

$E\subset\tilde{E}:=\bigcup_{1\leq i\leq n}G_{Q_{i}}(F_{i})\cap
B_{n}(x_{i},\delta_{i}),$

where $Q_{i}=Q_{x_{i}},F_{i}=F_{x_{i}}$, and $\delta_{i}=\delta_{x_{i}}$.

By Lebesgue’s covering theorem and the fact covering dimension is a
topological invariant,

$\dim[G_{Q_{i}}(F_{i})\cap\overline{B_{n}(x_{i},\delta_{i})}]=d.$

By Theorem 7.1, $\dim\tilde{E}=d$; thus, $\dim E\leq\dim\tilde{E}\leq d$. ∎

######  Lemma 7.4.

Let $E\subset\mathbb{R}^{n}$ be compact and let $(H(x))_{x\in E}$ be a
nontrivial, Glaeser-stable, proper M-bundle. The mapping $\varphi:E\to
2^{Gr(n,d)}$ by $x\mapsto\hat{H}(x)$ is lower semi-continuous.

###### Proof.

Let $O\subset Gr(n,d)$ be open and suppose $O\cap\hat{H}(x)\neq\emptyset$ for
some $x\in E$. In particular, let $W_{0}\in O\cap\hat{H}(x)$.

Since $W_{0}\in\hat{H}(x)$, there exists $(Q,P)\in H(x)$ such that
$W_{0}=Q\oplus\nabla P(x_{Q})$ and choose $\epsilon>0$ such that

$B_{Gr}(W_{0},\epsilon):=\\{W\in
Gr(n,d):\text{dist}(W,W_{0})<\epsilon\\}\subset O.$

By Lemma 4.4, there exist $\delta>0$ and $F\in
C^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$ such that $G_{Q}(F)\supset E\cap
B(x,\delta)$ and $J_{x_{Q}}F=P$, thus $W=Q\oplus\nabla F(x_{Q})$. Furthermore,

$J_{y_{Q}}F\in H^{Q}(y)\text{ for all }y\in B(x,\delta).$ (70)

The mapping which takes an $(n-d)\times d$ matrix $M$ to $W_{M}:=Q\oplus M\in
Gr(n,d)$ is continuous in $M$. Thus, there exists $\eta>0$ such that
$\|M-\nabla F(x_{Q})\|<\eta$ implies $W_{M}\in B_{Gr}(W_{0},\epsilon)$.

Since $\nabla F(x_{Q})$ is continuous in $x_{Q}$, there exists
$0<\delta^{\prime}<\delta$ such that $|x_{Q}^{\prime}-x_{Q}|<\delta^{\prime}$
implies $\|\nabla F(x_{Q})-\nabla F(x^{\prime}_{Q})\|<\eta$. Observe that
$(Q,\nabla F(x^{\prime}_{Q}))\in H(x^{\prime})$ by (70), so $Q\oplus\nabla
F(x^{\prime}_{Q})\in\hat{H}(x^{\prime})$.

Suppose $|x^{\prime}-x|<\delta^{\prime}$ for $x^{\prime}\in E$. Then,
$|x_{Q}^{\prime}-x_{Q}|<\delta$, so by the above reasoning there exists
$W^{\prime}\in\hat{H}(x^{\prime})$ (namely, $Q\oplus\nabla F(x^{\prime}_{Q})$)
such that $W^{\prime}\in B(W,\epsilon)\subset O$. In particular,
$\hat{H}(x^{\prime})\cap O\neq\emptyset$, proving our claim. ∎

In the following lemma, we restrict ourselves to M-bundles arising naturally
from manifold extension, with no further constraints. However, one may be able
to replicate our analysis for other M-bundles analogously.

######  Lemma 7.5.

Let $E\subset\mathbb{R}^{n}$ be compact and consider the M-bundle
$(H(x))_{x\in E}$, where $H(x)=H_{l^{*}}(x)$, the $l^{*}$-th Glaeser
refinement of (14).

Let $x\in E$. Then, there exist orthonormal
$v_{1},\ldots,v_{l}\in\mathbb{R}^{n}$ such that

$\hat{H}(x)=\\{W\in Gr(n,d):v_{1},\ldots,v_{l}\in W\\}.$ (71)

In particular, $\hat{H}(x)$ is a submanifold of $Gr(n,d)$ which is
homeomorphic to $G(n-l,d-l)$.

