2 Preliminaries
For the reader’s convenience, we briefly review some of the basic concepts and material concerning differential calculus in locally convex spaces, smooth manifolds and infinite-dimensional Lie groups.
The notion of
C
r
-map between locally convex spaces is that in Bastiani’s sense, also known as Keller’s
C
c
r
-map [12] (see [14, 11, 15, 6, 10, 8] for streamlined expositions, cf. also [4]).
For
C
r
-maps on suitable non-open domains, see [10, 20].
2.1
Let
E
,
F
be locally convex topological vector spaces,
U
⊆
E
open and
f
:
U
→
F
a map.
Then the derivative of 𝑓 at 𝑥 in the direction of ℎ is defined as
d
f
(
x
,
h
)
:=
(
D
h
f
)
(
x
)
:=
lim
t
→
0
1
t
(
f
(
x
+
t
h
)
-
f
(
x
)
)
if the limit exists.
The function 𝑓 is said to be differentiable at 𝑥 if
d
f
(
x
,
h
)
exists for all
h
∈
E
.
The function 𝑓 is said to be continuously differentiable or
C
1
if 𝑓 is continuous and differentiable at all points of 𝑈 and
d
f
:
U
×
E
→
F
,
(
x
,
h
)
↦
d
f
(
x
,
h
)
is a continuous map.
The function 𝑓 is said to be a
C
r
-map if 𝑓 is
C
1
and
d
f
is a
C
r
-
1
-map, and
C
∞
(or smooth) if 𝑓 is
C
r
for all
r
∈
N
.
A continuous map
f
:
U
→
F
is
C
r
if and only if the iterated directional derivatives
d
(
k
)
f
(
x
,
h
1
,
…
,
h
k
)
:=
(
D
h
k
…
D
h
1
f
)
(
x
)
exist for all
k
∈
N
with
k
≤
r
,
x
∈
U
and
h
1
,
…
,
h
k
∈
E
, and define continuous functions
d
(
k
)
f
:
U
×
E
k
→
F
.
More generally, consider a locally convex subset
U
⊆
E
with dense interior.
A continuous map
f
:
U
→
F
is said to be
C
r
if
f
|
U
∘
:
U
∘
→
F
is
C
r
and each of the maps
d
(
k
)
(
f
|
U
∘
)
:
U
∘
×
E
k
→
F
admits a (necessarily unique) continuous extension
d
(
k
)
f
:
U
×
E
k
→
F
.
2.2
We shall also need
C
r
,
s
-maps as developed in [1, 3].
Let
E
1
,
E
2
and 𝐹 be locally convex spaces,
U
,
V
open subsets of
E
1
,
E
2
, respectively, and
r
,
s
∈
N
0
∪
{
∞
}
.
A mapping
f
:
U
×
V
→
F
is said to be a
C
r
,
s
-map if, for all
i
,
j
∈
N
0
such that
i
≤
r
,
j
≤
s
, the iterated directional derivative
d
(
i
,
j
)
f
(
x
,
y
,
w
1
,
…
,
w
i
,
v
1
,
…
,
v
j
)
:=
(
D
(
w
i
,
0
)
⋯
D
(
w
1
,
0
)
D
(
0
,
v
j
)
⋯
D
(
0
,
v
1
)
f
)
(
x
,
y
)
exists for all
x
∈
U
,
y
∈
V
,
w
1
,
…
,
w
i
∈
E
1
,
v
1
,
…
,
v
j
∈
E
2
, and
d
(
i
,
j
)
f
:
U
×
V
×
E
1
i
×
E
2
j
→
F
,
(
x
,
y
,
w
1
,
…
,
w
i
,
v
1
,
…
,
v
j
)
↦
(
D
(
w
i
,
0
)
⋯
D
(
w
1
,
0
)
D
(
0
,
v
j
)
⋯
D
(
0
,
v
1
)
f
)
(
x
,
y
)
is continuous.
More generally, it is useful to have a definition of
C
r
,
s
-maps on not-necessarily open domains available.
If 𝑈 and 𝑉 are locally convex subsets with dense interior of
E
1
and
E
2
, respectively, and
r
,
s
∈
N
0
∪
{
∞
}
, then we say that
f
:
U
×
V
→
F
is a
C
r
,
s
-map if 𝑓 is continuous and
f
|
U
0
×
V
0
:
U
0
×
V
0
→
F
is a
C
r
,
s
-map and, for all
i
,
j
∈
N
0
such that
i
≤
r
,
j
≤
s
, the map
d
(
i
,
j
)
(
f
|
U
0
×
V
0
)
:
U
0
×
V
0
×
E
1
i
×
E
2
j
→
F
admits a continuous extension
d
(
i
,
j
)
f
:
U
×
V
×
E
1
i
×
E
2
j
→
F
.
2.3
Since the composition of
C
r
maps between open subsets of locally convex spaces is a
C
r
map, we can define
C
r
-manifolds 𝑀 as in the finite-dimensional case (see [14, 11, 15, 6, 10]). A smooth manifold modelled on a locally convex topological vector space𝐸 is a Hausdorff topological space 𝑀, together with a maximal set 𝒜 of homeomorphisms (charts)
φ
:
U
→
V
from open subsets of 𝑀 onto open subsets of 𝐸 such that the domains cover 𝑀 and the transition maps
φ
∘
ψ
-
1
are smooth on their domain for all
φ
,
ψ
∈
A
.
If the transition maps
φ
∘
ψ
-
1
are just
C
r
on their domain for all
φ
,
ψ
∈
A
, then it is said to be a
C
r
-manifold.
If each 𝑉 is merely a locally convex subset of 𝐸 with dense interior, then 𝑀 is said to be a manifold with rough boundary.
If each 𝑉 is open, then 𝑀 is an ordinary manifold without boundary.
If each 𝑉 is relatively open in a closed halfspace
λ
-
1
(
[
0
,
∞
[
)
, where
λ
∈
E
′
(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 is a relatively open subset of
λ
1
-
1
(
[
0
,
∞
[
)
∩
⋯
∩
λ
n
-
1
(
[
0
,
∞
[
)
, for suitable
n
∈
N
(which may depend on 𝜑) and linearly independent
λ
1
,
…
,
λ
n
∈
E
′
.
Products of manifolds and smoothness of maps between manifolds are defined also as in the finite-dimensional case.
A subset
N
⊆
M
is said to be a submanifold of 𝑀 if there exists a closed vector subspace
F
⊆
E
and, for each
x
∈
N
, there exists an 𝐸-chart
(
U
,
φ
)
of 𝑀 with
x
∈
U
and
φ
(
U
∩
N
)
=
φ
(
U
)
∩
F
.
