# Extending Whitney’s extension theorem: nonlinear function spaces

David Michael Roberts111The University of Adelaide, SA 5005, Australia,
<EMAIL_ADDRESS>Alexander Schmeding222TU Berlin, Straße des 17.
Juni 136, 10623 Berlin, Germany<EMAIL_ADDRESS>
###### Abstract

This article shows that there is a continuous extension operator for
compactly-supported smooth sections of vector bundles on possibly non-compact
smooth manifolds, where the closed set to which sections are restricted
satisfy a mild restriction on possible boundary cusps. These function spaces
are locally convex but in general not Fréchet and so one cannot use the
existing theory for Fréchet spaces of smooth functions. Further, in the
global, nonlinear case of _all_ smooth functions between possibly non-compact
smooth manifolds, we prove that the analogous restriction map between locally
convex manifolds of smooth functions is a submersion in the sense of admitting
continuous local splittings on charts, under the assumption the closed set is
either a submanifold with corners, or is locally convex with rough boundary
and compact. In all these results the spaces of functions are spaces of smooth
functions, and not spaces of Whitney jets.

Keywords: Whitney extension theorem, smooth functions on closed domain,
Whitney jet, polynomial cusps, Fréchet space, submersion, manifolds with
corners, manifolds with rough boundary, manifold of mappings, exponential law

MSC2010: 58D15 (primary), 46T10, 58C07, 54C35, 46A04, 46A13, 53C21 (secondary)

###### Contents

  1. 1 Preliminaries and Notation
  2. 2 Whitney’s extension theorem for linear spaces of functions
  3. 3 The cusp condition
  4. 4 Proof of Theorem A
    1. Smooth bundle sections on a closed set without narrow fjords
    2. Interlude: Patching by partition of unity
    3. Global extensions of bundle sections on a closed set
  5. 5 Manifolds of mappings for manifolds with rough boundary
  6. 6 Submanifolds with rough boundary and the proof of Theorem B
  7. A  Essentials on infinite-dimensional calculus and function spaces
    1. Topologies on function spaces with non-compact source
  8. B  The space of (smooth) Whitney jets

## Introduction and statement of results

The classical Whitney extension theorem [31] states that for a closed subset
of euclidean space, every Whitney jet (i.e. a family of continuous functions
which yields a (formal) Taylor expansion) can be extended to a smooth mapping
on the whole euclidean space. This leads immediately to the question, when the
extension can be realised as a continuous linear operator between the Fréchet
spaces of Whitney jets and smooth functions. In the one-dimensional case the
problem is solved by Whitney [32] (see [5, Section 2.4] for an exposition),
but it is known that not every closed set in the multidimensional case admits
a linear extension operator (cf. [9, p. 123]). This situation stimulated
research to identify conditions on the closed set which would guarantee the
existence of a linear extension operator. We refer the reader to [32] and the
monograph [5, 6] as well as the references given there for more information on
these results and related questions for other classes of functions. In
particular, for Banach spaces of $C^{m}$ functions the results of Fefferman
[8] give a much stronger result, since the set to which functions are
restricted can be _any_ subset of Euclidean space. In [21, Remark 22.18],
Kriegl and Michor note the problem of constructing, in the so-called
convenient calculus333Convenient calculus is one generalisation of calculus
for infinite-dimensional spaces beyond the realm of Banach spaces., a
continuous extension operator for even smooth one-variable functions on
general subsets. Kriegl does prove, in [20, Theorem 1.8], that subsets of a
convenient vector space with _smooth_ boundary admit a continuous extension
operator for functions valued in a convenient vector space.

In the present article we wish to establish a suitable version of these
results for manifolds. However, since spaces of Whitney jets are already quite
technical in the vector space setting, we would rather like to replace them by
a space of smooth functions. It is known that this can be done under suitable
assumptions on the closed set. Namely, for Hölder domains with dense interior
one gets an extension operator for vector spaces of smooth functions instead
of Whitney jet (see e.g. [10, Theorem 2.1]). Replacing vector spaces of smooth
mappings by manifolds of mappings, our aim is to prove that the restriction of
smooth functions to a "suitably nice" subset is a submersion of the infinite
dimensional manifolds (i.e. it locally is a projection onto a splitting
subspace [14]). Since the Whitney extension theorem generalises to paracompact
manifolds by the usual partition of unity argument, the problem addressed here
is twofold: Firstly, one needs to formulate conditions on the closed set (in
the manifold) such that a linear extension operator exists. Secondly, we would
like to establish differentiability and the submersion property on manifolds
of mappings, which are in general not even Fréchet spaces. To this end we base
our investigation on the so called Bastiani calculus [3] to make sense of
(differentiable) manifolds in this setting (a short overview is included in
Appendix A for the reader’s convenience). Results of the type established in
the present paper have been used in [28] to prove that mapping groupoids
between Lie groupoids are (infinite-dimensional) Lie groupoids.

We describe our results now in more detail. Fix a pair of finite-dimensional
manifolds $M,N$ with $M$ being a $\sigma$-compact and equipped with a
Riemannian metric. Let $C\subset M$ be a closed set satisfying a cusp
condition, defined below in Definition 3.1. This condition allows general
Lipschitz domains, but also much rougher boundary conditions, for instance
Koch snowflake-like sets.

Since the interior $C^{\circ}$ is dense in $C$, it makes sense to define
differentiable mappings as those which are differentiable on the interior and
whose differentials extend continuously (this is the setup considered in
[10]). Note that at this stage we do not assume that $C$ carries any
submanifold structure of its own, whence smoothness is only a meaningful
concept because we can test in charts of the manifolds $M$ and $N$ which do
not have a boundary. In this sense, one can talk about smooth (compactly
supported) sections on $C$ with values in vector bundles over $M$. We then
discuss a natural locally convex structure on the space $\Gamma_{c}(C,E)$ of
all such sections such that the Whitney extension theorem yields our first
result.

#### Theorem A

_Let $E\to M$ be a (finite-rank) vector bundle and $C\subset M$ a closed set
satisfying the cusp condition. The restriction map
$\operatorname{res}_{C}\colon\Gamma_{c}(M,E)\to\Gamma_{c}(C,E),\sigma\mapsto\sigma|_{C}$
on compactly-supported smooth sections has a continuous linear splitting._

We wish to give a global version of Theorem A in terms of manifolds of
mappings, i.e. for the map $\operatorname{res}^{M}_{C}\colon
C^{\infty}(M,N)\rightarrow C^{\infty}(C,N)$. Recall that for a smooth manifold
$M$ (possibly with corners), the space $C^{\infty}_{\text{fS}}(M,N)$ of smooth
mappings with the fine very strong topology (see [17] and [25], where the fS-
topology is called $\mathcal{FD}$-topology) can be turned into an infinite
dimensional manifold. If $M$ is compact the fine very strong topology
coincides with the well known compact open $C^{\infty}$-topology. Hence if the
subset $C$ is a manifold, smoothness of $\operatorname{res}_{C}^{M}$ is a well
defined concept. We prove in Section 6 that Theorem A yields local sections of
$\operatorname{res}_{C}^{M}$ if $C$ is a submanifold with corners of $M$. If
$C$ is compact, we can even relax the condition and allow _submanifolds with
rough boundary_ , a definition introduced by Karl-Hermann Neeb [15]. We
establish in Proposition 6.5 that closed submanifolds with rough boundary
satisfy the necessary cusp condition, so as to make Theorem A applicable. Note
however, that for this generalised boundary one has first to establish the
manifold structure of the manifold of mappings. We give the construction in
Section 5 and remark that it might be of independent interest as it is to the
best of our knowledge the only source currently available for the
construction. Thus our next main result can be formulated as follows:

#### Theorem B

_For $C\subset M$ a submanifold with corners, or compact and a submanifold
with rough boundary, then the restriction map
$\operatorname{res}_{C}^{M}\colon C^{\infty}_{\text{fS}}(M,N)\rightarrow
C^{\infty}_{\text{fS}}(C,N)$ is a submersion of locally convex manifolds._

Recall that for infinite-dimensional manifolds whose model spaces are more
general than Banach spaces, a submersion is a map that locally, in submersion
charts, looks like a projection out of a product. This is a stronger condition
that the map on tangent spaces being a split surjection (cf. [14] for a
detailed study). Looking at charts, the map in Theorem B splits locally on the
spaces of compactly-supported sections of certain vector bundles by Theorem A.

We remark here that Theorem B does not imply that $\operatorname{res}_{C}^{M}$
is surjective as not necessarily all smooth functions on closed submanifolds
with (rough) boundary will admit extensions to the ambient manifold (compare
[22, Corollary 6.27]). A simple example is the case where $M=S^{2}$, $C\subset
S^{2}$ is a closed equatorial ‘belt’ and $N=S^{1}$. A map $C\to S^{1}$ cannot
extend to $S^{2}$ if has non-zero winding number.

Finally, we look at nested closed subsets which satisfy the assumptions of
Theorem B. The following corollary allows a more general result.

#### Corollary C

_With the manifolds $M,N$ as above and closed sets $C\subset D\subset M$ which
both satisfy the assumptions of Theorem B, the restriction map_

$\operatorname{res}_{C}^{D}\colon C^{\infty}_{\text{fS}}(D,N)\to
C^{\infty}_{\text{fS}}(C,N),\quad f\mapsto f|_{C},$

_is a submersion of locally convex manifolds._

The proof follows again by looking at charts (of the manifold of mappings),
and by composing the extension operator $\Gamma_{c}(C,E)\to\Gamma_{c}(M,E)$
with the restriction $\operatorname{res}^{M}_{C^{\prime}}$. It is elementary
that this composite is locally a splitting of the restriction map
$\operatorname{res}_{C}^{C^{\prime}}$.

A more specific corollary applies the above collection of results to closed
sets that are geodesically strongly convex444Recall that a subset $S$ is
strongly convex if for every pair $p,q\in S$ there is a unique shortest
geodesic segement $\overline{pq}$ connecting $p$ and $q$ such that
$\overline{pq}\subseteq S$., for example closures $\overline{U_{i\ldots j}}$
of iterated finite intersections $U_{i\ldots j}=U_{i}\cap\ldots\cap U_{j}$ of
geodesically strongly convex charts $U_{i}$. Such closed sets satisfy the
required cusp condition and we prove in Lemma 6.3 that they are submanifolds
with rough boundary. Thus Corollary C immediately yields.

#### Corollary D

_Let $M$ be a smooth Riemannian manifold with geodesically strongly convex
compact sets $C\subset D\subset M$ and $N$ another smooth manifold. Then the
restriction map_

$\operatorname{res}_{C}^{D}\colon C^{\infty}(D,N)\to C^{\infty}(C,N)$

_is a submersion of Fréchet manifolds._

A close analogue of this corollary was stated as [28, Proposition 3], with
only a rough sketch of a proof, ignoring the function space topologies, and
also allowing $M$ to be a manifold with corners. This hypothesis was only
included so as to recover Corollary D, where a closed set
$\overline{U_{i\ldots j}}$ played the rôle of the ambient “manifold with
corners” $M$. However, _it is not necessarily the case_ that these closed sets
are manifolds with corners, as defined in [25, § 2.4] (see also Definition 5.1
below). Thus Corollary D should be taken to replace [28, Proposition 3].

One can ask the obvious questions as to how much further the results here can
be pushed, especially in light of the results of Frerick on general sets
satisfying the cusp condition. The biggest obstacle in pursuing this, is to
define the relevant locally convex topologies or manifold structures in the
linear and non-linear cases respectively. In light of this, an extension of
the results in the present paper might be possible but there seems to be no
straightforward way to do this.

A brief outline of the paper is as follows. In Section 1 we give basic notions
that are needed for the paper, relegating most technical results for infinite-
dimensional calculus and manifolds to Appendix A. Section 2 gives the
necessary ingredients to build up to Theorem A, namely various bits of
extension theory and patching results in the linear setting, and these are
assembled in Section 4. We then recall (from the forthcoming [15, Chapter
1.4]) the fundamentals of the theory of manifolds with rough boundary in
Section 5 and construct the smooth manifolds of maps in that case. In Section
6 we then finally prove Theorem B. Appendix B is a summary of the theory of
Whitney jets, for ease of reference.

##  1 Preliminaries and Notation

We wish to study an extension operator between spaces of smooth functions on
manifolds. In the end, we will see that, as for the vector space case, an
extension operator for functions defined on a “suitably nice” subset $C$ of a
manifold $M$ to smooth functions on the whole manifold exists. Further, we
want to establish that the _restriction of $N$-valued functions_ is a
submersion in the sense of [14].

###### 1.1. (Notation)

We write $\mathbb{N}\colonequals\\{1,2,\ldots\\}$ and
$\mathbb{N}_{0}\colonequals\mathbb{N}\cup\\{0\\}$.

Frequently we will use standard multiindex notation to denote (iterated)
partial derivatives of a (smooth) function $f\colon\mathbb{R}^{d}\supseteq
U\rightarrow\mathbb{R}^{m}$ as $\partial^{\alpha}f$ for
$\alpha\in\mathbb{N}_{0}^{d}$ (see B.1).

###### 1.2. (Conventions)

For a subset $S$ of a topological space we denote by $S^{\circ}$ its interior.
We say that a subset $C$ of a topological space is _regular_ , if $C^{\circ}$
is dense in $C$. We note that closed subsets satisfying the cusp condition to
be defined below are always regular.

Further, every finite-dimensional manifold considered in the following will
always be assumed to be Hausdorff and $\sigma$-compact.

We say $M$ is a _Banach_ (or _Fréchet_) manifold if all its modelling spaces
are Banach (or Fréchet) spaces. In general, infinite-dimensional manifolds
will not required to be $\sigma$-compact or paracompact.

We consider functions on non-open sets following [33] (where these mappings
are used to define manifolds with boundary). Further we frequently have need
for smooth functions on possibly infinite-dimensional manifolds (think
manifold of mappings). To this end we base our investigation on the so called
Bastiani calculus [3] which readily generalises beyond the realm of Banach
spaces (cf. Appendix A for a short introduction).

###### Definition 1.3.

Let $E,F$ be locally convex spaces and $C\subseteq E$ be a set with dense
interior. A continuous mapping $f\colon C\rightarrow F$ is called $C^{1}$-map
if $f\big{|}_{C^{\circ}}$ is $C^{1}$ in the sense of Bastiani calculus and the
derivative $d(f\big{|}_{C^{\circ}})$ extends (necessarily uniquely) to a
continuous mapping $df\colon C\times E\rightarrow F$.

Similarly we say $f$ is $C^{k}$ for $k\in\mathbb{N}\cup\\{\infty\\}$ if
$f\big{|}_{C^{\circ}}$ is $C^{k}$ and the iterated differentials extend
(uniquely) to all of $C$. We say $f$ is _smooth (or $C^{\infty}$)_ if $f$ is
$C^{k}$ for every $k\in\mathbb{N}$ and write $C^{\infty}(C,F)$ for the set of
all smooth maps on $C$.

We have the following version of the chain rule:

###### Lemma 1.4. ([33, Remark 5] and [1, Lemma 3.17])

Let $C\subseteq E$ and $D\subseteq F$ be regular subsets of locally convex
spaces $E,F$ and $H$ be another locally convex space. Consider $C^{k}$
mappings $f\colon C\rightarrow D\subseteq F$ and $g\colon D\rightarrow H$ then
$g\circ f$ is a $C^{k}$ mapping if one of the following conditions is
satisfied

  1. 1.

$f(C^{\circ})\subseteq D^{\circ}$ (no condition on $C$ and $D$),

  2. 2.

$C,D$ are locally convex sets, i.e. every point has a neighborhood in the set
which is convex (no condition on $f$ and $g$).

Note that for an open set $C\cap U$ is a regular set if $C$ is regular. Thus
the chain rule allows us to make sense of $C^{k}$-mappings on regular subsets
of smooth manifolds without boundary.

###### Definition 1.5.

Let $C\subseteq M$ be a regular subset of a manifold without boundary. A
continuous map $f\colon C\rightarrow N$ into a manifold $N$ without boundary
is a $C^{k}$-mapping if for every $x\in C$ there is a pair of charts
$(\varphi,U),(\psi,V)$ such that $\psi\circ f\circ\varphi^{-1}|_{\varphi(U\cap
C)}$ makes sense and is a $C^{k}$-mapping.

Clearly by Lemma 1.4 condition 1. this definition is independent of the choice
of charts. However, we note that many of the familiar rules of calculus are no
longer valid for $C^{k}$-mappings on sets with dense interior which are not
locally convex. In any case, these results are not needed to treat spaces of
sections as locally convex spaces in Section 2 below and to prove Theorem A.

To retain the “usual behaviour” of differentiable functions (most importantly,
the Mean Value Theorem, and hence the chain rule) one needs to require in
addition that the set is locally convex. This will be important to establish
the global setting required in Theorem B. Namely, the usual rules of calculus
enable the construction of manifolds of mappings as outlined in Section 5.