###### Proof.

For $Q\in O(n)$ define

$\hat{H}^{Q}(x):=\\{Q\oplus\nabla P(x_{Q}):P\in H^{Q}(x)\\}\subset\hat{H}(x).$
(72)

Define $\varphi_{Q}:\mathbb{R}^{(n-d)\times d}\to Gr(n,d)$ by

$\varphi_{Q}(M)=Q\oplus M,$

the graph of the linear map represented by $M$ in $Q$-coordinates. This is
well known to be a coordinate chart for $Gr(n,d)$ (see Example 1.36 of [18],
for example).

We claim that

$\hat{H}^{Q}(x)=\hat{H}(x)\cap\varphi_{Q}(\mathbb{R}^{(n-d)\times d}).$

By definition, $\hat{H}^{Q}(x)\subset\varphi_{Q}(\mathbb{R}^{(n-d)\times d})$,
so by (72),
$\hat{H}^{Q}(x)\subset\hat{H}(x)\cap\varphi(\mathbb{R}^{(n-d)\times d})$.

Now let $W\in\hat{H}(x)\cap\varphi_{Q}(\mathbb{R}^{(n-d)\times d})$ and choose
$(Q^{\prime},P^{\prime})\in H(x)$ such that $Q^{\prime}\oplus\nabla
P^{\prime}(x_{Q^{\prime}})=W$. Since $(H(x))_{x\in E}$ is consistent and $W$
is the graph of a function in $Q$-coordinates,
$(Q,P):=J_{x,Q}(Q^{\prime},P^{\prime})\in H(x)$ and by Remark 3.2,
$Q\oplus\nabla P(x_{Q})=W$. Thus, $W\in\hat{H}^{Q}(x)$, proving our claim.

Fix $Q\in O(n)$ such that $H^{Q}(x)$ is nonempty. Since $H^{Q}(x)$ is an
affine space and evaluation of the gradient of a polynomial at $x_{Q}$ is a
linear map, $\varphi_{Q}^{-1}(\hat{H}(x))\subset\mathbb{R}^{(n-d)\times d}$ is
an affine space. This is consistent with (71), but we will need more.

As in the beginning of the proof of Lemma 4.4, choose $\delta>0$ so that all
$y\mapsto y_{Q}$ is injective on $E\cap B_{n}(x,\delta)$. Let
$E^{\prime}=\\{y_{Q}\in\mathbb{R}^{d}:y\in B_{n}(x,\delta)\\}$ and
$f=(f_{1},\ldots,f_{n-d})$, where $f:E^{\prime}\to\mathbb{R}^{n-d}$ by
$y_{Q}\mapsto y_{Q}^{\perp}$.

Observe that Glaeser refinement of M-bundles treats the different components
$P_{1},\ldots,P_{n-d}$ of polynomials
$P=(P_{1},\ldots,P_{n-d})\in\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}^{n-d})$
independently.

That is, one may compute $H(x)$ from $H_{0}(x)$ through separate Glaeser
refinements of the M-bundles $(H^{i}(t))_{t\in E^{\prime}}$, where

$(H^{i})^{Q}(y):=\\{P\in\mathcal{P}^{m}(\mathbb{R}^{d},\mathbb{R}):P(y_{Q})=f_{i}(y_{Q})\\}.$

It is a standard result that the fibers of (usual) bundles defined in this way
are translates of ideals in $\mathcal{P}(\mathbb{R}^{d},\mathbb{R})$, and that
this property is preserved under Glaeser refinement, provided the fiber
remains nonempty (see Lemma 2.1 in [7]). Furthermore, this ideal depends only
on $E^{\prime}$. This result carries over trivially to the case of M-bundles,
thus

$(H^{i})^{Q}(x)=g_{i}(x)+I(x),$

where $g_{i}(x)\in\mathcal{P}(\mathbb{R}^{d},\mathbb{R})$ and
$I(x)\subset\mathcal{P}(\mathbb{R}^{d},\mathbb{R})$ is an ideal.