A mapping
f
:
M
→
N
between
C
k
-manifolds is said to be
C
k
if, for each
x
∈
M
and each chart
(
V
,
ψ
)
on 𝑁 with
f
(
x
)
∈
V
, there is a chart
(
U
,
φ
)
on 𝑀 with
x
∈
U
,
f
(
U
)
⊆
V
, and
ψ
∘
f
∘
φ
-
1
is
C
k
.
We will denote by
C
k
(
M
,
N
)
the space of all
C
k
-mappings from 𝑀 to 𝑁.
A
C
k
-mapping
f
:
M
→
N
is said to be a
C
k
-diffeomorphism if
f
-
1
:
N
→
M
exists and is also
C
k
.
Two manifolds are said to be diffeomorphic if there exists a diffeomorphism between them.
2.4
If 𝑀 is a finite-dimensional smooth manifold with rough boundary and 𝐿 a compact smooth manifold with rough boundary such that
L
⊆
M
as a set, the inclusion map
L
→
M
is smooth and the inclusion map
L
0
→
L
is smooth (where
L
0
is the interior of 𝐿 relative 𝑀), then we say that 𝐿 is well located in 𝑀.
If 𝑀 is locally compact, then the interiors
L
0
of well-located compact smooth manifolds
L
⊆
M
with rough boundary form an open cover of 𝑀.
In fact, given
x
∈
M
and a chart
ϕ
:
U
→
V
around 𝑥 onto a locally convex subset
V
⊆
R
n
with dense interior, we can choose a compact neighbourhood 𝐾 of
ϕ
(
x
)
in 𝑉 and take
L
:=
ϕ
-
1
(
K
)
.
2.5
A Lie group𝐺 is a group, equipped with a smooth manifold structure modelled on a locally convex space 𝐸 such that the group operations are smooth maps.
We write
1
∈
G
for the identity element and, respectively,
λ
g
(
x
)
=
g
x
and
ρ
g
(
x
)
=
x
g
for the left and right multiplication by 𝑔 on 𝐺.
As in finite dimensions, the tangent space
L
(
G
)
:=
T
1
(
G
)
≅
E
at the identity element of a Lie group 𝐺 can be made a topological Lie algebra via the identification with the Lie algebra of left invariant vector fields on 𝐺.
We recall that a vector field𝑋 on a locally convex Lie group 𝐺 is said to be left invariant if
X
∘
λ
g
=
T
λ
g
∘
X
as mappings
G
→
T
G
.
Then each
x
∈
T
1
(
G
)
corresponds to a unique left invariant vector field
x
l
with
x
l
(
g
)
:=
T
1
λ
g
(
x
)
,
g
∈
G
.
The space of left invariant vector fields is closed under the Lie bracket of vector fields, hence inherits a Lie algebra structure.
In this sense, we obtain on
g
:=
T
1
(
G
)
a continuous Lie bracket (see [10]) which is uniquely determined by
[
x
,
y
]
l
=
[
x
l
,
y
l
]
for
x
,
y
∈
g
.
The Lie group 𝐺 is said to have an exponential function if, for each
x
∈
g
, the initial value problem
γ
(
0
)
=
1
,
γ
′
(
t
)
=
T
1
λ
γ
(
t
)
.
x
has a solution
γ
x
∈
C
∞
(
R
,
G
)
and the function
exp
G
:
g
→
G
,
x
↦
γ
x
(
1
)
is smooth.
To each morphism
φ
:
G
→
H
of Lie groups, we further associate its tangent map
L
(
φ
)
:=
T
1
(
φ
)
:
L
(
G
)
→
L
(
H
)
, and the usual argument with related vector fields implies that
L
(
φ
)
is a homomorphism of Lie algebras.
2.6
Let 𝐺 be a Lie group with Lie algebra 𝔤.
For each
g
∈
G
, we define the conjugation or inner automorphism by the map
c
g
:
G
→
G
,
x
↦
g
x
g
-
1
.
This defines a smooth action of 𝐺 on itself by automorphisms, hence induces continuous linear automorphisms
Ad
(
g
)
:=
L
(
c
g
)
:
g
→
g
.
Thus the adjoint representation
Ad
:
G
→
Aut
(
g
)
is given by
Ad
(
g
)
=
T
1
(
c
g
)
:
g
→
g
for
g
∈
G
.
By the definition of the Lie functor,
Ad
(
g
)
is a Lie algebra homomorphism.
We also define for
x
∈
g
a linear map
ad
(
x
)
:
g
→
g
,
ad
x
(
y
)
:=
T
1
(
Ad
∙
(
y
)
)
(
x
)
.
2.7
Let 𝐺 be a Lie group with Lie algebra 𝔤 and
k
∈
N
0
∪
{
∞
}
.
The tangent map
T
(
m
G
)
of the multiplication map
m
G
:
G
×
G
→
G
defines a Lie group structure on the tangent bundle
T
G
([18] and cf. [10]).
Iterating this procedure, we obtain a Lie group structure on all higher tangent bundles
T
n
G
.
For each
n
∈
N
0
, we thus obtain topological groups
C
(
T
n
M
,
T
n
G
)
c
.
o
.
.
We also observe that, for two smooth maps
f
1
,
f
2
:
M
→
G
, the functoriality of 𝑇 yields
T
(
f
1
⋅
f
2
)
=
T
(
m
G
∘
(
f
1
,
f
2
)
)
=
T
(
m
G
)
∘
(
T
f
1
,
T
f
2
)
=
T
f
1
⋅
T
f
2
.
Therefore, the inclusion map
C
k
(
M
,
G
)
↪
∏
n
=
0
k
C
(
T
n
M
,
T
n
G
)
c
.
o
.
from (1.1) is a group homomorphism so that the inverse image of the product topology from the right-hand side is a group topology on
C
k
(
M
,
G
)
and thus turns
C
k
(
M
,
G
)
into a topological group, even if 𝑀 and 𝐺 are infinite-dimensional.
Let 𝑋, 𝑌 and 𝐸 be locally convex spaces, and
U
⊆
X
and
V
⊆
Y
locally convex subsets with dense interior.
Then the following holds (cf. [10]).
For every
r
≥
s
, the inclusion map
C
r
(
U
,
E
)
→
C
s
(
U
,
E
)
is a continuous linear map.
The topology on
C
∞
(
U
,
E
)
is initial with respect to the family of inclusion maps
C
∞
(
U
,
E
)
→
C
k
(
U
,
E
)
, where
k
∈
N
0
.
Furthermore,
C
∞
(
U
,
E
)
=
lim
←
C
k
(
U
,
E
)
.