##  2 Whitney’s extension theorem for linear spaces of functions

The aim of this section is to recall the Whitney extension theorem in the
vector space case. Further, we discuss conditions under which the space of
Whitney jets can be identified with spaces of smooth functions on a closed
regular set. In this section we let $C\subseteq\mathbb{R}^{d},d\in\mathbb{N}$
be a regular closed set.

###### 2.1. (Ideals of functions vanishing on closed sets)

Let $m\in\mathbb{N}$ and $W\subseteq\mathbb{R}^{d}$ be an open neighborhood of
the regular closed set $C$. We consider

$\mathcal{I}_{C}(W,\mathbb{R}^{m})\colonequals\\{g\in
C^{\infty}(W,\mathbb{R}^{m})\mid\partial^{\alpha}g|_{C}\equiv
0\quad\forall\alpha\in\mathbb{N}_{0}^{d}\\}.$

Since the operators $\partial^{\alpha}\colon
C^{\infty}_{\text{co}}(U,\mathbb{R}^{m})\rightarrow
C_{\text{co}}(U,\mathbb{R}^{m}),f\mapsto\partial^{\alpha}f$ and
$\operatorname{ev}_{x}\colon
C_{\text{co}}(W,\mathbb{R}^{m})\rightarrow\mathbb{R}^{m},f\mapsto f(x)$ are
continuous linear (cf. [1, Definition 2.5 and Proposition 3.20] with respect
to the compact open $C^{\infty}$-topology (cf. Appendix A),

$\mathcal{I}_{C}(W,\mathbb{R}^{m})=\bigcap_{\alpha\in\mathbb{N}_{0}^{d}}\bigcap_{x\in
C}(\operatorname{ev}_{x}\circ\partial^{\alpha})^{-1}(0)$

is a closed vector subspace of the Fréchet space
$C^{\infty}_{co}(U,\mathbb{R}^{m})$. Indeed, if we denote by
$\mathcal{E}(C,\mathbb{R}^{m})$ the $\mathbb{R}^{m}$-valued Whitney jets on
$C$ (see Appendix B), we can view $\mathcal{I}_{C}(W,\mathbb{R}^{m})$ as the
kernel of the linear restriction map $r_{W}\colon
C^{\infty}(W,\mathbb{R}^{m})\rightarrow\mathcal{E}(C,\mathbb{R}^{m}),g\mapsto(\partial^{\alpha}g)_{\alpha}$.
Recall from [9, p. 126] that $r_{W}$ is continuous if $m=1$.555Indeed the
article claims this only for $W=\mathbb{R}^{d}$ but continuity follows
directly from the remarks above Definition 2.1 in loc.cit. as explained in
B.6. Identifying $C^{\infty}_{co}(W,\mathbb{R}^{m})\cong
C^{\infty}_{co}(W,\mathbb{R})^{m}$ (cf. [12, Lemma 3.4]) we obtain continuity
of $r_{W}$ for arbitrary neighborhoods $W$ and $m\in\mathbb{N}$.

###### 2.2. (Whitney extension theorem [31, Theorem 1], or [9, Theorem 2.2]
for a modern introduction)

The following sequence of Fréchet spaces is exact:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{C}(W,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\infty}_{\text{co}}(W,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}(C,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(1)

###### Remark 2.3.

Recall that in the category of locally convex spaces, a sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of continuous linear maps is _exact_ if it satisfies both of the following
conditions

  1. 1.

_algebraically exact_ , i.e. images of maps coincide with kernels of the next
map,

  2. 2.

_topologically exact_ , i.e. $i$ and $q$ are open mappings onto their images.

If $A$, $B$ and $C$ are Fréchet spaces topological exactness follows from
algebraic exactness by virtue of the open mapping theorem; for general locally
convex spaces this is not the case (cf. e.g. [30]).

Note that the Whitney extension theorem in general requires only a closed set
$C$ and not (as we required) a closed and regular set. However, in our
approach we will replace the space of Whitney jets by a space of smooth
functions on a closed set. Here the regularity assumption comes into play (cf.
Appendix A) and we will now construct a mapping which deals with the
identification:

###### 2.4.

Consider the mapping

$D\colon
C^{\infty}(C,\mathbb{R}^{m})\rightarrow\prod_{\alpha\in\mathbb{N}_{0}^{d}}C_{\text{co}}(C,\mathbb{R}^{m}),\quad
f\mapsto(\partial^{\alpha}f)_{\alpha}.$

Then $D$ makes sense by our definition of $C^{\infty}(C,\mathbb{R}^{m})$ and
is injective and linear. Arguing as in [9, Section 2] the image of $D$ is a
closed subspace of the Fréchet space
$\prod_{\alpha\in\mathbb{N}_{0}^{d}}C_{\text{co}}(C,\mathbb{R}^{m})$ (note
that we have compact convergence of functions and all derivatives on the dense
interior of $C$!).

As the mapping $D$ takes a smooth function on $C$ to a jet expansion (i.e. its
family of derivatives), one is tempted to think that $D$ takes its image in
the space $\mathcal{E}(C,\mathbb{R}^{m})$ of Whitney jets. However, this is
wrong in general as the following example from [4, Example 2.18] shows:

###### Example 2.5.

Let $C$ be the complement of the open subset $\\{(x,y)\in\mathbb{R}^{2}\mid
0<y<\exp(-1/x^{2}),x>0\\}$. Then $C$ is a closed regular set and we define a
function $f\in C^{\infty}(C,\mathbb{R}^{2})$ as follows

$f(x,y)=\begin{cases}\exp(-1/x^{2})&\text{ if }x>0,\ y\geq\exp(-1/x^{2})\\\
0&\text{ otherwise }\end{cases}$

A computation of
$\left(f\\!\left(x,\exp(-1/x^{2})\right)-f(x,0)\right)/(\exp(-1/x^{2})-0)=1$
shows that $f$ cannot be extended to a smooth function on $\mathbb{R}^{2}$.
Thus in particular, the image of $f$ under the mapping $D$ from 2.4 is not a
Whitney jet.

As a consequence $D$ can take its image in the space of Whitney jets only if
every smooth function on $C$ extends to a smooth function on an open
neighborhood of $C$. It turns out that the non existence of extensions is tied
to the exponential type cusps of the set $X$ in the example. Prohibiting such
inward cusps, which we shall call _narrow fjords_ , ensures that every smooth
function can indeed be extended.

###### Definition 2.6. ([4, 2.16.1])

Let $A$ be a regular closed subset of $\mathbb{R}^{d}$. We say $A$ has _no
narrow fjords_ if for all $a\in A$ exists an integer $p$, a compact
neighborhood $K$ of $a$ in $A$ and a constant $C>0$ such that any $x,y\in K$
can be joined by a rectifiable path $\gamma$ lying inside $A^{\circ}$, except
perhaps for finitely many points, and the length $\ell(\gamma)$ of $\gamma$
satisfies

$\lVert x-y\rVert\geq C\,\ell(\gamma)^{p}.$

Note that this definition gives control over how fast the width of fjords can
shrink as one moves inwards along them, see Figure 1. Further, the no narrow
fjords condition is closely related to the conditions called
$C$-quasiconvexity and the $(C,\omega)$-convexity from [5, Definition 2.63].

$A$$a$$K$$x$$y$$\gamma$ Figure 1: No narrow fjords condition

###### Example 2.7.

Let $A$ be a regular closed set. Recall that the open set $A^{\circ}$ satsifes
the _bounded turning condition_ if there is a constant $C>0$ such that for all
$x,y\in A$, there is a rectifiable path $\gamma$ from $x$ to $y$ such that
$\lVert x-y\rVert\geq C\,\ell(\gamma)$. If $A^{\circ}$ satisfies the bounded
turning condition then it has no narrow fjords. Any uniform domain [23] (see,
for example, [26, Definition 2.2] for an updated formulation) satisfies the
bounded turning condition, which includes all Hölder domains and NTA domains
("non-tangentially accessible domains" as introduced by [18]), and so the
closures of all these sets all have no narrow fjords.

###### 2.8.

Let now $C$ be a regular closed set with no narrow fjords. Then [4,
Proposition 2.16] asserts that $D\colon
C^{\infty}(C,\mathbb{R}^{m})\rightarrow\bigoplus_{\alpha\in\mathbb{N}_{0}^{d}}C_{\text{co}}(C,\mathbb{R}^{m})$
takes its image in $\mathcal{E}(C,\mathbb{R}^{m})$. As a consequence of the
Whitney extension theorem 2.2, every element in $C^{\infty}(C,\mathbb{R}^{m})$
extends to a smooth map on $\mathbb{R}^{d}$, whence the image of $D$ coincides
with the space of Whitney jets $\mathcal{E}(C,\mathbb{R}^{m})$ (see Definition
B.10). Thus we topologize $C^{\infty}(C,\mathbb{R}^{m})$ with the
identification topology induced by $D$, turning it into a Fréchet space
isomorphic to the space of Whitney jets on $C$. In particular, the exact
sequence (1) yields an exact sequence of Fréchet spaces

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{C}(W,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\infty}_{\text{co}}(W,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\infty}(C,\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(2)

In the next section we are going to investigate _outward_ cusp conditions on
the boundary of closed subsets and show how they can be transferred to
Riemannian manifolds.

##  3 The cusp condition

In the last section we have already encountered a cusp condition preventing
the occurrence of certain (inward) cusps on the boundary of the closed set on
which we are working. The key functional-analytic result we use to extend
sections is due to Frerick in [9]. It uses a metric condition on a closed
domain $F$ in $\mathbb{R}^{n}$ to ensure there is a continuous extension
operator for Whitney jets on $F$ to smooth functions on $\mathbb{R}^{n}$. The
following definition abstracts the hypothesis from [9, Theorem 3.16] and from
Definition 2.6 so as to apply to closed sets in a metric space more general
than $\mathbb{R}^{n}$.

###### Definition 3.1.

Let $(M,d)$ be a locally compact metric space $(M,d)$. A closed set $F\subset
M$ has

  1. 1.

_no narrow fjords_ if for each $x\in F$ there exists $p\in\mathbb{N}$,
$K\subseteq F$ a compact neighborhood of $x$ and $D>0$ such that all $y,z\in
K$ can be joined by a rectifiable curve $\gamma$ lying inside $F^{\circ}$,
except perhaps for finitely many points, such that its length
$\ell_{d}(\gamma)$ satisfies $d(y,z)\geq D\,\ell_{d}(\gamma)^{p}$.

  2. 2.

_at worst polynomial outward cusps_ if for all compact $K\subset M$ there
exist $\varepsilon_{0},\rho>0$ and $r\geq 1$ such that for all $z\in
K\cap\partial F$ and $0<\varepsilon<\varepsilon_{0}$ there is an $x\in F$ with
$d(x,z)<\varepsilon$ such that if $d(x,y)<\rho\varepsilon^{r}$ then $y\in F$
and $d(z,y)<\varepsilon$.

If $F$ has at worst polynomial outward cusps and no narrow fjords we simply
say that $F$ _satisfies the cusp condition_.

$M$$F$$\ldots$$K$$z$$\varepsilon_{0}$$\varepsilon$$x$$\rho\varepsilon^{r}$
Figure 2: Polynomial outward cusps

In the case that $r=1$, the condition on outward polynomial cusps is sometimes
called the (interior) corkscrew condition [16, p. 123], and so our polynomial
cusps can be seen as corkscrews with nonlinear growth.

###### Remark 3.2.

In Definition 3.1.2, if the constants $\varepsilon_{0},\rho,r$ work for the
compact set $K$, then so do smaller such constants, and if
$\varepsilon_{0}\leq 1$ then we can also increase $r$. Putting this together,
we can assume that $\varepsilon_{0}=\rho<1$ and increase $r$ as needed, and as
a result can replace $\rho\varepsilon^{r}$ by $\varepsilon^{r+1}$. Hence we
can, without loss of generality, assume that $\rho=1$ and $r\geq 2$.

###### Example 3.3.

Every Lipschitz domain satisfies the cusp condition, as do Hölder domains and
NTA domains. The compact subset of $\mathbb{R}^{2}$ whose boundary is the Koch
curve satisfies the cusp condition.

Recall that $\mathcal{E}(F)$ denotes the space of Whitney jets on the closed
set $F$. In the following Theorem, $\mathbb{R}^{n}$ is taken with the
Euclidean metric.

###### Theorem 3.4. ([9, Theorem 3.16])

Let $F\subset\mathbb{R}^{n}$ be closed and have at worst polynomial outward
cusps. Then the surjective map $C^{\infty}(\mathbb{R}^{n})\to\mathcal{E}(F)$
of Fréchet spaces has a continuous splitting.

Moreover, Theorem 3.4 combined with 2.8 yields the following Corollary which
generalises [10, Theorem 2.1].

###### Corollary 3.5.

Let $F\subset\mathbb{R}^{n}$ be closed and satisfy the cusp condition, then
the surjective map $C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(F,\mathbb{R})$ of
Fréchet spaces has a continuous splitting.

We want to be able to sensibly transfer both Frerick’s Theorem and Corollary
3.5 in Euclidean space to a Riemannian manifold, so we will need a result that
allows change of metric. The following result is stated in more generality
than we need, since it should be of independent interest.

###### Lemma 3.6.

Let $(M,d_{1})$ be a locally compact, complete metric space, $F\subset M$ be
closed and let $F$ have at worst polynomial outward cusps using the metric
$d_{1}$. If $d_{2}$ is another metric on $M$ that is locally bi-Hölder
equivalent to $d_{1}$, then $F$ has at worst polynomial outward cusps using
the metric $d_{2}$.

###### Proof.

Let $K\subset M$ be any compact set and $\varepsilon_{0,1}$, $\rho_{1}$ and
$r_{1}$ be the constants guaranteed to exist for $K$ by virtue of $F$
satisfying Definition 3.1.2 for $d_{1}$. By Remark 3.2 we will assume
$\rho_{1}=1$, $r_{1}\geq 2$ and $\varepsilon_{0,1}<1$.

Define the compact set

$N\colonequals\overline{\\{x\in M\mid d_{1}(F\cap K,x)<2\text{ and
}d_{2}(F\cap K,x)<2\\}}.$

Now as $d_{1}$ and $d_{2}$ are locally bi-Hölder equivalent there are
constants $C\geq 1$ and $0<\alpha\leq 1$ such that

$\frac{1}{C}\,d_{1}(a,b)^{\frac{1}{\alpha}}\leq d_{2}(a,b)\leq
C\,d_{1}(a,b)^{\alpha}$

for all $a,b\in N$.

Take

$\displaystyle\varepsilon_{0,2}$
$\displaystyle\colonequals\min\\{C\,\varepsilon_{0,1}^{\alpha},{\textstyle\frac{1}{2}}\\},$
$\displaystyle\rho$
$\displaystyle\colonequals\frac{1}{C^{1+r_{1}/\alpha^{2}}},\text{ and}$
$\displaystyle r_{2}$ $\displaystyle\geq r_{1}/\alpha^{2}\quad\text{such that
}\rho\,\varepsilon_{0,2}^{r_{2}}\leq\varepsilon_{0,1}$

to be the putative uniform constants required so that $F$ satisfies Definition
3.1.2 for the metric $d_{2}$. Note that since $\varepsilon_{0,2}<1$ it does
makes sense to enlarge $r_{2}$ until the upper bound on
$\rho\,\varepsilon_{0,2}^{r_{2}}$ is satisfied.

Let $z\in\partial F\cap K$ be arbitrary, and take any $\varepsilon_{2}$ such
that $0<\varepsilon_{2}<\varepsilon_{0,2}$. Define
$\varepsilon_{1}=(\varepsilon_{2}/C)^{\frac{1}{\alpha}}$. Since
$\varepsilon_{1}=(\varepsilon_{2}/C)^{\frac{1}{\alpha}}<(\varepsilon_{0,2}/C)^{\frac{1}{\alpha}}=\varepsilon_{0,1}$
then there is an $x\in F$ satisfying $d_{1}(x,z)<\varepsilon_{1}$ such that

$d_{1}(x,y)<\varepsilon_{1}^{r}\quad\Rightarrow\quad
d_{1}(z,y)<\varepsilon_{1}\,\text{ and }\,y\in F.$

Note that as $z\in K$ and $d_{1}(x,z)<\varepsilon_{1}<\varepsilon_{0,1}<2$, we
have $x\in N$. Hence $d_{2}(x,z)\leq
C\,d_{1}(x,z)^{\alpha}<\,\varepsilon_{1}^{\alpha}=\varepsilon_{2}$, as
required.