Let

$I^{\prime}(x)=\\{v\in\mathbb{R}^{d}:\exists P\in I(x),\nabla P(x)=v\\}.$

$I^{\prime}(x)$ is a subspace of $\mathbb{R}^{d}$; thus, there exist
orthonormal $w_{1},\ldots,w_{l}\in\mathbb{R}^{d}$ such that

$I^{\prime}(x)=Span(w_{1},\ldots,w_{l})^{\perp}=\\{v\in\mathbb{R}^{d}:\nabla
P(x)\cdot Span(w_{1},\ldots,w_{l})=0\text{ for all }P\in I(x)\\}.$

By rotation in $\mathbb{R}^{d}$, we may assume
$w_{1}=e_{1},\ldots,w_{l}=e_{l}$.

Thus,

$\\{\nabla P:P\in I(x)\\}=\\{v\in\mathbb{R}^{d}:\Pi_{l}v=0\\},$

where $\Pi_{l}:\mathbb{R}^{d}\to\mathbb{R}^{l}$ is the orthogonal projection
onto the first $l$ coordinates.

Furthermore,

$\\{\nabla
P:P\in\bar{H}^{j}(x)\\}=\\{v\in\mathbb{R}^{d}:\Pi_{l}v=(\partial_{1}f_{j}(x),\ldots,\partial_{l}f_{j}(x))\\}.$

Putting it all together,

$\hat{H}(x)=\\{I_{d}\oplus M:M\in\mathbb{R}^{(n-d)\times d},\text{ the first
}l\text{ columns of }M\text{ are
}(\partial_{1}f_{j}(x))_{j=1}^{n-d},\ldots(\partial_{l}f_{j}(x))_{j=1}^{n-d}\\}.$

Thus, (71) holds with

$v_{i}=\begin{bmatrix}e_{i}\\\
\partial_{i}f_{j}(x))_{j=1}^{n-d}\end{bmatrix},1\leq i\leq l$

for elements of $\varphi_{Q}(\mathbb{R}^{(n-d)\times d})$; it remains to
extend the conclusion to all of $\hat{H}(x)$.

Let $W_{0}\in\hat{H}(x)\setminus\varphi_{Q}(\mathbb{R}^{(n-d)\times d})$.
Choose $(Q^{\prime},P^{\prime})\in H(x)$ such that
$W_{0}=Q^{\prime}\oplus\nabla P(x_{Q^{\prime}})$. We may take $Q^{\prime}$ so
that $\|Q-Q^{\prime}\|<\eta$ for arbitrarily small $\eta>0$ since the
condition that $W_{0}\notin\varphi_{Q}(\mathbb{R}^{(n-d)\times d})$ is
equivalent to $W_{0}$ and $Q\mathbb{R}^{n-d}$ having nontrivial intersection.

Let $B\subset\mathbb{R}^{(n-d)\times d}$ be a small ball centered at $\nabla
P(x_{Q})$ for some $P\in H^{Q}(x)$ so that the entirety of $\nabla(H^{Q}(x))$
may be deduced from $B\cap\nabla(H^{Q}(x))$ and the fact $H^{Q}(x)$ is an
affine space. Taking $\eta$ and $B$ small enough,
$\varphi_{Q}(B)\subset\varphi_{Q^{\prime}}(\mathbb{R}^{(n-d)\times d})$. Thus,

$\hat{H}^{Q^{\prime}}(x)\cap\varphi_{Q}(B)=\\{W\in\varphi_{Q^{\prime}}(x):v_{1},\ldots,v_{l}\in
W\\}\cap\varphi_{Q}(B).$

By the prior reasoning for $Q$ applied to $Q^{\prime}$,

$\hat{H}^{Q^{\prime}}(x)=\\{W\in\varphi_{Q^{\prime}}(\mathbb{R}^{(n-d)\times
d}):v_{1}^{\prime},\ldots,v_{l^{\prime}}^{\prime}\in W\\}$

for some $v_{1}^{\prime},\ldots,v_{l^{\prime}}^{\prime}\in\mathbb{R}^{n}$.
However, by construction, we must have that the span of
$v_{1}^{\prime},\ldots,v_{l^{\prime}}^{\prime}$ is equal to the span of
$v_{1},\ldots,v_{l}$, and one may deduce whether $W_{0}\in\hat{H}(x)$ through
whether it contains $v_{1},\ldots,v_{l}$.

One may take $v_{1},\ldots,v_{l}$ to be orthonormal by the Gram-Schmidt
process. ∎

######  Lemma 7.6.

Let $E\subset\mathbb{R}^{n}$ be compact and let $(H(x))_{x\in E}$ be a
nontrivial, Glaeser-stable, proper, consistent M-bundle. Then,
$(\hat{H}(x))_{x\in E}$ is equi-$LC^{d-1}$.