Accordingly,
C
k
(
U
,
C
∞
(
V
,
E
)
)
=
lim
←
C
k
(
U
,
C
r
(
V
,
E
)
)
.
For every
k
∈
N
0
, the topology on
C
k
+
1
(
U
,
E
)
is initial with respect to the inclusion map
C
k
+
1
(
U
,
E
)
→
C
(
U
,
E
)
together with the mapping
C
k
+
1
(
U
,
E
)
→
C
k
(
U
×
X
,
E
)
,
γ
↦
d
γ
.
3 Regular Lie groups and the fundamental theorem
In this section, we discuss the
C
k
-regularity concept.
After recalling some definitions and results (mainly from [19, 15, 9, 10]), we shall introduce a version of the fundamental theorem for 𝔤-valued functions (Theorem 3.24).
Definition 3.1
The Maurer–Cartan form
κ
G
∈
Ω
1
(
G
,
g
)
is the unique left invariant 𝔤-valued 1-form on 𝐺 with
κ
G
,
1
=
id
g
, i.e.,
κ
G
(
x
l
)
=
x
for each
x
∈
g
.
The logarithmic derivative of a map 𝑓 can be described as a pullback of the Maurer–Cartan form.
Definition 3.2
Let 𝑀 be a smooth manifold (with boundary), and let 𝐺 be a Lie group with Lie algebra 𝔤 and Maurer–Cartan form
κ
G
∈
Ω
1
(
G
,
g
)
.
For an element
f
∈
C
1
(
M
,
G
)
, we call
δ
(
f
)
:=
f
∗
κ
G
=
:
f
-
1
⋅
d
f
∈
Ω
C
0
1
(
M
,
g
)
the (left) logarithmic derivative of 𝑓.
Remark 3.3
Let 𝐸 be a locally convex space and 𝑀 a smooth finite-dimensional manifold (possibly with boundary).
We write
Ω
C
r
1
(
M
,
E
)
for the space of 𝐸-valued 1-forms on 𝑀 defining
C
r
-functions
T
M
→
E
.
The space of 𝐸-valued smooth 1-forms will be denoted by
Ω
1
(
M
,
E
)
.
We endow
Ω
C
r
1
(
M
,
E
)
with the topology induced by the embedding
Ω
C
r
1
(
M
,
E
)
↪
C
r
(
T
M
,
E
)
,
where
T
M
is the tangent bundle and
C
r
(
T
M
,
E
)
is endowed with the compact open
C
r
- topology, so that
Ω
C
r
1
(
M
,
E
)
is a closed subspace of
C
r
(
T
M
,
E
)
.
The space
Ω
1
(
M
,
E
)
is endowed with the topology induced by the diagonal embedding
Ω
1
(
M
,
E
)
↪
∏
r
=
1
∞
Ω
C
r
1
(
M
,
E
)
.
The left logarithmic derivative is a 𝔤-valued 1-form on 𝑀.
For
k
∈
N
∪
{
∞
}
, we thus obtain a map
δ
:
C
k
(
M
,
G
)
→
Ω
C
k
-
1
1
(
M
,
g
)
satisfying the following lemma.
Lemma 3.4
For
f
,
g
∈
C
k
(
M
,
G
)
with
k
≥
1
, the following assertions hold.
The map
f
-
1
:
M
→
G
,
m
↦
f
(
m
)
-
1
is
C
k
with
δ
(
f
-
1
)
=
-
Ad
(
f
)
δ
(
f
)
.
We have the following product and quotient rules:
δ
(
f
g
)
=
Ad
(
g
)
-
1
δ
(
f
)
+
δ
(
g
)
and
δ
(
f
g
-
1
)
=
Ad
(
g
)
(
δ
(
f
)
-
δ
(
g
)
)
.
From this, the next lemma easily follows.
Lemma 3.5
If 𝑀 is connected and
f
,
g
∈
C
k
(
M
,
G
)
, then
δ
(
f
)
=
δ
(
g
)
⇔
there exists
h
∈
G
such that
g
=
λ
h
∘
f
.
In particular,
δ
(
f
)
=
δ
(
g
)
and
f
(
m
0
)
=
g
(
m
0
)
for some
m
0
∈
M
imply
f
=
g
.
Definition 3.6
Definition 3.6 (Integrability and local integrability)
We say that
α
∈
Ω
C
0
1
(
M
,
g
)
is integrable if there exists a
C
1
-function
f
:
M
→
G
with
δ
(
f
)
=
α
.
We say that 𝛼 is locally integrable if each point
m
∈
M
has an open neighbourhood 𝑈 such that
α
|
U
is integrable.
Remark 3.7
Using induction on 𝑘, we can prove that if
α
∈
Ω
C
k
1
(
M
,
g
)
is integrable and
α
=
δ
f
with a
C
1
-function
f
:
M
→
G
, then we have that 𝑓 is
C
k
+
1
.
In the following, we frequently abbreviate
I
:=
[
0
,
1
]
.
Definition 3.8
Definition 3.8 (Left product integral and left evolution)
Let
ξ
:
I
→
L
(
G
)
be a continuous curve, defined on an interval
I
⊆
R
.
If
γ
:
I
→
G
is a
C
1
-curve such that
δ
(
γ
)
=
ξ
, we say that 𝛾 is a left product integral for 𝜉.
If
γ
(
0
)
=
1
, we call 𝛾 the left evolution of 𝜉 and write
Evol
G
(
ξ
)
:=
γ
.
Definition 3.9
Definition 3.9 (
C
k
-regular Lie group)
Let
k
∈
N
0
∪
{
∞
}
.
A Lie group 𝐺 with Lie algebra 𝔤 is said to be
C
k
-regular if, for each
ξ
∈
C
k
(
I
,
g
)
, the initial value problem
(3.1)
γ
(
0
)
=
1
,
δ
(
γ
)
=
ξ
has a solution
γ
=
γ
ξ
∈
C
k
+
1
(
I
,
G
)
, and the evolution map
evol
G
:
C
k
(
I
,
g
)
→
G
,
ξ
↦
γ
ξ
(
1
)
is smooth.
We recall from Lemma 3.5 that the solutions of (3.1) are unique whenever they exist.
If 𝐺 is
C
k
-regular, we write
Evol
G
:
C
k
(
I
,
g
)
→
C
k
+
1
(
I
,
G
)
,
ξ
↦
γ
ξ
for the corresponding map on the level of Lie group-valued curves.
The group 𝐺 is said to be regular if it is
C
∞
-regular.
If
k
∈
N
0
∪
{
∞
}
and 𝐺 is
C
k
-regular, then 𝐺 is
C
l
-regular for all
l
∈
N
0
∪
{
∞
}
such that
l
≥
k
(see [9]).