Now take $y\in M$ such that $d_{2}(x,y)<\rho\,\varepsilon_{2}^{r_{2}}$. Then
$d_{2}(y,z)\leq
d_{2}(y,x)+d_{2}(x,z)<\rho\,\varepsilon_{2}^{r_{2}}+\varepsilon_{2}<\varepsilon_{0,1}+1<2$,
and so $y\in N$. So we can calculate that

$\displaystyle d_{1}(x,y)$
$\displaystyle\leq\big{(}C\,d_{2}(x,y)\big{)}^{\alpha}$
$\displaystyle<\left(C\,\rho_{2}\right)^{\alpha}\varepsilon_{2}^{r_{2}\alpha}$
$\displaystyle=\left(\frac{\varepsilon_{2}^{\alpha^{2}r_{2}/r_{1}}}{C}\right)^{\frac{r_{1}}{\alpha}}$
$\displaystyle\leq\left(\frac{\varepsilon_{2}}{C}\right)^{\frac{r_{1}}{\alpha}}=\varepsilon_{1}^{r_{1}}$

where we have used that $\alpha^{2}r_{2}\geq r_{1}$ and $\varepsilon_{2}<1$.
Using the cusp condition for $K$ in $d_{1}$,

$\displaystyle d_{1}(z,y)$ $\displaystyle<\varepsilon_{1}\quad(\text{and }y\in
F)$ $\displaystyle\Rightarrow d_{2}(z,y)$ $\displaystyle\leq
C\,d_{1}(z,y)^{\alpha}$
$\displaystyle<C\,\varepsilon_{1}^{\alpha}=\varepsilon_{2}.$

Hence $F$ has at worst polynomial cusps for $d_{2}$. ∎

Note that if we have _uniformly_ bi-Hölder equivalent metrics then we can
dispense with the assumption of completeness; the proof goes through the same
without the need to define the compact set $N$.

We also have the following simple result for transferring the other half of
the cusp condition.

###### Lemma 3.7.

Let $(M,d_{1})$ be a locally compact, complete metric space, $F\subset M$ be
closed and let $F$ have no narrow fjords using the metric $d_{1}$. Then if
$d_{2}$ is another metric on $M$ that is locally bi-Lipschitz to $d_{1}$, then
$F$ has no narrow fjords using the metric $d_{2}$.

This follows once recalling that rectifiable paths can be taken to be
Lipschitz functions $I\to M$.

###### Corollary 3.8.

Take a manifold $M$ with a continous Riemannian metric $g$, and a locally bi-
Lipschitz chart, $\phi\colon U\xrightarrow{\sim}\mathbb{R}^{n}$ on $M$. Here
$\mathbb{R}^{n}$ is given the Euclidean metric, and $U$ the restriction of the
geodesic metric $d^{g}$ on $(M,g)$. Then if $C\subset M$ is closed and
satisfies the cusp condition for the metric $d^{g}$, then $F=\phi(C\cap
U)\subset\mathbb{R}^{n}$ satisfies the cusp condition in the Euclidean metric.

###### Remark 3.9.

Note that by standard arguments666see eg. the answer by Benoît Kloeckner at
https://mathoverflow.net/a/236851/, every $C^{1}$ manifold with a continuous
Riemannian metric $g$ has an atlas of charts that are locally bi-Lipschitz to
Euclidean space, hence _a fortiori_ locally bi-Hölder.

We can apply this (perhaps overly general) result to our setup, namely where
we take a relatively compact smooth chart $U$ on the smooth manifold $M$.

Thus we have a commutative diagram of Fréchet spaces (cf. 2.8 and Appendix B
for a description of the topologies)

$\textstyle{C^{\infty}_{\text{co}}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C^{\infty}_{\text{co}}(\mathbb{R}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{C^{\infty}(C\cap
U)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C^{\infty}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$Theorem
3.4

where the vertical arrows are surjective, and a continuous section of the
restriction map $C^{\infty}(\mathbb{R}^{n})\to C^{\infty}(F)$. Thus:

###### Lemma 3.10.

Let $C\subset M$ be a closed set satisfying the cusp condition and
$U\xrightarrow{\simeq}\mathbb{R}^{n}$ be a smooth chart on $M$. Then the
restriction map $C^{\infty}_{\text{co}}(U)\to C^{\infty}(C\cap U)$ of Fréchet
spaces has a continuous section.

##  4 Proof of Theorem A

In this section we provide the necessary details for the proof of Theorem A
from the introduction. As a first step, we consider spaces of sections on
certain closed regular subsets of a Riemannian manifold. After these sections
have been discussed, it will turn out that we only need to collect the bits
and pieces from the previous sections to obtain the result. Throughout this
section we fix the following data:

###### 4.1.

In the following $M$ will be a $d$-dimensional $\sigma$-compact manifold with
a fixed choice of Riemannian metric $g$, $E\rightarrow M$ a rank $m$-vector
bundle and $C\subseteq M$ a closed subset which satisfies the cusp condition
with respect to the geodesic metric $d^{g}$.

Further, we choose and fix auxiliary data as outlined in 4.4. In particular,
denote the locally finite atlas by $(U_{i},\varphi_{i})_{i\in I}$,
$C_{i}=U_{i}\cap C$ and the relatively compact charts by $V_{i}\subseteq
U_{i}$ which satisfy $C\subseteq\bigcup_{i\in I}V_{i}$ .

The main idea of the proof of Theorem A is as follows: We take a section and
use local triviality of the bundle to cut it into pieces which can be extended
due to the cusp condition. Then we reassemble the pieces into a section by
using a classical local to global approach with a partition of unity. In the
next subsections we provide the necessary tools: First we define the spaces of
sections, then we prepare the local to global result.

### Smooth bundle sections on a closed set without narrow fjords

Our first task is to construct a suitable topology for the vector space of
sections into $E$ on $C$.

###### Definition 4.2.

For a closed regular set $C$ which has no narrow fjords we define

$\displaystyle\Gamma_{c}(C,E)\colonequals\\{\sigma\in
C^{\infty}(C,E)\mid\pi_{E}\circ\sigma=\operatorname{id}_{C}\text{ and
}\operatorname{supp}\sigma\text{ is compact}\\}$

the _compactly supported smooth sections on $C$_. Further, define

$\displaystyle\mathcal{I}_{c}(C,E)\colonequals\\{\sigma\in\Gamma_{c}(M,E)\mid
T^{k}_{x}\sigma=0\quad\forall x\in C,k\in\mathbb{N}\\}$

the subspace of all compactly supported sections vanishing (with all their
derivatives) on $C$.777Here we use the notation $T^{k}=T\circ T\cdots\circ
T(k$ times) to denote the $k$-fold iterated tangent functor $T$. Note that
unpacking the definition of the iterated tangent functors (see e.g. [11, Lemma
1.14] for a local version) the vanishing of all iterated tangent functors at a
point is equivalent to the vanishing of all iterated partial derivatives in
any chart containing the point.

###### Remark 4.3.

Clearly the pointwise operations turn $\Gamma_{c}(C,E)$ and
$\mathcal{I}_{c}(C,E)$ into vector spaces. One can argue as in the vector
space case to see that $\mathcal{I}_{c}(C,E)$ is a closed subspace of
$\Gamma_{c}(M,E)$ (with the fine very strong topology). Indeed using an atlas
of $M$ we can use Lemma A.5 and Remark A.6 to rewrite the problem in charts,
where closedness follows from the argument in 2.2. (Avoiding localisation in
charts, one can alternatively use Lemma A.5 together with [12, Lemma 3.8].)

###### 4.4.

Let $\mathcal{U}=(U_{i},\varphi_{i})_{i\in I}$ be a locally finite atlas of
relatively compact charts of $M$ such that $\varphi_{i}(U_{i})=\mathbb{R}^{d}$
and there is a collection $\mathcal{V}$ of open sets
$V_{i}\subseteq\overline{V}_{i}\subseteq U_{i},\ i\in I$ with

  * •

$C\subseteq\bigcup_{i\in I}V_{i}$

  * •

$(\chi_{i})_{i\in I}$ is a smooth partition of unity with
$\operatorname{supp}\chi_{i}\subseteq V_{i}$

We set $C_{i}\colonequals C\cap U_{i}$ for $i\in I$ and note that
$\varphi_{i}(C_{i})\subseteq\varphi_{i}(U_{i})=\mathbb{R}^{d}$ is closed.

The following proposition is (apart from the topological assertions and the
fact that we are working with smooth functions and not jets) is a folklore
fact which easily follows from the Whitney extension theorem 2.2 in charts and
a gluing argument. Since this argument will be the basis of our construction
we give full details.

###### Proposition 4.5. (Whitney extension theorem for sections on a manifold)

The linear restriction map
$\operatorname{res}_{C}\colon\Gamma_{c}(M,E)\rightarrow\Gamma_{c}(C,E)$ is
surjective and endows $\Gamma_{c}(C,E)$ with a quotient topology such that

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma_{c}(M,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{C}}$$\scriptstyle{\rho_{\mathcal{U}}}$$\textstyle{\Gamma_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{i}\mathcal{I}_{C_{i}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{i}C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\bigoplus_{i}C^{\infty}(C_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
(3)

is commutative with exact rows in the category of locally convex spaces.
Here $r\colon\Gamma_{c}(C,E)\rightarrow\bigoplus_{i\in
I}C^{\infty}(C_{i},\mathbb{R}^{m})$ sends $f\mapsto(\operatorname{pr}_{2}\circ
T\varphi_{i}\circ f|_{C_{i}})_{i\in I}$.

###### Proof.

Let us first deal with the lower row: Since $\varphi_{i}$ is a diffeomorphism,
we can use precomposition by $\varphi_{i}$ to identify
$C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})\cong
C^{\infty}_{\text{co}}(\varphi_{i}(U_{i}),\mathbb{R}^{m})$ and
$C^{\infty}(C_{i},\mathbb{R}^{m})\cong
C^{\infty}(\varphi_{i}(C_{i}),\mathbb{R}^{m})$. Now
$F_{i}:=\varphi_{i}(C_{i})$ is a closed subset of the ambient space and
$\varphi_{i}(U_{i})$ is an open neighborhood of $F_{i}$. Moreover, since $C$
has no narrow fjords, Lemma 3.7 implies that $F_{i}$ has no narrow fjords,
whence 2.8 yields for every $i\in I$ an exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{I}_{C_{i}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
101.29742pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
101.29742pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
160.73889pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 177.4408pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
177.4408pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{C^{\infty}(C_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
253.52863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
253.52863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces\ignorespaces}}}},$

where we set
$\mathcal{I}_{C_{i}}(U_{i},\mathbb{R}^{m})\cong\mathcal{I}_{F_{i}}(\varphi_{i}(U_{i}),\mathbb{R}^{m})$,
$C^{\infty}(C_{i},\mathbb{R}^{m})\cong C^{\infty}(F_{i},\mathbb{R}^{m})$ and
suppress the identifications in the notation. Using that taking countable
direct sums in the category of locally convex spaces is exact, we see that the
lower row of (3) is exact.

By Lemma A.5 we have canonical embeddings $\rho_{\mathcal{U}}$ of
$\Gamma_{c}(M,E)$ into $\bigoplus_{i}\Gamma(E|_{U_{i}})$ and
$\rho_{\mathcal{V}}$ of $\Gamma_{c}(M,E)$ into
$\bigoplus_{i}\Gamma(E|_{V_{i}})$. We identify $\Gamma(E|_{U_{i}})\cong
C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})$ as in Remark A.6 and suppress
this in the notation. Since $\mathcal{I}_{c}(C,E)$ is clearly contained in the
kernel of $\operatorname{res}_{C}$, we obtain a commutative diagram of vector
spaces:

$\displaystyle\begin{aligned} \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
38.35234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
38.35234pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{I}_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
58.11334pt\raise-20.25185pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\rho_{\mathcal{U}}|_{C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
58.11334pt\raise-29.46484pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
118.96797pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
118.96797pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma_{c}(M,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
175.40599pt\raise 5.19028pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\operatorname{res}_{C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 205.97742pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
141.513pt\raise-20.25185pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\rho_{\mathcal{U}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
141.513pt\raise-29.46484pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
205.97742pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Gamma_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
227.05786pt\raise-20.25185pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.41405pt\hbox{$\scriptstyle{r=(r_{C_{i}})_{i\in I}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
227.05786pt\raise-29.46484pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
284.3164pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-5.5pt\raise-40.50371pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\bigoplus_{i}\mathcal{I}_{C_{i}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
110.72668pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
110.72668pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigoplus_{i}C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
179.59741pt\raise-35.31621pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
196.29932pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
196.29932pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigoplus_{i}C^{\infty}(C_{i},\mathbb{R}^{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
281.8164pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
281.8164pt\raise-40.50371pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces\ignorespaces}}}}\end{aligned}$
(4)

Here $r_{C_{i}}(f)\colonequals f|_{C_{i}},\ i\in I$ and
$\rho_{\mathcal{U}}|_{C}$ is induced from $\rho_{\mathcal{U}}$ and realises
$\mathcal{I}_{c}(C,E)$ as the closed subspace $\\{(c_{i})_{i\in
I}\in\bigoplus_{i}\mathcal{I}_{C_{i}}(U_{i},\mathbb{R}^{m})\mid
c_{i}|_{U_{i}\cap U_{j}}=\Phi_{ij}(\operatorname{id}_{M},c_{j})|_{U_{i}\cap
U_{j}}\\}$.888Recall that point evaluations and postcomposition with fixed
smooth functions are continuous in the compact open $C^{\infty}$-topology (see
e.g. [1]). An easy adaption of the argument in [29, proof of Lemma 3.21 (b)]
shows that the subspace indeed is a closed subspace of the direct sum. Note
that apart from the space $\Gamma_{c}(C,E)$ which is not yet topologised, (4)
is a commutative diagram in the category of locally convex spaces

####  $\operatorname{res}_{C}$ is surjective.

Consider $f\in\Gamma_{c}(C,E)$ and choose $(g_{i})_{i\in I}\in\bigoplus_{i\in
I}C^{\infty}(U_{i},\mathbb{R}^{m})$ with $q((g_{i})_{i})=r(f)$. In general
$(g_{i})_{i\in I}$ will not be contained in the image of $\rho_{\mathcal{U}}$,
but we see that

$g_{i}|_{C_{i}\cap C_{j}}=f|_{C_{i}\cap C_{j}}=g_{j}|_{C_{i}\cap
C_{j}}\quad\text{for every }i,j\in I.$ (5)

Using the partition of unity from 4.4, we construct smooth functions on $M$ by
continuing $\chi_{j}\cdot g_{j}|_{M\setminus U_{j}}\equiv 0$. Hence

$h_{i}\colonequals\sum_{j\in I}(\chi_{j}\cdot g_{j})|_{V_{i}}\in
C^{\infty}(V_{i},\mathbb{R}^{m}),\quad i\in I,$

By construction $(h_{i})_{i\in I}\in\bigoplus_{i\in
I}C^{\infty}(V_{i},\mathbb{R}^{m})$ and we have $h_{i}|_{V_{i}\cap
V_{j}}=h_{j}|_{V_{i}\cap V_{j}}$ for every pair $(i,j)\in I^{2}$. Thus
$(h_{i})_{i\in I}$ is contained in the image of $\rho_{\mathcal{V}}$ and we
can choose $h\in\Gamma_{c}(M,E)$ with $\rho_{V}(h)=(h_{i})_{i\in I}$. Now (5)
implies that $h_{i}|_{V_{i}\cap C_{j}}=f|_{V_{i}\cap C_{j}}$. As the $V_{i}$
cover $C$ (see 4.4), we see that $\operatorname{res}_{C}(h)=f$. Thus
$\operatorname{res}_{C}$ is surjective and we can endow $\Gamma_{c}(C,E)$ with
the quotient topology, thus turning it into a locally convex space.

####  $r$ is continuous with respect to the quotient topology

Follows directly from the commutativity of (4) and the definition of the
quotient topology. Note that $r$ is linear, whence (3) indeed is a commutative
diagram in the category of locally convex spaces.

#### The upper row of diagram (3) is exact.

In Remark 4.3 we have seen that $\mathcal{I}_{c}(C,E)$ is a closed subspace
and we know that $\operatorname{res}_{C}$ is surjective, open and continuous.
Hence we only need to prove that its kernel coincides with
$\mathcal{I}_{c}(C,E)$. Consider $g\in\ker(\operatorname{res}_{C})$. Since
$\rho_{\mathcal{U}}$ is injective, the commutativity of (4) implies that
$\rho_{\mathcal{U}}(g)$ is contained in the kernel of $q$ and by exactness of
the bottom row and the definition of $\rho_{\mathcal{U}}|_{C}$ we must have
$g\in\mathcal{I}_{c}(C,E)$. The converse inclusion is trivial and in
conclusion (3) is exact in the category of locally convex spaces. Finally we
remark that this implies that $\Gamma_{c}(C,E)$ is a Hausdorff space (as the
quotient of a Hausdorff space modulo a closed linear subspace). ∎

An important ingredient in the proof of the last lemma was the local to global
argument using a partition of unity. We will see in Lemma 4.11 that this
construction is continuous with respect to the function space topologies.

###### Lemma 4.6.

The map
$r\colon\Gamma_{c}(C,E)\rightarrow\bigoplus_{i}C^{\infty}(C_{i},\mathbb{R}^{m})$
is injective and its image is the closed subspace

$\mathcal{A}\colonequals\left\\{(h_{i})\in\bigoplus_{i\in
I}C^{\infty}(C_{i},\mathbb{R}^{m})\middle|h_{i}|_{C_{i}\cap
C_{j}}=\Phi_{ij}(\operatorname{id}_{C_{j}},h_{j})|_{C_{i}\cap C_{j}}\forall
i,j\in\mathbb{N}\right\\}.$

If $C$ is compact, $r$ induces an isomorphism of locally convex spaces
$\Gamma_{c}(C,E)\cong\mathcal{A}$.