###### Proof.

Let $W_{0}\in Gr(n,d)$ and let $U\ni W_{0}$ be an open subset of $Gr(n,d)$.
Choose $Q\in O(n)$ such that $W_{0}=Q(\mathbb{R}^{d})=Q\oplus 0$.

As in the proof of Lemma 7.5, we have a coordinate chart
$\varphi_{Q}:\mathbb{R}^{(n-d)\times d}\to Gr(n,d)$. Pick
$V=\varphi_{Q}(B_{(n-d)\times d}(0,\delta))$ where $\delta>0$ is chosen small
enough that $B_{(n-d)\times d}(0,\delta)\subset\varphi_{Q}^{-1}(U)$.

Let $x\in E$ such that $W_{0}\in\hat{H}(x)$. We would like to show that every
continuous image of $S^{m}$ ($m\leq n$) in $\hat{H}(x)\cap V$ is contractible
in $\hat{H}(x)\cap U$. Since $\varphi_{Q}$ is a homeomorphism and $V\subset
U$, it suffices to show that every continuous image of $S^{m}$ ($m\leq n$) in
$\varphi_{Q}^{-1}(\hat{H}(x))\cap B_{(n-d)\times d}(0,\delta))$ is
contractible in $\varphi_{Q}^{-1}(\hat{H}(x))\cap B_{(n-d)\times
d}(0,\delta))$. Again returning to the proof of Lemma 7.5, we know that
$\varphi_{Q}^{-1}(\hat{H}(x))$ is an affine space; thus the desired property
holds for its intersection with a ball. ∎

Totalling the results of this section we have removed the second condition
from Theorem 3.4 and proven the following:

######  Theorem 7.7.

Fix positive integers $m,n$ and let $E\subset\mathbb{R}^{n}$ be compact. Let
$\mathcal{H}_{l^{*}}=(H_{l^{*}}(x))_{x\in E}$ be the $l^{*}$-th Glaeser
refinement of (14).

Then there exists a $C^{m}$, 1-dimensional manifold $\mathcal{M}$ such that
$E\subset\mathcal{M}\subset\mathbb{R}^{n}$.

Furthermore, if $\Pi\in\hat{H}_{l^{*}}(x)$ and $\widehat{\mathcal{H}_{l^{*}}}$
has a section $f$ such that $f(x)=\Pi$, then for all $(Q,P)\in H_{l^{*}}(x)$
satisfying $Q\oplus\nabla P(x_{Q})=\Pi$, one may take $\mathcal{M}$ such that

$\mathcal{M}\cap B(x,\delta)=G_{Q}(F)\cap B(x,\delta)$

for some $\delta>0$ and $F\in C^{m}(\mathbb{R}^{1},\mathbb{R}^{n-1})$
satisfying $J_{x_{Q}}F=P$.

###### Proof.

We are in nearly in position to apply Theorem 3.4; all that is left is to show
that $(\hat{H}_{l^{*}}(x))_{x\in E}$ has a section. To do this, we would like
to apply Theorem 7.2 in the case where $X=E,A=\emptyset,D=d-1,Y=Gr(n,d)$, and
$\phi:x\mapsto\hat{H}(x)$.

By Lemma 7.5, each $\hat{H}_{l^{*}}(x)$ is homeomorphic to either $Gr(n,1)$
(which is in turn homeomorphic to the real projective space,
$\mathbb{RP}^{n-1}$) or $Gr(n,0)$ (a point). In either case,
$\hat{H}_{l^{*}}(x)$ is path-connected so $S^{0}$ is contractible in it.
Therefore, every $\hat{H}_{l^{*}}(x)$ is $C^{0}$-contractible.

The remainder of the hypotheses are satisfied directly by Lemmas 7.3, 7.4, and
7.6.

∎

Theorem 7.2 only provides sufficient conditions for a selection to exist.
There are plenty of circumstances in which there is a manifold extension and
the hypotheses of Theorem 7.2 do not apply, for instance, if $E$ is already a
manifold of dimension at least 2. It may be interesting to determine necessary
and sufficient conditions for $(\hat{H}(x))_{x\in E}$ to have a section.

Acknowledgment. The author would like to thank Allen Hatcher for helpful
communications regarding this work.

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