Proposition 3.10
Let 𝐺 be a connected, simply connected real Lie group, and let 𝐻 be a regular Lie group.
Then every continuous Lie algebra homomorphism
ψ
:
L
(
G
)
→
L
(
H
)
integrates to a smooth group homomorphism
φ
:
G
→
H
such that
L
(
φ
)
=
ψ
.
Proof
For the proof, we refer to [10].
∎
For the concept of Mackey completeness, see [13].
Proposition 3.11
Let 𝑀 be a locally compact smooth manifold with rough boundary and 𝐸 a locally convex space.
Then
C
k
(
M
,
E
)
is a locally convex space, and the evaluation map
ε
:
C
k
(
M
,
E
)
×
M
→
E
is
C
∞
,
k
.
If 𝐸 is Mackey complete, then
C
k
(
M
,
E
)
is Mackey complete.
Proof
All the spaces
C
(
T
n
M
,
T
n
E
)
c
are locally convex.
Therefore, the corresponding product topology is locally convex, and hence
C
k
(
M
,
E
)
is a locally convex space.
The continuity of the evaluation map follows from the continuity of the evaluation map for the compact-open topology because the topology on
C
k
(
M
,
E
)
is finer.
Next we observe that directional derivatives exist and lead to a map
d
ev
:
C
k
(
M
,
E
)
2
×
T
(
M
)
→
E
,
(
(
f
,
ξ
)
,
v
m
)
↦
ξ
(
m
)
+
T
m
(
f
)
v
m
whose continuity follows from the first step, applied to the evaluation map of
C
k
(
T
M
,
E
)
.
Hence
ev
is
C
1
, and iteration of this argument yields
C
k
.
Thus
ev
is
C
0
,
k
, in particular.
Since
ev
is linear in its first argument, it follows that
ev
is
C
∞
,
k
(see [1, Lemma 39]).
In view of [3, Proposition 4.5], we have
C
∞
(
I
,
C
k
(
M
,
E
)
)
≅
C
∞
,
k
(
I
×
M
,
E
)
≅
C
k
(
M
,
C
∞
(
I
,
E
)
)
,
and if 𝐸 is Mackey complete, then we have an integration map
C
k
(
M
,
C
∞
(
I
,
E
)
)
→
C
k
(
M
,
E
)
,
ξ
↦
∫
0
1
∙
d
t
∘
ξ
,
which implies that each
C
k
-curve with values in
C
k
(
M
,
E
)
has a Riemann integral, i.e., that
C
k
(
M
,
E
)
is Mackey complete.
∎
Theorem 3.12
Let 𝑀 be a
C
∞
-manifold with rough boundary, 𝐺 a Lie group with Lie algebra 𝔤, and let
k
∈
N
0
∪
{
∞
}
.
If 𝑀 is compact,
C
k
(
M
,
G
)
carries a Lie group structure for which any 𝔤-chart
(
ϕ
G
,
U
G
)
of 𝐺 yields a
C
k
(
M
,
g
)
-chart
(
ϕ
,
U
)
with
U
:=
{
f
∈
C
k
(
M
,
G
)
:
f
(
M
)
⊆
U
G
}
,
ϕ
(
f
)
:=
ϕ
G
∘
f
.
Proof
For the existence of the Lie group structure with the given charts and Lie algebra, cf. [7].
∎
Lemma 3.13
Let 𝐺 be a Lie group modelled on a locally convex space 𝐸, and let 𝑀 be a compact manifold (possibly with boundary) and
k
∈
N
0
∪
{
∞
}
.
Then the evaluation map
ε
:
C
k
(
M
,
G
)
×
M
→
G
is
C
∞
,
k
.
Proof
It suffices to show that each
γ
∈
C
k
(
M
,
G
)
has an open neighbourhood
W
⊆
C
k
(
M
,
G
)
such that
ε
|
W
×
M
is
C
∞
,
k
.
Let
φ
:
U
→
V
⊆
E
be a chart for 𝐺 around
1
∈
G
such that
C
k
(
M
,
U
)
is open in
C
k
(
M
,
G
)
and
φ
*
:
C
k
(
M
,
U
)
→
C
k
(
M
,
V
)
⊆
C
k
(
M
,
E
)
is a chart of
C
k
(
M
,
G
)
.
Then
W
:=
γ
.
C
k
(
M
,
U
)
is an open neighbourhood of 𝛾 in
C
k
(
M
,
G
)
.
By Chain Rule 1 [3, Lemma 3.17],
ε
|
W
×
M
will be
C
∞
,
k
if we can show that the map
C
k
(
M
,
U
)
×
M
→
G
,
(
η
,
x
)
↦
ε
(
γ
.
η
,
x
)
=
γ
(
x
)
η
(
x
)
=
μ
(
γ
(
x
)
,
ε
(
η
,
x
)
)
is
C
∞
,
k
, where
μ
:
G
×
G
→
G
is the group multiplication which is smooth,
γ
(
x
)
is
C
k
in 𝑥 and
C
∞
,
k
in
(
η
,
x
)
.
By Chain Rule 2 [3, Lemma 3.18], we only need to show that
ε
U
:
C
k
(
M
,
U
)
×
M
→
U
⊆
G
,
(
η
,
x
)
↦
η
(
x
)
is
C
∞
,
k
.
Now we have a commutative diagram
where
ε
~
:
C
k
(
M
,
V
)
×
M
→
V
is a
C
∞
,
k
-map as a restriction of the
C
∞
,
k
-map
C
k
(
M
,
E
)
×
M
→
E
,
(
η
,
x
)
↦
η
(
x
)
(see Proposition 3.11).
The vertical arrows being charts, it follows that
ε
U
is
C
∞
,
k
.
∎
Proposition 3.14
Let 𝐺 be a Lie group, 𝑁 a smooth manifold, 𝑀 a compact smooth manifold (both possibly with rough boundary) and
r
,
k
∈
N
0
∪
{
∞
}
.
Then a map
f
:
N
→
C
k
(
M
,
G
)
is
C
r
if and only if
f
∧
:
N
×
M
→
G
is
C
r
,
k
.
Proof
Let
f
:
N
→
C
k
(
M
,
G
)
be
C
r
.
Then
f
∧
(
x
,
y
)
:=
f
(
x
)
(
y
)
=
ε
(
f
(
x
)
,
y
)
, where
ε
:
C
k
(
M
,
G
)
×
M
→
G
,
(
γ
,
y
)
↦
γ
(
y
)
is
C
∞
,
k
by Lemma 3.13.
Thus, by Chain Rule 1 [3, Lemma 3.17],
f
∧
is
C
r
,
k
.