###### Proof.

We already know that $r$ is continuous and it is clearly injective and takes
its image in $\mathcal{A}$. Now every $C_{i}\subseteq C$ is contained in the
compact set $\overline{U}_{i}$. Hence for a family $(f_{i})\in\mathcal{A}$,
the obvious mapping

$f\colon C\rightarrow E,\quad f(x)\colonequals
T\varphi_{i}^{-1}f_{i}(x),\text{ for }x\in C_{i}$

makes sense and is a compactly supported smooth section over $C$, i.e. it is
contained in $\Gamma_{c}(C,E)$. Hence $\mathcal{A}$ is the image of $r$. Again
since point evaluation and postcomposition by fixed smooth mappings are
continuous in the compact open $C^{\infty}$-topology, (2) shows that this is
also the case for the Fréchet topology on $C^{\infty}(C,\mathbb{R}^{m})$. An
easy adaptation of the argument in [29, proof of Lemma 3.21 (b)] establishes
closedness of $\mathcal{A}$.

Let us now assume that $C$ is compact. Then there are only finitely many $i\in
I$ such that $C_{i}\neq\emptyset$ we conclude that $\mathcal{A}$ is a Fréchet
space as a closed subspace of a finite product of such spaces. Furthermore,
$\Gamma_{c}(M,E)$ is isomorphic to a closed subspace (cf. Lema A.5) of the
webbed space $\bigoplus_{i}C^{\infty}_{\text{co}}(U_{i},\mathbb{R}^{m})$,
whence $\Gamma_{c}(C,E)$ is webbed as a qotient of a webbed space ([24, Lemma
24.28]). Now the open mapping theorem [24, 24.30] shows that $r$ is open as a
mapping into $\mathcal{A}$, whence $r$ induces the claimed isomorphism of
locally convex spaces. ∎

###### Remark 4.7.

Note that the topology on $\Gamma_{c}(C,E)$ does not automatically coincide
with the quotient topology of $\operatorname{res}_{C}$ if $C$ is not compact.
Studying the above proof, the open mapping theorem is not applicable since
$\mathcal{A}$ is not necessarily ultrabornological (as it is not clear that it
would be a limit subspace of the direct sum). In fact the authors do not know
whether the quotient topology may be properly finer in the non-compact case.

However, the problem mentioned in the last remark is not relevant for us,
since we will only consider sets which allow continuous extension operators.
In the presence of such a section, the two topologies coincide:

###### Lemma 4.8.

Assume that there exists a continuous section
$s\colon\mathcal{A}\rightarrow\Gamma_{c}(M,E)$ of the map
$r\circ\operatorname{res}_{C}$, then the quotient topology turns $r$ into an
isomorphism $\Gamma_{c}(C,E)\cong\mathcal{A}$.

###### Proof.

Since $r\circ\operatorname{res}_{C}$ is continuous surjective and admits a
(global) continuous section, it is a quotient map between locally convex
spaces. As $r^{-1}\circ(r\circ\operatorname{res}_{C})=\operatorname{res}_{C}$
we deduce that $r^{-1}\colon\mathcal{A}\rightarrow\Gamma_{c}(C,E)$ is
continuous, whence $r$ induces and an isomorphism of locally convex spaces
onto its image. ∎

Thus we in the situation of Theorem A (to be proved in the end of the
section), the topologies coincide.

### Interlude: Patching by partition of unity

In this interlude, we discuss continuity properties for the map which patches
mappings on a locally finite-covering by means of a partition of unity.

###### 4.9.

Recall that for a given compact subset $K$ of $M$ only finitely many members
of the locally finite open cover $\mathcal{U}$ have a non-trivial intersection
with $K$. Thus for each $i\in I$ we obtain a finite subset of $I$ by setting

$J_{i}\colonequals\\{j\in I\mid U_{j}\cap U_{i}\neq\emptyset\\}$

###### 4.10.

Fix $n\in\mathbb{N}$ and consider for $i\in I$ maps $f_{j}\in
C^{\infty}(U_{j},\mathbb{R}^{n})$ for $j\in J_{i}$. Multiplying with the
partition of unity 4.1, we obtain for every such pair a smooth mapping
$f_{ji}\colonequals\chi_{j}|_{V_{j}\cap U_{i}}\cdot f_{j}|_{V_{j}\cap U_{i}}$
defined on the (possibly empty) set $V_{j}\cap U_{i}$. Note that since
$\operatorname{supp}\chi_{j}\subseteq V_{j}$, the mapping vanishes in a
neighborhood of the boundary of $V_{j}\cap U_{i}$ in $U_{i}$. Thus we can
extend $f_{ji}$ by $0$ to a smooth map on all of $U_{i}$ (or by a similar
argument to all of $V_{i}$). In the following we will extend these mappings to
all of $U_{i}$ (or similarly to $V_{i}$) and suppress the extension in the
notation.

###### Lemma 4.11.

Using the notation from 4.10, the mixing map

$\displaystyle\mu\colon\bigoplus_{i\in
I}C^{\infty}_{co}(U_{i},\mathbb{R}^{n})$
$\displaystyle\longrightarrow\bigoplus_{i\in
I}C^{\infty}_{co}(V_{i},\mathbb{R}^{n})$ $\displaystyle(f_{i})_{i\in I}$
$\displaystyle\mapsto\left(\sum_{j\in J_{i}}(\chi_{j}|_{V_{i}\cap V_{j}})\cdot
f_{j}|_{V_{i}\cap V_{j}}\right)_{i\in I}$

is continuous linear. Its image is contained in the closed subspace
$A\colonequals\\{(g_{i})_{i\in I}\mid g_{j}|_{V_{i}\cap
V_{j}}=g_{i}|_{V_{i}\cap V_{j}}\forall i,j\in I\\}$.

###### Proof.

The mapping $\mu$ makes sense. As argued in 4.10, every component of
$\mu(f_{i})_{i\in I}$ is a smooth function as a finite sum of such functions.
Note that every $i\in I$ appears only in finitely many of the sets $J_{k},k\in
I$. Thus every $f_{i}$ appears at most in finitely many of the sums of the
definition of $\mu$, whence $\mu$ makes sense as a mapping between direct
sums. Clearly $\mu$ is linear.

####  $\mu$ takes its image in $A$

Observe that by construction we have $\operatorname{supp}\chi_{k}\subseteq
V_{k}$. Hence if $\chi_{k}$ does not vanish on $V_{i}\cap V_{j}$ we must have
$k\in J_{i}\cap J_{j}$. Thus the sum in $\sum_{k\in
J_{i}}(\chi_{k}|_{V_{k}\cap V_{i}})\cdot f_{k}|_{V_{i}\cap V_{k}}$ coincides
on $V_{i}\cap V_{j}$ (up to vanishing summands) with the one summing over
$J_{j}$. In conclusion, $\mu$ takes its image in $A$.

#### Continuity of the auxiliary mappings $m_{i}$

Let us first fix $i\in I$ and consider the linear map

$m_{i}\colon\bigoplus_{j\in
J_{i}}C^{\infty}_{co}(U_{j},\mathbb{R}^{n})\rightarrow
C^{\infty}(V_{i},\mathbb{R}^{n}),\quad(f_{j})_{j}\mapsto\sum_{j\in
J_{i}}(\chi_{j}|_{V_{i}\cap V_{j}})\cdot f_{j}|_{V_{i}\cap V_{j}}.$

As $J_{i}$ is finite and $C^{\infty}_{co}(V_{i},\mathbb{R}^{n})$ is a
topological vector space it clearly suffices to establish smoothness for all
of the mappings

$\displaystyle c_{j}\colon C^{\infty}_{co}(U_{j},\mathbb{R}^{n})$
$\displaystyle\rightarrow C^{\infty}_{co}(V_{i},\mathbb{R}^{n}),\quad j\in
J_{i}$ $\displaystyle f$ $\displaystyle\mapsto\chi_{j}|_{V_{i}\cap V_{j}}\cdot
f|_{V_{i}\cap V_{j}}.$

Recall that the space $C^{\infty}_{co}(U_{j},\mathbb{R}^{n})$ is a topological
$C^{\infty}_{co}(U_{j},\mathbb{R})$-module (see e.g. [13, Corollary F.13]).
Thus the map $\kappa_{j}(f)\colonequals\chi_{j}|_{U_{j}}\cdot f$ is
continuous, takes its image in the linear subspace
$C^{\infty}_{\overline{V}_{j}}(U_{j},\mathbb{R}^{n})\subseteq
C^{\infty}_{\text{co}}(U_{j},\mathbb{R}^{n})$ of smooth functions supported in
$\overline{V}_{j}$. Now [13, Lemma 4.24 and Lemma 4.6] extending functions in
$C^{\infty}_{\overline{V}_{j}}(U_{j},\mathbb{R}^{n})$ by $0$ to all of $M$ and
restricting then to $V_{i}\cap V_{j}$ yields a continuous linear map which,
composed with $\kappa_{j}$, coincides with $c_{j}$. We conclude that $c_{j}$
and thus $m_{i}$ is continuous linear.

#### Continuity of $m$

We define the mapping

$\tilde{\mu}\colonequals\oplus_{i\in I}m_{i}\colon\bigoplus_{i\in
I}\bigoplus_{j\in
J_{i}}C^{\infty}_{co}(U_{i},\mathbb{R}^{n})\rightarrow\bigoplus_{i\in
I}C^{\infty}_{co}(V_{i},\mathbb{R}^{n}).$

This mapping is continuous linear, since the mappings $m_{i}$ are so by the
previous step. It follows from the universal property of the locally convex
direct sum that

$\displaystyle B_{diag}\colon\bigoplus_{i\in
I}C^{\infty}(U_{i},\mathbb{R}^{n})$ $\displaystyle\rightarrow\bigoplus_{i\in
I}\bigoplus_{\\#\\{k\in I\mid i\in
J_{k}\\}}C^{\infty}(U_{i},\mathbb{R}^{n})\cong\bigoplus_{i\in
I}\bigoplus_{j\in J_{i}}C^{\infty}(U_{j},\mathbb{R}^{n})$
$\displaystyle(f_{i})_{i\in I}$ $\displaystyle\mapsto\quad(\oplus_{\\#\\{k\in
I\mid i\in J_{k}\\}}f_{i})_{i\in I}\quad\mapsto\quad((f_{j})_{j\in
J_{i}})_{i\in I}$

is continuous linear (where due to the construction, there is a bijection
between the index sets of both sums). Now, we have $\mu=\tilde{\mu}\circ
B_{diag}$ and thus $m$ is continuous linear as a composition of such mappings.
∎

### Global extensions of bundle sections on a closed set

We will now prove Theorem A from the introduction, whose statement we repeat
here for convenience.

###### Theorem 4.12. (Theorem A)

Let $C\subseteq M$ be a closed set satisfying the cusp condition. Then the
restriction map
$\operatorname{res}_{C}\colon\Gamma_{c}(M,E)\rightarrow\Gamma_{c}(C,E)$ admits
a continuous linear section $\mathcal{E}_{C}^{M}$.

###### Proof.

We use the notation and data introduced in 4.1. For the proof we consider a
commutative diagram of locally convex spaces (where the numbers indicate where
the (continuous) linear map was constructed):

$\textstyle{\Gamma_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{\eqref{eq:
ex:vssect}}$$\scriptstyle{\mathcal{E}_{C}^{M}}$$\textstyle{\bigoplus_{i\in
I}C^{\infty}(C_{i})^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$extensionLemma
4.13$\textstyle{\bigoplus_{i\in
I}C^{\infty}_{\text{co}}(U_{i})^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\stackrel{{\scriptstyle\mu|^{\operatorname{im}(\rho_{\mathcal{V}})}}}{{\text{Lemma
}\ref{lem:
sm:mix}}}}$$\scriptstyle{\mu}$$\textstyle{\Gamma_{c}(M,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{C}}$$\scriptstyle{\rho_{\mathcal{V}}}$$\textstyle{\operatorname{im}(\rho_{\mathcal{V}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inc$\textstyle{\bigoplus_{i\in
I}C^{\infty}_{\text{co}}(V_{i},\mathbb{R}^{m})}$ (6)

We postpone the proof of the commutativity of (6) to Lemma 4.13 below, where
also the extension map needed in the computation is defined.

Now $\mathcal{E}_{C}^{M}$ is defined via the right half of the diagram (using
that $\rho_{\mathcal{V}}$ is a topological embedding onto its image by Lemma
A.5). Since all the mappings in the definition are continuous and linear,
$\mathcal{E}_{C}^{M}$ is a continuous linear section of
$\operatorname{res}_{C}$. ∎

###### Lemma 4.13.

There exists a continuous linear extension map
$\varepsilon\colon\bigoplus_{i\in
I}C^{\infty}(C_{i})^{m}\rightarrow\bigoplus_{i\in
I}C^{\infty}_{\text{co}}(\mathbb{R}^{d})^{m}$ which makes (6) commutative.

###### Proof.

We construct continuous linear mappings $\varepsilon_{i}\colon
C^{\infty}(C_{i})^{m}\rightarrow C^{\infty}_{\text{co}}(\mathbb{R}^{d})^{m}$
and define $\varepsilon\colonequals\oplus_{i\in I}\varepsilon_{i}$. Thus
$\varepsilon$ will be continuous linear by the properties of the direct sum.
For the construction we distinguish two cases depending on $i\in I$:

#### Case 1: $C_{i}=\emptyset$.

Since the chart does not intersect the domain of our map, only have to extend
the empty function, whence $E_{i}$ is simply defined as the constant $0$-map
in this case.

#### Case 2: $C_{i}\neq\emptyset$.

Due to our setup, the sets $C_{i}$ satisfy the assumptions made in the
statement of Lemma 3.10. Hence in this case there is a continuous linear
extension operator $C^{\infty}(C_{i})\rightarrow C^{\infty}(U_{i})$. We define
$\varepsilon_{i}$ as the $m$-fold product of this extension operator.

This completes the construction of the extension map and all there is left is
to prove that the diagram (6) commutes. However, this is obvious from a
trivial calculation if one recalls the following facts:

  * •

For each pair $i,j\in I$ and $(f_{k})_{k}\in\operatorname{im}r$ we have
$f_{i}|_{C_{i}\cap C_{j}}=f_{j}|_{C_{i}\cap C_{j}}$,

  * •

the extension operators $\varepsilon_{i}$ do not change the map on the
$C_{i}$,

  * •

composition with $\rho_{V}^{-1}\mu$ is just mixing and restricting with a
partition of unity and then reconstruction via the sheaf property of smooth
maps.

Composing again with the restriction map, the outer square of (6) commutes. ∎

As a direct consequence of the above theorem, we obtain:

###### Corollary 4.14.

If $C\subset M$ is a closed subset which satisfies the cusp condition in the
geodesic metric on $M$, the exact sequence (3) splits, i.e. we have the
isomorphim of topological spaces
$\Gamma_{c}(M,E)\cong\mathcal{I}_{c}(C,N)\oplus\Gamma_{c}(C,N)$.

###### Remark 4.15.

As a consequence of Theorem 4.12 and Lemma 4.8 the map $r$ from Proposition
4.5 and (6) is a topological embedding onto a closed subspace for every closed
subset which satisfies the cusp condition.

##  5 Manifolds of mappings for manifolds with rough boundary

In this section we recall some essentials on manifolds with rough boundary
from [15, Chapter 1.4]. Then we recall the classical construction of manifolds
of mappings and how to apply them to the rough boundary case.

###### Definition 5.1.

We recall from [15] (cf. [1, Section 4]) that a _manifold with rough boundary_
modelled on a locally convex space $E$ is a Hausdorff topological space $M$
with an atlas of smoothly compatible homeomorphisms $\phi\colon U_{\phi}\to
V_{\phi}$ from open subsets $U_{\phi}$ of $M$ onto locally convex subsets
$V_{\phi}\subseteq E$ with dense interior (to distinguish from ordinary
manifold charts, they are also called _rough_ $E$-charts).

If $x\in M$ we call $x$ a _formal boundary point_ if there is a rough
$E$-chart $(U_{\varphi},\varphi)$ around $x$ such that
$\varphi(x)\in\partial\varphi(U_{\varphi})$. Denote by $\partial M$ the
_(formal) boundary_ of $M$, i.e. the set of formal boundary points of $M$.

If each $V_{\phi}$ is open, $M$ is an ordinary manifold (without boundary). If
each $V_{\phi}$ is relatively open in a closed hyperplane
$\lambda^{-1}\\!\left([0,\infty[\,\right)$, where $\lambda\in E^{\prime}$ (the
space of continuous linear functional on $E$), then $M$ is a _manifold with
smooth boundary_. In the case of a _manifold with corners_ , each $V_{\phi}$
is a relatively open subset of
$\lambda^{-1}_{1}\\!\left([0,\infty[\,\right)\cap\cdots\cap\lambda^{-1}_{n}\\!\left([0,\infty[\,\right)$,
for suitable $n\in\mathbb{N}$ (which may depend on $\phi$) and linearly
independent $\lambda_{1},\ldots,\lambda_{n}\in E^{\prime}$.

The boundary of manifolds with rough boundary is characterised by the
following.

###### Lemma 5.2.

Let $M$ be a manifold with rough boundary and $(U_{\varphi},\varphi)$,
$(U_{\psi},\psi)$ be rough $E$-charts around $x\in M$. Then
$\varphi(x)\in\partial\varphi(U_{\varphi})$ if and only if
$\psi(x)\in\partial\psi(U_{\psi})$.999A full proof is contained in the
forthcoming [15, Section 3]. However here is a rough sketch: Argue by
contradiction. In the chart where the image is in the boundary choose a convex
neighborhood $W$. Now apply the Hahn-Banach theorem to separate the image of
$x$ from the interior of $W$ by a functional $\lambda$. Taking the derivative
$\kappa$ of the change of charts, one derives a contradiction by considering
$\lambda\circ\kappa^{-1}$ on $\kappa(W)$.

In essence Lemma 5.2 shows that the formal boundary of $M$ arises from the
topological boundary of the images of charts in the model space.

###### 5.3.

By virtue of the chain rule Lemma 1.4 2\. we can define smooth mappings on
manifolds with rough boundary in the usual way.

Direct products of manifolds with rough boundary, tangent spaces and tangent
bundles101010The definition of tangent vectors at $x$ as equivalence classes
of vectors in the model space, i.e. $(\psi,x,v)\sim(\varphi,x,w)$ if and only
if $d(\psi\circ\varphi^{-1})(\varphi(x),v)=w$ where $\varphi,\psi$ are
manifold charts, makes sense for a manifold with rough boundary and yields the
usual results and identifications for iterated tangent bundles, cf. also [25,
Section 2]. as well as vector bundles may be defined as usual.

We recall the compact open $C^{\infty}$-topology on the space
$C^{\infty}(M,N)$ of smooth mappings from a manifold with rough boundary into
a manifold without boundary.

###### Definition 5.4.

Let $M,N$ be manifolds with rough boundary. We define compact-open
$C^{\infty}$-topology on $C^{\infty}(M,N)$ as the intial topology induced by
the mappings

$T^{k}\colon C^{\infty}(M,N)\rightarrow C_{\text{co}}(T^{k}M,T^{k}N),\quad
f\mapsto T^{k}f,\quad k\in\mathbb{N}_{0}$

where the right hand side carries the compact open topology. We let
$C^{\infty}_{\text{co}}(M,N)$ denote the space with the compact open
$C^{\infty}$-topology.

###### Remark 5.5.

If $N=\mathbb{R}^{m}$ the compact open $C^{\infty}$ topology coincides with
topology described in [1, Definition 3.21 and Section 4] by adapting the
argument of [11, Lemma 1.14] to manifolds with rough boundary. In addition, if
$M$ is a locally convex regular closed subset of $\mathbb{R}^{m}$ and
$N=\mathbb{R}^{m}$, the compact open $C^{\infty}$-topology coincides with the
identification topology from 2.8 (by an argument analogous to B.6).

###### Proposition 5.6.

Let $f\colon N\rightarrow B$ and $g\colon A\rightarrow M$ be smooth mappings
between finite-dimensional manifolds with rough boundary. Then

$f^{*}g_{*}\colon C^{\infty}_{\text{co}}(M,N)\rightarrow
C^{\infty}_{\text{co}}(A,B),\quad h\mapsto f\circ h\circ g$

is continuous.

###### Proof.

The usual proof for manifolds without boundary (see e.g. [34, Lemma 5.5.])
carries over without any changes. ∎

###### Lemma 5.7.

The initial topology turns the map

$\mathcal{T}\colon
C^{\infty}_{\text{co}}(M,N)\rightarrow\prod_{k\in\mathbb{N}_{0}}C_{\text{co}}(T^{k}M,T^{k}N),\quad
f\mapsto(T^{k}f)_{k\in\mathbb{N}_{0}}$

into a topological embedding with closed image.

###### Proof.

By definition of an initial topology, the map $\mathcal{T}$ is a topological
embedding. Let now $(T^{k}f^{\alpha})_{k\in\mathbb{N}_{0},\alpha\in I}$ be a
net in the image of $\mathcal{T}$ which converges to
$(f_{k})_{k\in\mathbb{N}_{0}}$. If we can prove that $f_{k}=T^{k}f_{0}$ holds
for every $k\in\mathbb{N}$ then the image of $\mathcal{T}$ is closed. Clearly
we can verify the formula $f_{k}=T^{k}f_{0}$ locally in charts. As the
sequence converges with respect to the compact open $C^{\infty}$-topology, the
usual inductive proof using the fundamental theorem of calculus ([11, Theorem
1.5 and Lemma 1.7] which is valid on locally convex regular subsets of
$\mathbb{R}^{d}$!) carries over without any changes, see e.g. [34, Lemma 5.13
and Theorem 5.14]. ∎

As a consequence of the above identification, we obtain the following
completeness and metrisation results (which are well known in the case of a
manifold without boundary).

###### Lemma 5.8.

Let $M,N$ be manifolds with rough boundary, such that $M$ is locally
compact111111In contrast to manifolds without boundary, manifolds with rough
boundary need not be locally compact. For example, recall that regular locally
convex subsets of $\mathbb{R}^{d}$ are in general not locally compact, e.g.
$\\{(0,0)\\}\cup\\{(x,y)\in\mathbb{R}^{2}\mid x>0\\}$ in $\mathbb{R}^{2}$. and
$\sigma$-compact and $N$ metrisable and modelled on a metrisable space. Then
$C^{\infty}_{\text{co}}(M,N)$ is metrisable.

###### Proof.

For convenience let $d$ be the dimension of $M$. Since $M$ is locally compact,
for every manifold chart $(\varphi,U)$ the domain $U$ is locally compact,
whence $\varphi(U)$ is locally compact subset of $\mathbb{R}^{d}$. As
$\mathbb{R}^{d}$ is second countable, $\varphi(U)$ is locally compact and
second countable, whence $\sigma$-compact. Using the canonical atlas for the
iterated tangent bundle, we see that locally over $U$ we get a bundle
trivialisation $T^{k}M\supseteq
T^{k}U\cong\varphi(U)\times(\mathbb{R}^{d})^{2^{k}-1}$. Thus $T^{k}M$ is
$\sigma$-compact for every $k\in\mathbb{N}_{0}$. Since $N$ is metrisable and
modelled on a metrisable space, $T^{k}N$ is metrisable by [7]. Thus the spaces
$C_{\text{co}}(T^{k}M,T^{k}N)$ are metrisable by [34, Proposition 5.10 e)]
whence the embedding from Lemma 5.7 indentifies $C^{\infty}_{\text{co}}(M,N)$
as a subspace of a metrisable space. ∎

###### Corollary 5.9.

Let $M$ be a locally compact manifold with rough boundary and $F$ a Fréchet
space. Then $C^{\infty}_{\text{co}}(M,F)$ is a Fréchet space.

###### Proof.

It is well known that the pointwise operations turn
$C^{\infty}_{\text{co}}(M,F)$ into a topological vector space (cf. [1, Section
2]). By Lemma 5.8 the space $C^{\infty}_{\text{co}}(M,F)$ is metrisable. In
addition, [7] implies that every $T^{k}M$ is metrisable, whence a $k$-space.
Now $T^{k}F\cong F^{2^{k}}$ is complete and we infer from [12, Remark 3.2 (a)]
that $C_{\text{co}}(T^{k}M,T^{k}F)$ is complete. Now Lemma 5.7 identifies
$C^{\infty}_{\text{co}}(M,F)$ with a closed subspace of the complete space
$\prod_{k\in\mathbb{N}_{0}}C_{\text{co}}(T^{k}M,T^{k}F)$, whence
$C^{\infty}_{\text{co}}(M,F)$ is a Fréchet space. ∎

Finally we turn to smooth sections of bundles over a manifold with rough
boundary.

###### 5.10.

Let $p\colon E\rightarrow M$ be a vector bundle with typical fibre $F$. Assume
that $F$ is a Fréchet space and $M$ is a compact manifold with rough boundary.
Then we define

$\Gamma(M,E)\colonequals\\{\sigma\in C^{\infty}(M,E)\mid p\circ\sigma=\\\
id_{M}\\},$

and endow it with the subspace topology induced by
$C^{\infty}_{\text{co}}(M,E)$.

Note that since $M$ is compact, we can choose an open cover
$(W_{i},\kappa_{i})_{1\leq i\leq n}$ of domains of bundle trivialisations for
$E$ and denote by $E|_{U_{i}}$ the restricted bundle over $U_{i}$. Then define
the map

$\rho\colon\Gamma(M,E)\rightarrow\prod_{1\leq i\leq
n}\Gamma(W_{i},E|_{W_{i})}),\quad\sigma\mapsto(\sigma|_{W_{i}})_{i}.$

By Lemma 5.6 the map $\rho$ is continuous (as each of its components are given
by mappings $(\iota_{i})^{*}$, where $\iota_{i}\colon W_{i}\rightarrow M$ is
the inclusion). Clearly $\rho$ is injective, linear and identifies
$\Gamma(M,E)$ with the subspace
$\mathcal{C}\colonequals\\{(\gamma_{i})_{i}\mid\gamma_{i}|_{W_{i}\cap
W_{j}}=\gamma_{j}|_{W_{i}\cap W_{j}}\\}$. Since the the evaluation map is
continuous by [1, Proposition 3.20], $\mathcal{C}$ is closed. Working with
subbasic neighborhoods, one can also prove that $\rho$ is open onto its image.
We refer to [34, Lemma 6.4] for details.

Now we obtain isomorphisms $\Gamma(W_{i},E|_{W_{i}})\rightarrow
C^{\infty}_{\text{co}}(W_{i},F),\sigma\mapsto\operatorname{pr}_{2}\circ\psi_{i}\circ\sigma$
of topological vector spaces. Thus Corollary 5.9 implies that $\Gamma(M,E)$ is
a Fréchet space as a closed subspace of a direct product of Fréchet spaces.

Having discussed the topology on the manifold of mappings, we will now
construct an infinite-dimensional manifold structure on
$C^{\infty}_{\text{co}}(M,N)$ for $M$ a compact manifold with rough boundary
and $N$ a manifold without boundary. If $M$ is a manifold with corners, such a
construction (even for $M$ non-compact) can be found in [25]. Our proof
follows the general idea of loc.cit. but we avoid using an instance of the so
called $\Omega$-Lemma. For the rest of this section $M$ will be a compact
manifold with rough boundary and $N$ will be a Fréchet manifold without
boundary which admits a local addition.

###### 5.11.

A manifold $N$ admits a local addition, if there is a _local addition_
$\Sigma_{N}$ on $N$, i.e. a smooth mapping $\Sigma_{N}\colon
TN\supseteq\Omega\rightarrow N$ on an open $0$-neighborhood $\Omega$, such
that

  1. 1.

$\Sigma_{N}\circ 0_{N}=\operatorname{id}_{N}$, where $0_{N}\colon N\rightarrow
TN$ is the $0$-section.

  2. 2.

$(\pi_{N}|_{\Omega},\Sigma_{N})\colon\Omega\rightarrow N\times N$ induces a
diffeomorphism onto an open neighborhood of the diagonal in $N\times N$.

Using the local addition, we obtain canonical charts for the mapping space:

###### 5.12. (Canonical charts)

For $f\in C^{\infty}(M,N)$ we let $f^{*}TN$ be the pullback bundle of the
tangent bundle of $N$ with associated bundle map $F\colon f^{*}TN\rightarrow
TN$. Via the bundle map $F$, we can identify the Fréchet space of sections
$\Gamma(M,f^{*}TN)$ (cf. 5.10) with

$C_{f}^{\infty}(M,TN)\colonequals\\{g\in C^{\infty}(M,N)\mid\pi_{N}\circ
g=f\\}\subseteq C^{\infty}_{\text{co}}(M,TN)$

in the subspace topology. In the following we will suppress this (harmless)
identification without further notice. Then define the sets

$\displaystyle V_{f}\colonequals\\{g\in C^{\infty}(M,E)\mid
g(M)\subseteq\Omega\\}\cap C^{\infty}_{f}(M,TN),$ $\displaystyle
U_{f}\colonequals\\{g\in
C^{\infty}(M,N)\mid(f,g)(M)\subseteq(\pi,\Sigma_{N})(\Omega)\\}\subseteq
C^{\infty}_{\text{co}}(M,N)$

and note that both are open in the compact open $C^{\infty}$-topology. Now
Proposition 5.6 implies that $\varphi_{f}\colon U_{f}\rightarrow
W_{f},g\mapsto(\pi_{N},\Sigma_{N})^{-1}\circ(f,g)$ is a homeomorphism with
inverse $\varphi_{f}^{-1}(\tau)=(\Sigma_{N})_{*}(\tau)$.

###### Theorem 5.13.

Let $M$ be a compact manifold with rough boundary and $N$ a metrisable
manifold modelled on a Fréchet space which admits a local addition. The atlas
of canonical charts $(U_{f},\varphi_{f})_{f\in C^{\infty}(M,N)}$ turns
$C^{\infty}_{\text{co}}(M,N)$ into a Fréchet manifold modelled on spaces of
sections $\Gamma(M,f^{*}TN)$. The manifold structure does not depend on the
choice of local addition.

###### Proof.

The compact open $C^{\infty}$-topology on $C^{\infty}(M,N)$ is clearly finer
than the compact open topology, whence $C^{\infty}_{\text{co}}(M,N)$ is
Hausdorff. In 5.12 we have already seen that the canonical charts form an
atlas of homeomorphisms. Hence we only have to prove that the change of charts
are smooth.

#### Change of charts formula.

To this end, one observes that $h\colonequals\varphi_{f}\circ\varphi_{g}^{-1}$
has an open domain, $O_{f,g}\subseteq\Gamma(M,g^{*}TN)$ for each pair $f,g\in
C^{\infty}(M,N)$. Let now $\tau$ be in $O_{f,g}$ and $x\in M$, then we obtain
the formula

$h\colon O_{f,g}\rightarrow\Gamma(M,f^{*}TN),\quad
h(\tau)(x)=(\pi_{N},\Sigma_{N})^{-1}(f(x),\Sigma_{N}\circ\tau(x))$ (7)

#### Localisation in charts

Choose an atlas of local trivialisations $(W_{i},\kappa_{i}^{f})_{1\leq i\leq
n}$, for the bundle $f^{*}TN$. Adjusting our choices if necessary, we may
assume that for each $1\leq i\leq n$ $W_{i}$ is

  1. 1.

a domain of a bundle trivialisations $(W_{i},\kappa_{i}^{g})$ of $g^{*}TN$

  2. 2.

the domain of a manifold chart $(W_{i},\psi_{i})$ of $M$.

Now construct the topological embedding
$\rho\colon\Gamma(M,f^{*}TN)\rightarrow\prod_{1\leq i\leq
n}C^{\infty}_{\text{co}}(W_{i},F)$ as in 5.10, where $F$ is the model space of
$N$. Since precomposition with a smooth is continuous in the compact open
$C^{\infty}$-topology by Proposition 5.6, we see that
$C^{\infty}_{\text{co}}(W_{i},F)\cong
C^{\infty}_{\text{co}}(\psi_{i}(W_{i}),F)$ as locally convex spaces (similarly
for $g^{*}TN$). We remark for later use that since $M$ is compact, $W_{i}$ and
also $\psi_{i}(W_{i})$ are locally compact.