Conversely, assume that
g
:=
f
∧
:
N
×
M
→
G
is a
C
r
,
k
-map.
Then the map
g
∨
=
(
f
∧
)
∨
=
f
is
C
r
if we can show that each
x
0
∈
N
has an open neighbourhood
W
⊆
N
such that
g
∨
|
W
is
C
r
.
To achieve this, let
φ
:
U
→
V
⊆
E
be a chart of 𝐺 around 1.
The set
P
:=
{
(
x
,
y
)
∈
N
×
M
:
g
(
x
,
y
)
g
(
x
0
,
y
)
-
1
∈
U
}
is open in
N
×
M
and contains
{
x
0
}
×
M
.
Because
{
x
0
}
and 𝑀 are compact, the Wallace lemma (see [5, Theorem 3.2.10]) provides an open neighbourhood
W
⊆
N
of
x
0
such that
W
×
M
⊆
P
.
The map
h
:
W
×
M
→
U
⊆
G
,
(
x
,
y
)
↦
g
(
x
,
y
)
g
(
x
0
,
y
)
-
1
is
C
r
,
k
by Chain Rules 1 and 2 [3, Lemma 3.17 and Lemma 3.18] because
g
(
x
,
y
)
,
g
(
x
0
,
y
)
are
C
r
,
k
in
(
x
,
y
)
and
h
(
x
,
y
)
=
ν
(
g
(
x
,
y
)
,
g
(
x
0
,
y
)
)
,
where
ν
:
G
×
G
→
G
,
(
a
,
b
)
↦
a
b
-
1
is smooth.
We claim that
h
∨
:
W
→
C
k
(
M
,
U
)
,
x
↦
h
(
x
,
∙
)
is
C
r
.
If this is true, then also
g
∨
|
W
is
C
r
because
g
∨
(
x
)
=
h
∨
(
x
)
.
γ
=
(
ρ
γ
∘
h
∨
)
(
x
)
with
γ
:=
g
(
x
0
,
∙
)
∈
C
k
(
M
,
G
)
and the right translation
ρ
γ
:
C
k
(
M
,
G
)
→
C
k
(
M
,
G
)
,
η
↦
η
.
γ
is smooth.
To prove the claim, consider the commutative diagram
where
φ
∘
h
|
W
×
M
:
W
×
M
→
V
⊆
E
is
C
r
,
k
by definition of
C
r
,
k
-maps between manifolds and
(
φ
∘
h
|
W
×
M
)
∨
is
C
r
by the vector-valued exponential law in the locally compact case [3, Theorem 4.6].
Thus
h
∨
=
(
φ
*
)
-
1
∘
(
φ
∘
h
|
W
×
M
)
∨
is
C
r
as well.
∎
Lemma 3.15
Lemma 3.15 ([9, Lemma 2.2])
A map
f
:
M
→
C
k
+
1
(
I
,
G
)
is
C
r
if and only if 𝑓 is
C
r
as a map to
C
(
I
,
G
)
and
D
∘
f
:
M
→
C
k
(
I
,
T
G
)
is
C
r
, where
D
:
C
k
+
1
(
I
,
G
)
→
C
k
(
I
,
T
G
)
,
γ
↦
γ
′
.
Proposition 3.16
Proposition 3.16 ([9, Theorem A])
Let G be a Lie group with Lie algebra 𝔤.
If 𝐺 is
C
k
-regular, then the map
Evol
G
:
C
k
(
I
,
g
)
→
C
k
+
1
(
I
,
G
)
is smooth.
Lemma 3.17
Let
k
≥
2
.
For each
f
∈
C
k
(
M
,
G
)
, the 1-form
α
:=
δ
(
f
)
satisfies the Maurer–Cartan equation
d
α
+
1
2
[
α
,
α
]
=
0
.
Proof
We can proceed as in the case of smooth 1-forms treated in [10].
First we show that
κ
G
=
δ
(
id
G
)
satisfies the Maurer–Cartan equation.
It suffices to evaluate
d
α
on left invariant vector fields
x
l
,
y
l
, where
x
,
y
∈
g
.
Since
κ
G
(
z
l
)
is constant for each
z
∈
g
, we have
d
κ
G
(
x
l
,
y
l
)
=
x
l
κ
G
(
y
l
)
-
y
l
κ
G
(
x
l
)
-
κ
G
(
[
x
l
,
y
l
]
)
=
-
κ
G
(
[
x
,
y
]
l
)
=
-
[
x
,
y
]
=
-
1
2
[
κ
G
,
κ
G
]
(
x
l
,
y
l
)
.
Therefore,
α
=
f
*
κ
G
satisfies
d
α
=
f
*
d
κ
G
=
-
1
2
f
*
[
κ
G
,
κ
G
]
=
-
1
2
[
f
*
κ
G
,
f
*
κ
G
]
=
-
1
2
[
α
,
α
]
,
which is the Maurer–Cartan equation.
∎
Remark 3.18
Assume that 𝐺 is
C
k
-regular.
For
ξ
∈
C
k
(
I
,
g
)
,
0
≤
s
≤
1
and
η
(
t
)
:=
γ
ξ
(
s
t
)
, we have
δ
(
η
)
(
t
)
=
s
ξ
(
s
t
)
.
Therefore, we obtain with
S
:
C
k
(
I
,
g
)
×
I
→
C
k
(
I
,
g
)
,
S
(
ξ
,
s
)
(
t
)
:=
s
ξ
(
s
t
)
the relation
Evol
G
(
ξ
)
(
s
)
=
γ
ξ
(
s
)
=
evol
G
(
S
(
ξ
,
s
)
)
.
Lemma 3.19
Lemma 3.19 ([10])
If 𝐺 is
C
k
-regular,
x
∈
g
and
ξ
∈
C
k
(
[
0
,
1
]
,
g
)
, then the initial value problem
η
′
(
t
)
=
[
η
(
t
)
,
ξ
(
t
)
]
,
η
(
0
)
=
x
has a unique solution
η
:
[
0
,
1
]
→
g
given by
η
(
t
)
=
Ad
(
γ
ξ
(
t
)
)
-
1
x
.
Lemma 3.20
Consider a 𝔤-valued 1-form on
I
2
of class
C
1
,
α
=
v
d
x
+
w
d
y
∈
Ω
C
1
1
(
I
2
,
g
)
with
v
,
w
∈
C
1
(
I
2
,
g
)
.
𝛼 satisfies the Maurer–Cartan equation if and only if
(3.2)
∂
v
∂
y
-
∂
w
∂
x
=
[
v
,
w
]
.