#### Smoothness via the exponential law.

Now $\rho$ is a topological embedding with closed image, whence $h$ will be
smooth if and only if $\rho\circ h$ is smooth and this is the case if and only
if for each $1\leq i\leq n$ the mapping

$h_{i}\colon O_{f,g}\rightarrow
C^{\infty}(W_{i},F),\tau\mapsto\operatorname{pr}_{2}\circ\kappa_{i}\circ
h(\tau)|_{W_{i}}$

is smooth. Now we recall that $O_{f,g}\subseteq\Gamma(M,g^{*}TN)$ is open and
the spaces $\Gamma(M,g^{*}TN)$ and $C^{\infty}(W_{i},F)$ are Fréchet spaces by
5.10. Since $W_{i}$ is a manifold with rough boundary (being an open subset of
$M$) the exponential law [1, Theorem B] for smooth mappings on manifolds with
rough boundary yields: The $h_{i}$ (and thus $h$) are smooth if and only if
the mapping

$h_{i}^{\vee}\colon O_{f,g}\times W_{i}\rightarrow F,\quad
h_{i}^{\vee}(\tau,x)\colonequals h_{i}(\tau)(x)$

is smooth. However, (7) allows us to write

$h_{i}^{\vee}(\tau)(x)=\operatorname{pr}_{2}\circ(\pi_{N},\Sigma_{N})^{-1}\circ(f,\Sigma_{N}\circ(\operatorname{ev}(\tau|_{W_{i}}))(x),$
(8)

where $\operatorname{ev}\colon\Gamma(W_{i},g^{*}TN|_{W_{i}})\times
W_{i}\rightarrow g^{*}TN|_{W_{i}},(f,x)\mapsto f(x)$ is the evaluation map.
Using that $g^{*}TN|_{W_{i}}$ is trivial we identify
$\Gamma(W_{i},g^{*}TN|_{W_{i}})\cong
C^{\infty}_{\text{co}}(\psi_{i}(W_{i}),F)$ and deduce from [1, Proposition
3.20] that $\operatorname{ev}$ is smooth. As the restriction of $\tau$ is
smooth by 5.10, we deduce from (8) that $h_{i}^{\vee}$ is smooth as a
composition of smooth functions. Summing up the change of charts are smooth
and the canonical charts form indeed a smooth atlas turning $C^{\infty}(M,N)$
into a Fréchet manifold

#### The construction is independent of the choice of local addition.

Replacing the local addition $\Sigma_{N}$ by $\tilde{\Sigma}_{N}$, the change
of charts formula (7) shows that the change of charts between a chart with
respect to $\Sigma_{N}$ and with respect to $\tilde{\Sigma}_{N}$ will be
smooth. Hence the manifold structure does not depend on the choice of local
addition. ∎

###### Remark 5.14.

A crucial ingredient in the proof of Theorem 5.13 was compactness of $M$ to
endow the function space $C^{\infty}(M,N)$ with the compact open $C^{\infty}$
topology and to use the exponential law instead of the so called
$\Omega$-Lemma [25, 8.7]. Though the authors believe that for non-compact $M$,
the space $C^{\infty}(M,N)$ can be endowed with a manifold structure along the
lines described in [25], this would involve two significant steps: One has to
define a version of the fine very strong topology for mapping spaces on non-
compact manifolds of mappings and reprove the results outlined in [17] (mostly
trivial with the notable exception of continuity of the composition). Then one
needs an analogue of the $\Omega$-Lemma for manifolds with rough boundary
(which will be contained in [15], due to H. Glöckner, private communications).

##  6 Submanifolds with rough boundary and the proof of Theorem B

In this section we establish the global version of the splitting of spaces of
compactly supported sections. Our aim is to construct submersions between the
infinite-dimensional manifolds of mappings. To this end we need to clarify
first our concept of a submanifold with boundary sitting inside of manifolds
without boundary. If we require the submanifold with rough boundary to be
closed, then it will automatically have no narrow fjords. The authors believe
that they will also automatically satisfy the cusp condition but were not able
to prove the latter statement.

###### Definition 6.1. (Submanifold with rough boundary)

Let $M$ be a finite-dimensional manifold (possibly with rough boundary). A
subset $S\subseteq M$ is called _(embedded) submanifold with rough boundary_
of $M$ if for every $p\in S$ there is a chart $(U_{p},\varphi_{p})$ of $M$
with $p\in U_{p}$ and $\varphi_{p}(p)=0$ and a regular locally convex subset
$R_{p}$ such that $\varphi_{p}(S\cap U_{p})=\varphi_{p}(U_{p})\cap R_{p}$.

If for every $p\in S$ the regular locally convex set is a relatively open set
in a quadrant (cf. Definition 5.1), then we say that $S$ is an (embedded)
_submanifold with corners_.

If in addition $S$ is a closed subset which satisfies the above conditions, we
say that $S$ is a _closed submanifold with rough boundary_ (or _with corners_
, respectively).

###### Remark 6.2.

  1. 1.

A submanifold with rough boundary inherits the structure of a manifold with
rough boundary from the ambient manifold and this structure turns the
inclusion $\iota_{S}\colon S\rightarrow M$ into a smooth embedding. Thus
submanifolds with rough boundary as defined here are initial submanifolds,
i.e. a mapping $f\colon N\rightarrow S\subseteq M$ between manifolds with
rough boundary is smooth as a map into $S$ if and only if it is smooth as a
map into $M$. Note that closed submanifolds of $\sigma$-compact manifolds are
again $\sigma$-compact.

  2. 2.

We remark that our definition of an embedded submanifold with corners is a
special case of a submanifold with corners as in [25, 2.5]. Since we are only
interested in a very specialised case, we do not need the more general
definition. In particular, we refrain from defining submanifolds of lower
dimension (which could be done as usual but is not needed here).

  3. 3.

Due to our definition an embedded submanifold with rough boundary $S\subseteq
M$ is regular: Consider $x\in\partial S$ and let $(U_{\varphi},\varphi)$ be a
submanifold chart. Then $\varphi(x)\in\varphi(U_{\varphi}\cap
S)=\varphi(U_{\varphi})\cap C$ for a regular locally convex set $C$. Hence
$W\colonequals\varphi^{-1}(\varphi(U_{\varphi}\cap C^{\circ}))\subseteq S$ is
an open set in $M$, whence contained in the interior of $S$. Choosing a
sequence in W we can approximate $\varphi(x)$, whence
$x\in\overline{S^{\circ}}$. Thus $S$ is regular and we see in addition that
$x\in\partial S$ entails $\varphi(x)\in\partial\varphi(U_{\varphi})$.

Before we continue, let us construct a class of examples for submanifolds with
rough boundary of a Riemannian manifold which will be used to prove Corollary
D from the introduction.

###### Lemma 6.3.

Let $M$ be a Riemannian manifold and $C$ be a regular closed subset which is
strongly convex, i.e. for every $p,q\in C$ exists a unique minimal geodesic
segment $\overline{pq}$ connecting $p$ and $q$ such that
$\overline{pq}\subseteq C$. Then $C$ is a submanifold of $M$ with rough
boundary.

###### Proof.

By standard Riemannian geometry, we can choose for every $p\in C$ an open
$0$-neighborhood $W_{p}\subseteq T_{p}M$ such that the restriction of the
Riemannian exponential map $\exp_{p}\colonequals\exp|_{W_{p}}\colon
W_{p}\rightarrow M$ induces a diffeomorphism onto an (open) $p$-neighborhood
in $M$. We will show that the manifold charts $\exp_{p}$ induce suitable
submanifold charts with rough boundary.

Clearly if $p\in C^{\circ}$ we can just shrink $W_{p}$ to obtain such a
submanifold chart. For $q\in\partial C$ (the boundary of $C$) we have to work
harder. Define for $q\in\partial C$ the set

$V_{q}\colonequals\\{\lambda w\in T_{q}M\mid 0<\lambda<\infty,w\in W_{q}\text{
and }\exp_{q}(w)\in C^{\circ}\\}.$

We observe that
$V_{q}=\bigcup_{0<\lambda<\infty}\lambda\cdot\exp_{q}^{-1}(C^{\circ})$ is an
open subset of $T_{q}M$. Now we exploit the geometric properties of strongly
convex sets following [19], where these sets are called
"konvex".121212Loc.cit. assumes that $M$ is a complete Riemannian manifold. We
do not assume completeness as the parts of [19] needed here do not rely on the
completeness of $M$. In fact, the geodesic segments needed in the proofs exist
since we are working in the strongly convex set $C$. As $C$ is strongly
convex, also the interior $C^{\circ}$ is strongly convex [19, Korollar 4.5.1],
whence it is also "schwach konvex" (weakly convex) in the terminology of
loc.cit.. Now regularity of $C$ implies that $q\in\partial C=\partial
C^{\circ}$. We can thus copy the argument in the proof of [19, 4.9.2] verbatim
(note that the geodesics occuring there are only needed locally in a small
neighborhood around $q$!) to establish that $V_{q}$ is a convex cone in
$T_{q}M$ whose tip is $0_{q}$. In particular $\overline{V}_{q}$ is a convex
cone, i.e. a closed subset with dense interior that is (locally) convex.

#### Claim:

$\exp_{q}(W_{q}\cap\overline{V}_{q})=C\cap\exp_{q}(W_{q})$. If this is true
then $\exp_{q}$ restricts to a submanifold chart (with rough boundary) for $C$
around $q$ as $W_{q}\cap\overline{V}_{q}$ has dense interior (namely
$W_{q}\cap V_{q}$) and is locally convex as an intersection of two (locally)
convex sets in $T_{p}M$. We conclude that $C$ is a closed embedded submanifold
with rough boundary of the Riemannian manifold $M$.

#### Proof of the claim:

Let us first observe that $\exp_{q}(V_{q}\cap W_{q})=\exp_{q}(W_{q})\cap
C^{\circ}$ by construction of $V_{q}$ (and the diffeomorphism property of
$\exp_{q}$). Now let $x\in W_{q}\cap\overline{V}_{q}$. Since the interior of
this subset is the dense set $W_{q}\cap V_{q}$, we can choose and fix a
sequence $(x_{n})_{n\in\mathbb{N}}\subseteq V_{q}\cap W_{q}$ with
$\lim_{n}x_{n}=x$. By continuity of $\exp_{q}$ and since $\exp_{q}(x_{n})\in
C^{\circ}\cap\exp_{q}(W_{q})$ we have
$\exp_{q}(x)\in\exp_{q}(W_{q})\cap\overline{C^{\circ}}=\exp_{q}(W_{q})\cap C$.
Conversely, if $p_{n}\in C^{\circ}\cap\exp_{q}(W_{q})$ is a sequence
converging to $p\in C\cap\exp_{q}(W_{q})$ we use continuity of $\exp_{q}^{-1}$
to see that $p\in\exp_{q}(W_{q}\cap\overline{V}_{q})$. Summing up, the claim
follows. ∎

###### Remark 6.4.

Actually, strongly convex subsets have very nice boundary behaviour. For
example, it is known that they have Lipschitz boundary (cf. e.g. [2]).
However, we are not aware of another source in the literature where
submanifold charts of the above kind are explicitely constructed.

In light of Example 3.3 this implies that a strongly convex regular closed
subset satisfies the cusp condition.

Encouraged by these results, we shall now prove that every closed submanifold
with rough boundary satsifies the cusp condition, Definition 3.1.

###### Proposition 6.5.

Let $M$ be a Riemannian manifold and $C$ be a closed submanifold with rough
boundary. Then $C$ satisfies the cusp condition.

###### Proof.

Following Remark 6.2 we already know that $C$ is a regular closed set. We
first have to check the no narrow fjord condition. To this end, fix $x\in C$
together with a manifold chart $\varphi\colon
U\rightarrow\varphi(U)\subseteq\mathbb{R}^{n}$ around $x$ and $\varphi(U\cap
C)=\varphi(U)\cap R$ for a suitable regular locally convex set $R$. Shrinking
the chart if necessary, we can assume that it is bi-Lipschitz with respect to
the geodesic length metric and the Euclidean metric (cf. Remark 3.9). Then
arguing as in Lemma 3.7, it suffices to check the no narrow fjords condition
for $\varphi(x)$ as an element of the locally convex subset $\varphi(U)\cap
R\subseteq\mathbb{R}^{n}$ (note that this set need not be closed whence it
does not make sense to say that it has no narrow fjords!)

Now since $U\cap C$ is the intersection of an open and a closed subset of a
locally compact space, it is locally compact, whence $\varphi(U\cap
C)=\varphi(U)\cap R$ is locally compact. We can thus choose a compact
neighborhood $K$ of $\varphi(x)$ in $\varphi(U)\cap R$. Then local convexity
of $\varphi(U)\cap R$ allows us to choose a neighborhood $\varphi(x)\in
W\subseteq K$ which is convex. Denote now by
$K_{\varphi(x)}=\overline{W}\subseteq K$ the closure of $W$. It is again a
convex set and compact by construction. Now an easy but tedious computation
involving metric estimates, convexity of $K_{\varphi(x)}$ and the boundary
behavior observation $\partial(\varphi(U)\cap R)\cap
K_{\varphi(x)}\subseteq\partial K_{\varphi(x)}$ yields the constants needed to
verify the no narrow fjords condition for $K_{\varphi(x)}$. Alternatively,
observe that convex sets have Lipschitz boundary (cf. again [2]) which implies
that $K_{\varphi(x)}$ has no narrow fjords.

To check the outward polynomial cusp condition, we use again the compact,
convex neighborhood $K_{\varphi(x)}$ of $\varphi(x)$ in $\varphi(C\cap U)$,
and a diffeomorphism $\varphi(U)\simeq\mathbb{R}^{n}$ (which is locally bi-
Lipschitz). Since the image of the compact convex neighborhood with Lipschitz
boundary yields the required estimates for the polynomial outward cusp
condition, the diffeomorphism transfers them from $\varphi(U)$ to
$\mathbb{R}^{n}$ (and so satisfying Frerick’s version of the definition).
Invoking Lemma 3.6, we see that $C\cap U$ satisfies the condition. Hence $C$
has at worst polynomial outward cusps, completing the proof. ∎

For regular closed subsets which are at the same time submanifolds with rough
boundary we prove now that for every vector bundle the space of sections from
Definition 4.2 can canonically be identified with the sections of the
corresponding pullback bundle over the submanifold with rough boundary.

###### Proposition 6.6.

Let $\pi_{E}\colon E\rightarrow M$ be a rank $k$ vector bundle over a manifold
without boundary. Let $C\subseteq M$ be a closed submanifold with rough
boundary. Assume in addition that

  1. 1.

$C$ is a submanifold with corners, or

  2. 2.

$C$ is compact.

Then the pullback
$p_{C}\colon\Gamma_{c}(M,E)\rightarrow\Gamma_{c}(C,\iota_{C}^{*}E),\
\sigma\mapsto\sigma\circ\iota_{C}$ by the inclusion $\iota_{C}\colon
C\rightarrow M$ is a linear quotient map.

###### Proof.

Observe that since $C\subseteq M$ is closed, the inclusion $\iota_{C}$ is a
proper mapping (i.e. preimages of compact sets are compact). Thus by our
definition of submanifold with rough boundary (or with corners), every section
$\sigma\in\Gamma_{c}(M,E)$ induces a smooth pullback section
$\sigma\circ\iota_{C}\in\Gamma_{c}(C,\iota_{C}^{*}E)$ and the pullback map
$p_{C}$ makes sense and is clearly linear. Further, taking canonical
identifications, a section $\tau\in\Gamma_{c}(C,\iota_{C}^{*}E)$ clearly
coincides with a smooth mapping $\tilde{\tau}\colon C\rightarrow E$ with
compact support such that $\pi_{E}\circ\tilde{\tau}=C$ (here we use
$\tilde{\tau}$ to mark the difference in the codomain). Now Proposition 6.5
implies that $C$ has no narrow fjords, whence by Proposition 4.5 there is
$\hat{\tau}\in\Gamma_{c}(M,E)$ which restricts to $\tilde{\tau}$ on $C$. We
deduce that $p_{C}(\hat{\tau})=\tau$, whence $p_{C}$ is surjective.

To establish continuity we have to distinguish the two cases, due to the
difference in the function space topologies.

  1. 1.

($C$ is a submanifold with corners) The map $\iota_{C}^{*}\colon
C^{\infty}_{\text{fS}}(M,E)\rightarrow C^{\infty}_{\text{fS}}(C,E),f\mapsto
f\circ\iota_{C}$ is continuous by [25, Theorem 7.3] (and even smooth) as
$\iota_{C}$ is proper. Consider the linear subspace
$\mathcal{D}_{\iota_{C}}(C,E)\colonequals\\{g\in
C^{\infty}(C,E)\mid\pi_{E}\circ g=\iota_{C},g\equiv 0\text{ off some compact
set in }C\\}$ of $C^{\infty}_{fS}(C,E)$. It is easy to see that
$\iota_{C}^{*}$ restricts to a continuous mapping
$I\colon\Gamma_{c}(M,E)\rightarrow\mathcal{D}_{\iota_{C}}(C,E).$ However, due
to the definition of the pullback bundle (see [25, 1.18 and 1.19]), the space
$\mathcal{D}_{\iota_{C}}(C,E)$ is isomorphic as a linear and topological space
to $\Gamma_{c}(C,\iota_{C}^{*}(E))$, composing $I$ with this isomorphism we
obtain $p_{C}$ which is thus continuous.

  2. 2.

($C$ is compact) Since $M$ is $\sigma$-compact, we can choose and fix a
locally finite (countable) atlas of bundle trivialisations
$(W_{i},\kappa_{i})_{i\in\mathbb{N}}$ for $E$ such that every $W_{i}$ is
relatively compact. Since $C$ is compact, only finitely many $W_{i}$ intersect
$C$. After reordering, we may assume that $W_{i}\cap C\neq\emptyset$ iff
$1\leq i\leq n$ for some $n\in\mathbb{N}$. Observe that since $W_{i}$ is an
open subset of $M$ and $C$ is an embedded submanifold (with rough boundary),
for $i\leq n$, the set $\iota_{C}^{-1}(W_{i})=W_{i}\cap C$ is a submanifold
with rough boundary of $W_{i}$ (as submanifold charts $\varphi\colon
U\rightarrow\varphi(U)$ are bijections, we find $\varphi(W_{i}\cap U\cap
C)=\varphi(W_{i}\cap U)\cap R$). Hence Proposition 5.6 implies that
$p_{i}\colon C^{\infty}_{\text{co}}(W_{i},\mathbb{R}^{k})\rightarrow
C^{\infty}_{\text{co}}(W_{i}\cap C,\mathbb{R}^{k}),f\mapsto f|_{C\cap W_{i}}$
is continuous linear for $i\leq n$. Thus we obtain a continuous (linear) map
$q\colon\bigoplus_{i\in\mathbb{N}}C^{\infty}_{\text{co}}(W_{i},\mathbb{R}^{d})\rightarrow\bigoplus_{i\leq
n}C^{\infty}_{\text{co}}(W_{i}\cap
C,\mathbb{R}^{d}),(f_{i})_{i}\mapsto(p_{i}(f_{i}))_{1\leq i\leq n}$. Following
Lemma A.5 and 5.10 we obtain a commutative diagram

$\textstyle{\Gamma_{c}(M,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{p_{C}}$$\textstyle{\displaystyle\bigoplus_{i\in\mathbb{N}}\Gamma(W_{i},E|_{W_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\displaystyle\bigoplus_{i\in\mathbb{N}}C^{\infty}_{\text{co}}(W_{i},\mathbb{R}^{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\textstyle{\Gamma(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\displaystyle\bigoplus_{1\leq
i\leq n}\Gamma(W_{i}\cap C,(\iota_{C}^{*}E)|_{W_{i}\cap
C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\displaystyle\bigoplus_{1\leq
i\leq n}C^{\infty}(W_{i}\cap C,\mathbb{R}^{d})}$

where the the image the $\rho$ are topological embeddings with closed images.
Hence $p_{C}$ is continuous in this case.

Finally, let us establish that $p_{C}$ is a quotient map, i.e. $p_{C}$ is
open. To this end, recall from Appendix A that $\Gamma_{c}(M,E)$ is an
(LF)-space, i.e. webbed and ultrabornological. Further, if $C$ is a manifold
with corners, also $\Gamma_{c}(C,\iota_{C}^{*}E)$ is an (LF) space. If $C$ is
compact and a manifold with rough boundary, then $\Gamma(C,\iota_{C}^{*}E)$ is
even a Fréchet space by 5.10. In both cases, $p_{C}$ is open by the open
mapping theorem [24, 24.30]. ∎

###### Proposition 6.7.

Let $M$ be a manifold and $E\rightarrow M$ be a rank $k$ vector bundle. If $C$
is closed submanifold with rough boundary which satisfies the assumptions of
Proposition 6.6, then $\Gamma_{c}(C,E)=\Gamma_{c}(C,\iota_{C}^{*}E)$ as
locally convex vector spaces.

###### Proof.

Since $C$ is an embedded submanifold, a section in the pullback bundle is
smooth if and only if it is a smooth as a mapping $C\rightarrow E$. Thus as
sets we canonically identify $\Gamma_{c}(C,E)=\Gamma_{c}(C,\iota_{C}^{*}E)$.
Now Proposition 6.6 and Proposition 4.5 yield a commutative diagram

$\textstyle{\Gamma_{c}(M,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{C}}$$\scriptstyle{p_{C}}$$\textstyle{\Gamma_{c}(C,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\Gamma_{c}(C,\iota_{C}^{*}E)}$$\scriptstyle{\operatorname{id}}$

where the diagonal arrows are quotient mappings. ∎

###### 6.8. (Short reminder: Manifolds of mappings for non-compact source
manifolds)

Assume that $C$ is a (sub-)manifold with corners which is possibly non-
compact. Then the function space $C^{\infty}(C,N)$ can be endowed with an
infinite-dimensional manifold structure which constructed similarly to the
construction outlined in Section 5: One endows $C^{\infty}(C,N)$ withe the
$\mathcal{FD}$-topology described in [25] (a Whitney type topology). In the
boundaryless case [17], this topology is also called the fine very strong
topology and therefore we denote by $C^{\infty}_{\text{fS}}(C,N)$ the function
space with the fine very strong (=$\mathcal{FD}$-) topology.

Choosing a local addition on $N$, the construction of manifold charts is
completely analogous to the construction outlined in 5.12 with the notable
exception that one has to restrict to $\Gamma_{c}(C,f^{*}TN)$ and one has to
intersect $U_{f}$ with

$\\{g\in C^{\infty}(C,N)\mid\exists K\text{ compact, such that }\forall x\in
C\setminus K\ f(x)=g(x)\\}.$

The rest of the construction is completely analogous to the one outlined in
5.12 and yields the same structure as in Theorem 5.13 if $C$ is compact (note
that we will thus also write
$C^{\infty}_{fS}(C,N)=C^{\infty}_{\text{co}}(C,N)$ if $C$ is compact with
rough boundary).

We are now ready to prove Theorem B which we restate here for the reader’s
convenience. Recall that $M$ is equipped with a Riemannian metric.

###### Theorem 6.9.

For $C\subset M$ a submanifold with corners, or compact and a submanifold with
rough boundary, then the restriction map $\operatorname{res}_{C}^{M}\colon
C^{\infty}_{\text{fS}}(M,N)\rightarrow C^{\infty}_{\text{fS}}(C,N)$ is a
submersion of locally convex manifolds.

###### Proof.

Let $\iota_{C}\colon C\rightarrow M$ be the canonical inclusion, which is
smooth as $C$ is an embedded submanifold. Hence
$\operatorname{res}_{C}^{M}=\iota_{C}^{*}$ is smooth by [25, Theorem 7.3] (if
$C$ is a submanifold with corners). Since the compact-open $C^{\infty}$
topology is coarser than the fine very strong topology (cf. [17]), Proposition
5.6 implies that $\operatorname{res}_{C}^{M}$ is continuous if $C$ is compact
and a submanifold with rough boundary (note that $M$ might be non-compact).
Hence to establish smoothness and the submersion property, it suffices to
construct submersion charts for $\operatorname{res}_{C}^{M}$.

Let now $F\in C^{\infty}(M,N)$ and
$f\colonequals\operatorname{res}_{C}^{M}(F)$. Then we use that $C$ satisfies
the cusp condition and consider the canonical charts $(U_{F},\varphi_{F})$ and
$(U_{f},\varphi_{f})$ (cf. 5.12) to obtain a commutative diagram

$\textstyle{C^{\infty}_{\text{fS}}(M,N)\supseteq
U_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{C}^{M}}$$\scriptstyle{\varphi_{F}}$$\textstyle{\Gamma_{c}(M,F^{*}E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Corollary
4.14$\scriptstyle{\operatorname{res}_{C}}$$\textstyle{\mathcal{I}_{c}(C,M)\oplus\Gamma_{c}(C,F^{*}TN)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{2}}$$\textstyle{C^{\infty}_{\text{fS}}(C,N)\supseteq
U_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{f}}$$\textstyle{\Gamma_{c}(C,f^{*}TN)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$Proposition
6.7$\textstyle{\Gamma_{c}(C,F^{*}TN)}$

Observe that $\operatorname{res}_{C}^{M}$ is a smooth submersion as the
canonical charts conjugate it to a projection onto a complemented closed
subspace, which is continuous linear. ∎

Note that Corollary D from the introduction follows from the results in this
section and Corollary C in the wash as Lemma 6.3 asserts that strongly convex
regular closed subsets of Riemannian manifolds are submanifolds with rough
boundary.

## Acknowledgements

The first author thanks various commenters on MathOverflow, especially Uri
Bader, as well as Benoît Kloeckner for helpful discussions. The second author
thanks Helge Glöckner for useful comments on manifolds with rough boundary and
their spaces of mappings (in particular that Theorem 5.13 follows from the
exponential law). Both authors thank Seppo Hiltunen who made them aware of a
critical error in an earlier version of this paper.

##  Appendix A  Essentials on infinite-dimensional calculus and function
spaces

In this appendix we collect the necessary background on the theory of
manifolds that are modelled on locally convex spaces and how spaces of smooth
maps can be equipped with such a structure. Let us first recall some basic
facts concerning differential calculus in locally convex spaces.

### Calculus in locally convex spaces

We base our investigation on the so called Bastiani calculus [3] and our
exposition here follows [11, 27].

###### Definition A.1.

Let $E,F$ be locally convex spaces, $U\subseteq E$ be an open subset, $f\colon
U\rightarrow F$ a map and $r\in\mathbb{N}_{0}\cup\\{\infty\\}$. If it exists,
we define for $(x,h)\in U\times E$ the directional derivative

$df(x,h)\colonequals D_{h}f(x)\colonequals\lim_{t\rightarrow
0}t^{-1}\big{(}f(x+th)-f(x)\big{)}.$

We say that $f$ is $C^{r}$ if the iterated directional derivatives

$d^{(k)}f(x,y_{1},\ldots,y_{k})\colonequals(D_{y_{k}}D_{y_{k-1}}\cdots
D_{y_{1}}f)(x)$

exist for all $k\in\mathbb{N}_{0}$ such that $k\leq r$, $x\in U$ and
$y_{1},\ldots,y_{k}\in E$ and define continuous maps $d^{(k)}f\colon U\times
E^{k}\rightarrow F$. If $f$ is $C^{\infty}$ it is also called smooth. We
abbreviate $df\colonequals d^{(1)}f$ and for curves $c\colon I\rightarrow M$
on an interval $I$, we also write
$\dot{c}(t)\colonequals\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}c(t)\colonequals
dc(t,1)$.

We will frequently want to work with smooth mappings on non-open sets.
Contrary to the treatment in the main body of the text we restrict ourselves
here to regular sets which are locally convex.

###### Definition A.2. (Differentials on non-open sets)

  1. 1.

A subset $U$ of a locally convex space $E$ is _locally convex_ if every $x\in
U$ has a convex neighborhood $V$ in $U$.

  2. 2.

Let $U\subseteq E$ be a locally convex subset with dense interior and $F$ a
locally convex space. A continuous mapping $f\colon U\rightarrow F$ is called
$C^{r}$ if $f|_{U^{\circ}}\colon U^{\circ}\rightarrow F$ is $C^{r}$ and each
of the maps $d^{(k)}(f|_{U^{\circ}})\colon U^{\circ}\times E^{k}\rightarrow F$
admits a continuous extension $d^{(k)}f\colon U\times E^{k}\rightarrow F$
(which is then necessarily unique).

Note that for $C^{r}$-mappings on regular locally convex subsets the chain
rule holds (whereas it becomes false in general without requiring local
convexity, cf. Lemma 1.4). Hence there is an associated concept of locally
convex manifold with rough boundary.

### Topologies on function spaces with non-compact source

In this appendix we recall some basic facts on the topology of spaces of
smooth sections in vector bundles over a non-compact manifold. For the rest of
this section we let $M,N$ be finite-dimensional manifolds and $p\colon
E\rightarrow M$ be a vector bundle over $M$. Further, we denote by
$\Gamma(M,E)$ all smooth sections of the bundle and by
$\Gamma_{c}(M,E)\subseteq\Gamma(M,E)$ the space of compactly supported
sections.

###### A.3.

For the space of smooth mappings between manifolds with corners
$C^{\infty}(M,N)$ we consider the so called $\mathcal{FD}$-topology or _fine
very strong topology_ and write $C^{\infty}_{\text{fS}}(M,N)$ for the space
endowed with this topology. This is a Whitney type topology controlling
functions and their derivatives on locally finite families of compact sets.
Before we describe a basis of the fine very strong topology, we have to
construct a basis for the strong topology which we will then refine To this
end, we recall the construction of the so called basic neighborhoods (see
[17]). Consider $f$ smooth, $A$ compact, $\varepsilon>0$ together with a pair
of charts $(U,\psi)$ and $(V,\varphi)$ such that $A\subseteq V$ and $\psi\circ
f\circ\varphi^{-1}$ makes sense. Then we use multiindex notation B.1 to define
an _elementary $f$-neighborhood_

$\mathcal{N}^{r}\left(f;A,\varphi,\psi,\epsilon\right)\colonequals\left\\{\begin{subarray}{c}\displaystyle
g\in C^{\infty}(M,N),\quad\psi\circ g|_{A}\quad\text{ makes sense,}\\\
\displaystyle\sup_{\alpha\in\mathbb{N}_{0}^{d},|\alpha|<r}\sup_{x\in\varphi(A)}\lVert\partial^{\alpha}\psi\circ
f\circ\varphi^{-1}(x)-\partial^{\alpha}\psi\circ
g\circ\varphi^{-1}(x)\rVert<\varepsilon\end{subarray}\right\\},.$

A basic neighborhood of $f$ arises now as the intersection of (possibly
countably many) elementary neighborhoods
$\mathcal{N}^{r}\left(f;A_{i},\varphi_{i},\psi_{i},\epsilon_{i}\right)$ where
the family $(V_{i},\varphi_{i})_{i\in I}$ is locally finite. We remark that
basic neighborhoods form the basis of the very strong topology (see [17] for
more information). To obtain the fine very strong topology, one declares the
sets

$\\{g\in C^{\infty}(M,N)\mid\exists K\subseteq M\text{ compact such that
}\forall x\in M\setminus K,\ g(x)=f(x)\\}$ ($\star$)

to be open and constructs a subbase of the fine very strong topology as the
collection of sets $\eqref{eq: cpt:nbhd}$ (where $f\in C^{\infty}(M,N)$ and
the basic neighborhoods of the very strong topology.

Note that the fine very strong topology is only defined for manifolds without
boundary (and coincides with the $\mathcal{FD}$-topology, see [17, Appendix
C]) and we refer to [25] for more information on the $\mathcal{FD}$-topology
in the case of a manifold with corners.

If $M$ is compact, all topologies mentioned above coincides with the compact
open $C^{\infty}$-topology from Definition 5.4. Further, the fine-very strong
topology turns $C^{\infty}(M,N)$ into an infinite-dimensional manifold (cf.
[25] and [17]). If $N=\mathbb{R}^{n}$ then the pointwise operations turn
$C^{\infty}_{\text{fS}}(M,\mathbb{R}^{n})$ into a locally convex vector space
(which is in not a Fréchet space if $M$ is not compact).

We now turn to the space of compactly supported sections of a vector bundle.

###### A.4.

Compactly supported sections of a vector bundle Let $p\colon E\rightarrow M$
be a finite rank vector bundle over the finite dimensional manifold $M$
(possibly with corners). We consider three spaces of sections

$\displaystyle\Gamma(M,E)$ $\displaystyle\colonequals\\{f\in
C^{\infty}(M,E)\mid p\circ\sigma=\operatorname{id}_{M}\\},$
$\displaystyle\Gamma_{K}(M,E)$
$\displaystyle\colonequals\\{f\in\Gamma(M,E)\mid\text{supp }f\subseteq
K\text{for }K\subseteq M\text{ compact}\\},$ $\displaystyle\Gamma_{c}(M,E)$
$\displaystyle\colonequals\bigcup_{K\subseteq M\text{
compact}}\Gamma_{K}(M,E).$

Endow $\Gamma(M,E),\Gamma_{K}(M,E)$ and $\Gamma_{c}(M,E)$ with the subspace
topology from $C^{\infty}_{\text{fS}}(M,E)$ we obtain locally convex vector
spaces [25, Proposition 4.8 and Remark 4.11]. Moreover, we remark that by
compactness of $K$, the topology on $\Gamma_{K}(M,E)$ coincides with the
subspace topology induced by the compact open topology, whence one can prove
that $\Gamma_{K}(M,E)$ is a Fréchet space (cf. e.g. [12, Section 3.1])
Further, by [25, Proposition 4.8 and Remark 4.11] $\Gamma_{c}(M,E)$ is the
inductive limit (in the category of locally convex spaces) of the Fréchet
spaces $\Gamma_{K}(M,E)$, where $K$ runs through a compact exhaustion of $M$.
Thus $\Gamma_{c}(M,E)$ is a (LF)-space.

If $M$ is a manifold without boundary, a different description of the topology
on $\Gamma_{c}(M,E)$ is considered in [13, Appendix F]. By the following
lemma, this topology on $\Gamma_{c}(M,E)$ is equivalent to the fine very
strong subspace topology.

###### Lemma A.5.

If $M$ is a manifold without boundary, the following describe equivalent
locally convex topologies on $\Gamma_{c}(M,E)$:

  1. 1.

Give $\Gamma_{c}(M,E)\subseteq C^{\infty}_{\text{fS}}(M,E)$ the subspace
topology.

  2. 2.

Let $\mathcal{U}=\left\\{U_{i}\right\\}_{i\in I}$ be a locally finite cover of
$M$ by relatively compact open subsets $U_{i}\subseteq M$. Equip each
$\Gamma(U_{i},E|_{U_{i}})\subseteq C^{\infty}_{\text{co}}(U_{i},E|_{U_{i}})$
with the subspace topology, and give $\Gamma_{c}(M,E)$ the topology induced by

$\displaystyle\rho_{\mathcal{U}}\colon\Gamma_{c}(M,E)\to\bigoplus_{i\in
I}\Gamma(U_{i},E|_{U_{i}}),$

where the direct sum is given the box topology, and
$\rho_{i}\colon\Gamma_{c}(M,E)\to\Gamma(U_{i},E|_{U_{i}})$ is the restriction
map.