Suppose that 𝛼 satisfies the Maurer–Cartan equation.
Assume that 𝐺 is
C
k
-regular for some
k
∈
N
0
∪
{
∞
}
and 𝛼 of class
C
k
.
If
f
:
I
2
→
G
is
C
2
with
δ
(
f
)
(
∂
y
)
=
w
and
δ
(
f
)
(
∂
x
)
(
x
,
0
)
=
v
(
x
,
0
)
for all
x
∈
I
, then
δ
(
f
)
=
α
.
Assume that 𝐺 is
C
k
-regular for some
k
∈
N
0
∪
{
∞
}
and 𝛼 of class
C
k
+
2
.
Then the function
f
:
I
2
→
G
defined by
f
(
x
,
0
)
:=
γ
v
(
∙
,
0
)
(
x
)
and
f
(
x
,
y
)
:=
f
(
x
,
0
)
⋅
γ
w
(
x
,
∙
)
(
y
)
is
C
2
and satisfies
δ
(
f
)
=
α
.
Proof
(1) We can proceed as in the case of smooth 1-forms treated in [10] (and likewise in (2) (a)).
To evaluate the Maurer–Cartan equation for 𝛼, we first observe that
1
2
[
α
,
α
]
(
∂
∂
x
,
∂
∂
y
)
=
[
α
(
∂
∂
x
)
,
α
(
∂
∂
y
)
]
=
[
v
,
w
]
and obtain
d
α
+
1
2
[
α
,
α
]
=
∂
v
∂
y
d
y
∧
d
x
+
∂
w
∂
x
d
x
∧
d
y
+
[
v
,
w
]
d
x
∧
d
y
=
(
∂
w
∂
x
-
∂
v
∂
y
+
[
v
,
w
]
)
d
x
∧
d
y
.
(2) (a) We have
δ
f
=
v
^
d
x
+
w
d
y
with
v
^
(
x
,
0
)
=
v
(
x
,
0
)
for
x
∈
I
.
The Maurer–Cartan equation for
δ
f
reads
∂
v
^
∂
y
-
∂
w
∂
x
=
[
v
^
,
w
]
so that subtraction of this equation from (3.2) leads to
∂
(
v
-
v
^
)
∂
y
=
[
v
-
v
^
,
w
]
.
As
(
v
-
v
^
)
(
x
,
0
)
=
0
, the uniqueness assertion of Lemma 3.19, applied to
η
x
(
t
)
:=
(
v
-
v
^
)
(
x
,
t
)
,
implies that
(
v
-
v
^
)
(
x
,
y
)
=
0
for all
x
,
y
∈
I
, hence that
v
=
v
^
, which means that
δ
(
f
)
=
v
d
x
+
w
d
y
=
α
.
(b) Because
v
(
∙
,
0
)
∈
C
k
+
2
(
I
,
g
)
and 𝐺 is
C
k
+
2
-regular, we have
γ
v
(
∙
,
0
)
∈
C
k
+
3
(
I
,
G
)
.
Hence
I
2
→
G
,
(
x
,
y
)
↦
γ
v
(
∙
,
0
)
(
x
)
is a
C
k
+
3
-map and hence
C
2
.
By [3, Proposition 4.5], the map
w
∨
:
I
→
C
k
(
I
,
g
)
,
w
∨
(
x
)
(
y
)
:=
w
(
x
,
y
)
is
C
2
since 𝑤 is
C
k
+
2
and hence
C
2
,
k
.
Since
Evol
G
:
C
k
(
I
,
g
)
→
C
k
+
1
(
I
,
G
)
is smooth by Proposition 3.16, it follows that
Evol
G
∘
w
∨
:
I
→
C
k
+
1
(
I
,
G
)
,
x
↦
Evol
G
(
w
∨
(
x
)
)
=
γ
w
(
x
,
∙
)
is
C
2
.
Hence
(
Evol
G
∘
w
∨
)
∧
:
I
×
I
→
G
,
(
x
,
y
)
↦
γ
w
(
x
,
∙
)
(
y
)
is
C
2
,
k
+
1
by [3, Proposition 4.5].
We can also consider
w
∨
as a
C
1
-map to
C
k
+
1
(
I
,
g
)
.
Since 𝐺 is also
C
k
+
1
-regular, arguing as before, we see that
Evol
G
∘
w
∨
:
I
→
C
k
+
2
(
I
,
G
)
is
C
1
, whence
(
Evol
G
∘
w
∨
)
∧
is
C
1
,
k
+
2
using Proposition 3.14.
Being
C
2
,
k
+
1
(hence
C
2
,
1
) and
C
1
,
k
+
2
(hence
C
1
,
2
), the map
(
Evol
G
∘
w
∨
)
∧
is
C
2
in particular.
Hence
f
:
I
2
→
G
,
f
(
x
,
y
)
:=
γ
v
(
∙
,
0
)
(
x
)
γ
w
(
x
,
∙
)
(
y
)
is
C
2
.
Now (a) shows that
δ
(
f
)
=
α
.
∎
Lemma 3.21
Let
k
≥
2
, and let 𝑈 be a convex subset of the locally convex space 𝐸 with
U
∘
≠
∅
, 𝐺 a
C
k
-
2
-regular Lie group with Lie algebra 𝔤 and
α
∈
Ω
C
k
1
(
U
,
g
)
a
C
k
-differential form satisfying the Maurer–Cartan equation.
Then 𝛼 is integrable.
Proof
We may without loss of generality assume that
x
0
=
0
∈
U
.
For
x
∈
U
, we consider the
C
k
-curve
ξ
x
:
I
→
g
,
t
↦
α
t
x
(
x
)
.
The map
U
×
I
→
g
,
(
x
,
t
)
↦
ξ
x
(
t
)
is
C
k
, hence
C
2
,
k
-
2
.
Therefore, the map
U
→
C
k
-
2
(
I
,
g
)
,
x
↦
ξ
x
is
C
2
.
Hence the function
f
:
U
→
G
,
x
↦
evol
(
ξ
x
)
is
C
2
.
First we show that
f
(
s
x
)
=
γ
ξ
x
(
s
)
holds for each
s
∈
I
.
We have
S
(
s
,
ξ
x
)
(
t
)
=
s
ξ
x
(
s
t
)
=
α
s
t
x
(
s
x
)
=
ξ
s
x
(
t
)
and hence
f
(
s
x
)
=
γ
ξ
x
(
s
)
by Remark 3.18.
For
x
,
x
+
h
∈
U
, we consider the smooth map
β
:
I
×
I
→
U
,
(
s
,
t
)
↦
t
(
x
+
s
h
)
and the
C
2
-function
F
:=
f
∘
β
.
Then the preceding considerations imply
F
(
s
,
0
)
=
f
(
0
)
=
1
,
∂
F
∂
t
(
s
,
t
)
=
d
d
t
f
(
t
(
x
+
s
h
)
)
=
d
d
t
γ
ξ
x
+
s
h
(
t
)
=
F
(
s
,
t
)
ξ
x
+
s
h
(
t
)
=
F
(
s
,
t
)
α
t
(
x
+
s
h
)
(
x
+
s
h
)
=
F
(
s
,
t
)
(
β
∗
α
)
(
s
,
t
)
(
∂
∂
t
)
.
Also,
∂
F
∂
s
(
s
,
0
)
=
0
=
(
β
∗
α
)
(
s
,
0
)
(
∂
∂
s
)
.
As we have seen in Lemma 3.20 (b), these relations lead to
δ
(
F
)
=
β
∗
α
on
I
×
I
.
We therefore obtain
d
d
s
f
(
x
+
s
h
)
=
∂
F
∂
s
(
s
,
1
)
=
F
(
s
,
1
)
α
x
+
s
h
(
h
)
=
f
(
x
+
s
h
)
α
x
+
s
h
(
h
)
,
and for
s
=
0
, this leads to
T
x
(
f
)
(
h
)
=
f
(
x
)
α
x
(
h
)
so that
δ
(
f
)
=
α
.
∎
Proposition 3.22
Let 𝑀 be a connected smooth manifold with rough boundary, 𝐺 a Lie group with Lie algebra 𝔤, and
α
∈
Ω
C
0
1
(
M
,
g
)
a continuous 1-form.
If 𝛼 is locally integrable, then there exists a connected covering
q
:
M
^
→
M
such that
q
*
α
is integrable.
If, in addition, 𝑀 is simply connected, then 𝛼 is integrable.
Proof
For smooth 1-forms on manifolds without boundary, the result can be found in [10].
The proof applies without changes.
∎
Definition 3.23
For each locally integrable
α
∈
Ω
C
k
1
(
M
,
g
)
, the homomorphism
per
α
m
0
:
π
1
(
M
,
m
0
)
→
G
,
[
γ
]
↦
evol
G
(
γ
*
α
)
,
for each piecewise smooth loop
γ
:
I
→
M
in
m
0
, is called the period homomorphism of 𝛼 with respect to
m
0
.
Note that
evol
G
(
γ
*
α
)
always exists, no matter whether 𝐺 is
C
k
-regular or not.
In fact, if
q
:
M
~
→
M
is a universal covering,
γ
~
:
I
→
M
~
a lift of 𝛾 and
f
:
M
~
→
G
a
C
k
+
1
-function such that
δ
f
=
q
*
α
and
f
(
γ
~
(
0
)
)
=
1
, then
f
∘
γ
~
=
Evol
G
(
γ
*
α
)
.
Theorem 3.24
Theorem 3.24 (Fundamental theorem for 𝔤-valued functions)
Let 𝑀 be a smooth manifold (possibly with rough boundary and modelled on a locally convex space) and 𝐺 a Lie group with Lie algebra 𝔤.
Let
k
∈
N
∪
{
∞
}
.
Then the following assertions hold.
If
k
≥
2
, 𝐺 is
C
k
-
2
-regular and
α
∈
Ω
C
k
1
(
M
,
g
)
satisfies the Maurer–Cartan equation, then 𝛼 is locally integrable.
If 𝑀 is 1-connected and
α
∈
Ω
C
0
1
(
M
,
g
)
is locally integrable, then it is integrable.
Suppose that 𝑀 is connected, fix
m
0
∈
M
, and let
α
∈
Ω
C
0
1
(
M
,
g
)
such that 𝛼 is locally integrable.
Using piecewise smooth representatives of homotopy classes, we obtain a well-defined group homomorphism
per
α
m
0
:
π
1
(
M
,
m
0
)
→
G
,
[
γ
]
↦
evol
G
(
γ
*
α
)
,
and 𝛼 is integrable if and only if this homomorphism is trivial.
Proof
(1) If 𝛼 satisfies the Maurer–Cartan equation, then Lemma 3.21 implies its local integrability, provided 𝐺 is
C
k
-
2
-regular.
(2) See Proposition 3.22.
(3) For 𝑀 without boundary and 𝛼 a smooth 1-form, the result can be found in [10].
The proof carries over, except that we use a different argument to see that
q
*
α
is locally integrable if
q
:
M
~
→
M
is a universal covering and 𝛼 is locally integrable: if
x
∈
M
~
, then there exists a
C
1
-map
f
:
U
→
G
on an open
q
(
x
)
-neighbourhood
U
⊆
M
such that
δ
f
=
α
|
U
.
Then
q
-
1
(
U
)
is an open 𝑥-neighbourhood in
M
~
and
f
∘
q
:
q
-
1
(
U
)
→
G
has left logarithmic derivative
q
*
α
|
q
(
U
)
.
No recourse to the Maurer–Cartan equation (as in [10]) is necessary for this argument.
∎
Remark 3.25
If 𝑀 is one-dimensional, then each 𝔤-valued 2-form on 𝑀 vanishes so that
[
α
,
β
]
=
0
=
d
α
for
α
,
β
∈
Ω
C
1
1
(
M
,
g
)
.
Therefore, we have that all 1-forms
α
∈
Ω
C
1
1
(
M
,
g
)
trivially satisfy the Maurer–Cartan equation.
Lemma 3.26
Let 𝑀 be a smooth manifold with rough boundary, 𝑉 a locally convex topological vector space and
γ
:
[
0
,
1
]
→
M
a
C
s
+
1
-path with
s
∈
N
0
∪
{
∞
}
.
Then
ψ
:
Ω
C
s
1
(
M
,
V
)
→
C
s
(
[
0
,
1
]
,
V
)
,
ω
↦
γ
*
(
ω
)
is a smooth map.
Proof
It is well known that pullbacks along
C
s
-maps like
C
s
(
T
M
,
V
)
→
C
s
(
[
0
,
1
]
,
V
)
,
f
↦
f
∘
γ
′
are continuous (see, e.g., [10]).
Since 𝜓 is a restriction of the latter pullback, it is continuous.
Being linear and continuous, 𝜓 is smooth.
∎
Lemma 3.27
Let 𝑀 be a compact manifold, 𝑁 a locally convex manifold (both possibly with rough boundary), 𝐺 a Lie group with Lie algebra 𝔤, and let
i
:
C
r
(
M
,
G
)
→
C
(
M
,
G
)
be the inclusion map with
r
,
s
∈
N
0
∪
{
∞
}
,
r
≥
1
.