  3. 3.

For $K\subseteq M$ compact, give

$\displaystyle\Gamma_{K}(M,E)=\left\\{s\in\Gamma(M,E):\operatorname{supp}s\subseteq
K\right\\}\subseteq C^{\infty}(M,E)_{\text{co}}$

the subspace topology. Now equip $\Gamma_{c}(M,E)$ with the final topology
with respect to the inclusions
$\iota_{K}\colon\Gamma_{K}(M,E)\to\Gamma_{c}(M,E)$ as $K$ ranges through the
compact subsets of $M$.

###### Proof.

Note that the topology described in 3. is the inductive limit topology induced
by the inductive system $\\{\iota_{K}\\}_{K\subseteq M\text{ is compact}}$
(ordered by inclusion of compact sets). Hence the topologies described in 1.
and 3. coincide by A.4. However, also the topologies 2. and 3. coincide by
[13, F.19]. ∎

###### Remark A.6.

Assume that the sets $U_{i}$ from Lemma A.5 2\. are domains of manifold charts
$\varphi_{i}\colon U_{i}\rightarrow\mathbb{R}^{m}$ of $M$. Then we define for
each $X\in\Gamma(U_{i},E|_{U_{i}})$ the local representative

$X_{\varphi_{i}}\colonequals\operatorname{pr}_{2}\circ T\varphi_{i}\circ
X\varphi_{i}^{-1}\in C^{\infty}(\varphi_{i}(U_{i}),\mathbb{R}^{m}),$

where
$\operatorname{pr}_{2}\colon\varphi_{i}(U_{i})\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$
is the canonical projection. This mapping yields an isomorphism of locally
convex spaces

$\Gamma(U_{i},E|_{U_{i}})\rightarrow
C^{\infty}_{\text{co}}(\varphi_{i}(U_{i}),\mathbb{R}^{m}),\quad X\mapsto
X_{\varphi_{i}}.$

##  Appendix B  The space of (smooth) Whitney jets

In this appendix we recall some details from Whitney’s approach to the the
extension problem for smooth functions on a closed subset of $\mathbb{R}^{n}$.
Though the exposition in the main part of the article does not need these
results as such (since we will only cite their consequences from [9]). The
authors think that a quick recollection of these constructions will be
beneficial to understand the underlying ideas. Our exposition follows here [9,
Section 2] and mention that a more in depth treatment can be found in [5,
Chapter 2].

###### B.1.

Throughout this section we use standard multiindex notation,
i.e. $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{d})\in\mathbb{N}_{0}^{d}$,
$|\alpha|=\sum_{i=1}^{d}\alpha_{i}$, $\alpha!=\alpha_{1}!\cdots\alpha_{d}!$
and
$\partial^{\alpha}=\partial_{1}^{\alpha_{1}}\cdots\partial_{d}^{\alpha_{d}}$.

###### Definition B.2.

Let $K\subseteq\mathbb{R}^{d}$ be compact. For
$f=(f^{\alpha})\in\prod_{\alpha\in\mathbb{N}^{d}_{0}}C(K)$ and $x\in K$ we
define the formal Taylor polynomial

$\operatorname{Tay}_{x}^{m}f(y)\colonequals\sum_{|\alpha|\leq
m}\frac{f^{\alpha}(x)}{\alpha!}(y-x)^{\alpha}.$

The formal Taylor remainder $R^{m}_{x}f\in\prod_{|\alpha|\leq m}C(K)$ is then
defined by

$(R_{x}^{m}f)^{\alpha}\colonequals
f^{\alpha}-\partial^{\alpha}(\operatorname{Tay}_{x}^{m}f)|_{K}=f^{\alpha}-\operatorname{Tay}^{m-|\alpha|}_{x}(f^{\alpha+\beta})_{|\beta|\leq
m-\alpha}|_{K},\quad|\alpha|\leq m.$

###### B.3.

For $|\alpha|<m$ one easily checks the identities

$\displaystyle\partial^{\alpha}\operatorname{Tay}_{x}^{m}f=\operatorname{Tay}_{x}^{m-|\alpha|}((f^{\alpha+\beta})_{|\beta|\leq
m-|\alpha|})$ $\displaystyle
R_{x}^{m}(f)^{\alpha}=(R_{x}^{m-|\alpha|}(f^{\alpha+\beta})_{|\beta\leq
m-|\alpha|})^{0}$

We now define seminorms on the spaces of jets which allow us to define a
Fréchet topology on the space of Whitney jets (cf. Definition B.7 below).

###### Definition B.4.

For $f=(f^{\alpha})\in\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K)$ and
$m\in\mathbb{N}_{0}$ we define the seminorms

$\displaystyle|f|_{m,K}\colonequals\sup_{x\in K}\sup_{|\alpha|\leq
m}|f^{\alpha}(x)|.$

The seminorms $|\cdot|_{m,K}$ are closely connected to the compact open
$C^{\infty}$-topology on the space $C^{\infty}(U,\mathbb{R})$ as we will
discuss in B.6 below.

###### Definition B.5.

For $f=(f^{\alpha})\in\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K)$,
$m\in\mathbb{N}_{0}$ and $t>0$, we define

$\displaystyle q_{m}(f,t)$ $\displaystyle\colonequals q_{m}(K,f,t)$
$\displaystyle\colonequals\sup\\{|R_{x}^{m}(f)^{\alpha}(y)||y-x|^{|\alpha|-m}\colon
x,y\in K,0<|x-y|\leq t,|\alpha|\leq m\\}$

and

$\displaystyle\lVert\cdot\rVert_{m,K}\colonequals|\cdot|_{m,K}+\sup_{t}q_{m}(K,\cdot,t)$

###### B.6.

Let $U\subseteq\mathbb{R}^{d}$ be open and $K_{1}\subseteq
K_{2}\subseteq\ldots$ a fundamental sequence of compact sets.131313i.e.
$K_{1}\subseteq K_{2}\subseteq K_{3}\subseteq\ldots$ are compact and
$U=\bigcup_{l}K_{l}$. Then we recall from Remark 5.5 that the compact open
$C^{\infty}$-topology is initial with respect to the map

$\partial\colon
C^{\infty}(U,\mathbb{R})\rightarrow\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(U),\quad
f\mapsto(\partial^{\alpha}f).$

Pulling back the seminorms on the jet space by the mappings
$\partial(\cdot)|_{K_{l}}$, we obtain two families of seminorms:

  1. 1.

$\\{|\partial(\cdot)|_{K_{l}}|_{m,K_{l}}\mid
l\in\mathbb{N},m\in\mathbb{N}_{0}\\}$. These seminorms are the classical
seminorms which induce the compact open $C^{\infty}$-topology (cf. e.g. [12,
27]).

  2. 2.

$\\{\lVert\partial(\cdot)|_{K_{l}}\rVert_{m,K_{l}}\mid
l\in\mathbb{N},m\in\mathbb{N}_{0}\\}$. Also these seminorms induce the compact
open $C^{\infty}$-topology (this is easily seen by the usual estimate for the
$m$th Taylor remainder using the $m+1$st derivative on closed balls covering
$K_{l}$). Further we notice that $f\in\prod_{|\alpha|\leq m}C(U)$ is contained
in the image of $\partial$ if and only if $\lim_{t\rightarrow
0}q_{m}(K,f,t)=0$ for all $m\in\mathbb{N}_{0}$ and $K\subseteq U$ compact.

These considerations lead to the following definition.

###### Definition B.7.

Let $K\subseteq\mathbb{R}^{d}$ be compact. We say
$f\in\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K)$ is a _Whitney jet (of order
$\infty$)_, if $\lim_{t\rightarrow 0}q_{l}(K,f,t)=0$ for all $l\in\mathbb{N}$.
We denote the _Fréchet space of all Whitney jets (of order $\infty$)_ equipped
with the seminorms $\lVert\cdot\rVert_{l},l\in\mathbb{N}$ by $\mathcal{E}(K)$

###### B.8.

Since smooth functions on an open set $U$ restrict to Whitney jets on every
$K\subseteq U$ compact (cf. [9, p. 125]). We have as sets

$C^{\infty}(U,\mathbb{R})=\operatorname{proj}_{K_{n}}\mathcal{E}(K_{n})$ (9)

for every fundamental sequence of compact sets $K_{1}\subseteq
K_{2}\subseteq\ldots\subseteq U$. However, we can also view this limit in the
category of locally convex spaces and it follows from B.6 that the locally
convex topology on the left hand side of (9) coincides with the compact-open
$C^{\infty}$-topology described in Definition 5.4 (cf. Remark 5.5).

###### Definition B.9.

Let $U\subseteq\mathbb{R}^{d}$ be an open set with $(K_{l})_{l\in\mathbb{N}}$
a fundamental sequence of compact sets and $F\subseteq U$ a closed set. Then
we define the _space of Whitney jets on $F$_ as the projective limit
$\mathcal{E}(F)\colonequals\operatorname{proj}_{l\in\mathbb{N}}\mathcal{E}(F\cap
K_{l})$ (in the category of locally convex spaces).

Note that the definition of $\mathcal{E}(F)$ is independent of the choice of
fundamental sequence used to define it.

Using the canonical identification, we have
$C^{\infty}_{\text{co}}(U,\mathbb{R}^{m})\cong
C^{\infty}_{\text{co}}(U,\mathbb{R})^{m}$ as Fréchet spaces. Hence it makes
sense to define Whitney jets of vector valued functions as the Whitney jets of
the components of the functions:

###### Definition B.10.

Define $\mathcal{E}(K,\mathbb{R}^{d})$ to be the closed subspace of
$\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K,\mathbb{R}^{m})$ which corresponds to
$\mathcal{E}(K)^{m}$ under the identification
$\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K,\mathbb{R}^{m})\cong(\prod_{\alpha\in\mathbb{N}_{0}^{d}}C(K))^{m}$.
Similarly for $(K_{l})_{l}$ and $F$ as in Definition B.9 we define
$\mathcal{E}(F,\mathbb{R}^{m})\colonequals\operatorname{proj}_{l\in\mathbb{N}}\mathcal{E}(K_{l},\mathbb{R}^{m})$.

## References

  * [1] Alzaareer, H., Schmeding, A.: Differentiable mappings on products with different degrees of differentiability in the two factors.  Expo. Math. 33(2), 184–222 (2015).  URL https://doi.org/10.1016/j.exmath.2014.07.002
  * [2] Bangert, V.: Totally convex sets in complete Riemannian manifolds.  J. Differential Geom. 16(2), 333–345 (1981).  URL http://projecteuclid.org/euclid.jdg/1214436108
  * [3] Bastiani, A.: Applications différentiables et variétés différentiables de dimension infinie.  J. Analyse Math. 13, 1–114 (1964).  URL https://doi.org/10.1007/BF02786619
  * [4] Bierstone, E.: Differentiable functions.  Bol. Soc. Brasil. Mat. 11(2), 139–189 (1980).  10.1007/BF02584636.  URL https://doi.org/10.1007/BF02584636
  * [5] Brudnyi, A., Brudnyi, Y.: Methods of geometric analysis in extension and trace problems. Volume 1, _Monographs in Mathematics_ , vol. 102.  Birkhäuser/Springer Basel AG, Basel (2012).  URL https://doi.org/10.1007/978-3-0348-0209-3
  * [6] Brudnyi, A., Brudnyi, Y.: Methods of geometric analysis in extension and trace problems. Volume 2, _Monographs in Mathematics_ , vol. 103.  Birkhäuser/Springer Basel AG, Basel (2012).  URL https://doi.org/10.1007/978-3-0348-0212-3
  * [7] Etter, D., Griffin Jr., J.: On the metrisability of bundle space.  Proc. Amer. Math. Soc. 5, 466–467 (1954).  URL https://doi.org/10.1090/S0002-9939-1954-0063023-2
  * [8] Fefferman, C.: $C^{m}$ extension by linear operators.  Ann. of Math. (2) 166(3), 779–835 (2007).  URL https://doi.org/10.4007/annals.2007.166.779
  * [9] Frerick, L.: Extension operators for spaces of infinite differentiable Whitney jets.  J. Reine Angew. Math. 602, 123–154 (2007).  URL http://doi.org/10.1515/CRELLE.2007.005
  * [10] Frerick, L.: Stein’s extension operator for sets with ${\mathrm{Lip}}_{\gamma}$-boundary.  Analysis (Munich) 27(2-3), 251–259 (2007).  10.1524/anly.2007.27.2-3.251.  URL https://doi.org/10.1524/anly.2007.27.2-3.251
  * [11] Glöckner, H.: Infinite-dimensional Lie groups without completeness restrictions.  In: Geometry and analysis on finite- and infinite-dimensional Lie groups (B  ‘
---
e
dlewo, 2000), _Banach Center Publ._ , vol. 55, pp. 43–59. Polish Acad. Sci.,
Warsaw (2002).  URL https://doi.org/10.4064/bc55-0-3

  * [12] Glöckner, H.: Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups.  J. Funct. Anal. 194(2), 347–409 (2002).  URL https://doi.org/10.1006/jfan.2002.3942
  * [13] Glockner, H.: Lie groups over non-discrete topological fields (2004).  URL http://arxiv.org/abs/math/0408008
  * [14] Glockner, H.: Fundamentals of submersions and immersions between infinite-dimensional manifolds (2016).  URL http://arxiv.org/abs/1502.05795
  * [15] Glöckner, H., Neeb, K.H.: Infinite-dimensional Lie groups.  in preparation
  * [16] Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations.  Dover Publications, Inc., Mineola, NY (2006).  Unabridged republication of the 1993 original
  * [17] Hjelle, E.O., Schmeding, A.: Strong topologies for spaces of smooth maps with infinite-dimensional target.  Expo. Math. 35(1), 13–53 (2017).  URL http://doi.org/10.1016/j.exmath.2016.07.004
  * [18] Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in non-tangentially accessible domains.  Adv. Math. 46(1), 80–147 (1982)
  * [19] Karcher, H.: Schnittort und konvexe Mengen in vollständigen Riemannschen Mannigfaltigkeiten.  Math. Ann. 177, 105–121 (1968).  URL https://doi.org/10.1007/BF01350787
  * [20] Kriegl, A.: Remarks on germs in infinite dimensions.  Acta Math. Univ. Comenianae 66, 1–18 (1997)
  * [21] Kriegl, A., Michor, P.W.: The convenient setting of global analysis, _Mathematical Surveys and Monographs_ , vol. 53.  American Mathematical Society, Providence, RI (1997).  URL https://doi.org/10.1090/surv/053
  * [22] Lee, J.M.: Introduction to smooth manifolds, _Graduate Texts in Mathematics_ , vol. 218, second edn.  Springer, New York (2013)
  * [23] Martio, O., Sarvas, J.: Injectivity theorems in plane and space.  Ann. Acad. Sci. Fenn. Ser. A I Math. 4(2), 383–401 (1979).  10.5186/aasfm.1978-79.0413.  URL https://doi.org/10.5186/aasfm.1978-79.0413
  * [24] Meise, R., Vogt, D.: Introduction to functional analysis, _Oxford Graduate Texts in Mathematics_ , vol. 2.  The Clarendon Press, Oxford University Press, New York (1997)
  * [25] Michor, P.W.: Manifolds of differentiable mappings, _Shiva Mathematics Series_ , vol. 3.  Shiva Publishing Ltd., Nantwich (1980).  URL https://www.mat.univie.ac.at/m̃ichor/manifolds_of_differentiable_mappings.pdf
  * [26] Mourgoglou, M.: On harmonic measure and rectifiability in uniform domains.  J. Geom. Anal. (2018).  10.1007/s12220-018-0035-1.  URL https://doi.org/10.1007/s12220-018-0035-1
  * [27] Neeb, K.H.: Towards a Lie theory of locally convex groups.  Jpn. J. Math. 1(2), 291–468 (2006).  URL http://dx.doi.org/10.1007/s11537-006-0606-y
  * [28] Roberts, D.M., Vozzo, R.F.: The smooth Hom-stack of an orbifold.  In: D. Wood, J. de Gier, C. Praeger, T. Tao (eds.) 2016 MATRIX Annals, _MATRIX Book Series_ , vol. 1. Springer-Verlag (2018).  URL http://arxiv.org/abs/1610.05904
  * [29] Schmeding, A.: The diffeomorphism group of a non-compact orbifold.  Dissertationes Math. (Rozprawy Mat.) 507, 179 (2015).  URL http://dx.doi.org/10.4064/dm507-0-1
  * [30] Wengenroth, J.: Derived functors in functional analysis, _Lecture Notes in Mathematics_ , vol. 1810.  Springer-Verlag, Berlin (2003)
  * [31] Whitney, H.: Analytic extensions of differentiable functions defined in closed sets.  Trans. Am. Math. Soc. 36, 63–89 (1934).  URL https://doi.org/10.2307/1989708