A map
f
:
N
→
C
r
(
M
,
G
)
is
C
s
if and only if
i
∘
f
:
N
→
C
(
M
,
G
)
is
C
s
and
δ
∘
f
:
N
→
Ω
C
r
-
1
1
(
M
,
g
)
is
C
s
.
Proof
It is well known that 𝑖 is a smooth homomorphism of groups.
Also,
δ
:
C
r
(
M
,
G
)
→
Ω
C
r
-
1
1
(
M
,
g
)
is smooth (see [17, Proposition A.4]).
Hence if 𝑓 is
C
s
, then also the compositions
i
∘
f
and
δ
∘
f
are
C
s
.
Conversely, assume that
i
∘
f
and
δ
∘
f
are
C
s
.
Let
φ
:
U
→
V
be a chart for 𝐺 around 1 such that
φ
*
:=
C
r
(
M
,
φ
)
:
C
r
(
M
,
U
)
→
C
r
(
M
,
V
)
is a chart for
C
r
(
M
,
G
)
and
C
(
M
,
φ
)
a chart for
C
(
M
,
G
)
.
Because
i
∘
f
is continuous, after replacing 𝑁 by an open neighbourhood of a given point 𝑛 of 𝑁, we may assume that
f
(
N
)
f
-
1
(
n
)
⊆
C
(
M
,
U
)
.
It suffices to show that
g
:
N
→
C
r
(
M
,
G
)
,
x
↦
f
(
x
)
f
(
n
)
-
1
is
C
s
.
Let
π
:
T
M
→
M
be the canonical map.
Now note that
i
∘
g
=
ρ
c
∘
i
∘
f
is
C
s
, where we have abbreviated
c
:=
f
(
n
)
-
1
and the right translation
ρ
c
:
C
(
M
,
G
)
→
C
(
M
,
G
)
,
γ
↦
γ
c
is a smooth map.
Furthermore,
δ
∘
g
is
C
s
.
Indeed,
g
(
x
)
=
f
(
x
)
f
(
n
)
-
1
, where
f
(
x
)
,
f
(
n
)
-
1
∈
C
r
(
M
,
G
)
.
Hence
δ
(
g
(
x
)
)
=
Ad
(
f
(
n
)
)
.
(
δ
(
f
(
x
)
)
-
δ
(
f
(
n
)
-
1
)
)
, and
Ad
(
f
(
n
)
)
.
δ
(
f
(
n
)
-
1
)
is independent of 𝑥, hence
C
s
in 𝑥.
Also, we have that
Ad
(
f
(
n
)
)
.
δ
(
f
(
x
)
)
is
C
s
in 𝑥 since
δ
∘
f
:
N
→
Ω
C
r
-
1
1
(
M
,
g
)
⊆
C
r
-
1
(
T
M
,
g
)
is assumed
C
s
and
(
Ad
(
f
(
n
)
)
.
ω
)
.
(
v
)
=
Ad
(
f
(
n
)
(
π
(
v
)
)
)
ω
(
v
)
=
h
*
(
ω
)
(
v
)
,
where
ω
∈
Ω
C
r
-
1
1
(
M
,
g
)
,
v
∈
T
M
and
h
:
T
M
×
g
→
g
,
h
(
v
,
w
)
:=
Ad
(
f
(
n
)
(
π
(
v
)
)
)
w
is a
C
r
-function and linear in 𝜔, entailing that
h
*
:
C
r
-
1
(
T
M
,
g
)
→
C
r
-
1
(
T
M
,
g
)
,
h
*
(
w
)
(
v
,
w
)
:=
h
(
v
,
ω
(
v
)
)
is continuous linear, hence
C
s
.
Hence, without loss of generality,
f
(
N
)
⊆
C
r
(
M
,
U
)
.
As
i
∘
f
is
C
s
, the map
φ
*
∘
i
∘
f
:
N
→
C
(
M
,
V
)
is
C
s
.
We have
(
φ
*
∘
f
)
(
N
)
⊆
C
r
(
M
,
V
)
.
We show that
φ
*
∘
f
:
N
→
C
r
(
M
,
V
)
is
C
s
.
As a tool, consider the set
P
:=
{
(
x
,
y
)
∈
U
×
U
:
x
y
∈
U
}
which is open in
U
×
U
and contains
{
1
}
×
U
.
Thus
Q
:=
(
φ
×
φ
)
(
P
)
is open in
V
×
V
and contains
{
0
}
×
V
.
The map
ν
:
Q
→
V
,
ν
(
x
,
y
)
:=
φ
(
φ
-
1
(
x
)
φ
-
1
(
y
)
)
is smooth.
Also, the map
θ
:
V
×
E
→
E
,
(
x
,
u
)
↦
d
ν
(
x
,
0
;
0
,
u
)
is smooth, where 𝐸 is the modelling space of 𝑀, and we have
d
φ
(
x
.
v
)
=
θ
(
φ
(
x
)
,
d
φ
(
v
)
)
for
x
∈
U
,
v
∈
T
1
G
=
g
.
Let
j
:
C
r
(
M
,
g
)
→
C
(
M
,
g
)
be the inclusion map.
It is known that the map
(
j
,
d
)
:
C
r
(
M
,
g
)
→
C
(
M
,
g
)
×
C
r
-
1
(
T
M
,
g
)
,
γ
↦
(
γ
,
d
γ
)
is a linear topological embedding with closed image.
Hence
φ
*
∘
f
will be
C
s
if
j
∘
φ
*
∘
f
is
C
s
and
ψ
:=
d
∘
φ
*
∘
f
:
N
→
C
r
-
1
(
T
M
,
g
)
is
C
s
.
Now, as just observed,
j
∘
φ
*
∘
f
=
φ
*
∘
i
∘
f
is
C
s
.
By the exponential law [3, Proposition 4.5], 𝜓 will be
C
s
if
ψ
∧
:
N
×
T
M
→
g
is
C
s
,
r
-
1
.
But
ψ
∧
(
x
,
v
)
=
d
(
φ
∘
f
(
x
)
)
(
v
)
=
(
d
φ
∘
T
(
f
(
x
)
)
)
(
v
)
=
d
φ
(
f
(
x
)
(
π
(
v
)
)
.
δ
(
f
(
x
)
)
(
v
)
)
=
θ
(
φ
(
f
(
x
)
(
π
(
v
)
)
)
,
d
φ
(
(
δ
f
(
x
)
)
(
v
)
)
)
,
and 𝜃 is
C
∞
,
φ
(
f
(
x
)
(
π
(
v
)
)
)
=
(
φ
*
∘
f
)
∧
(
x
,
π
(
v
)
)
is
C
s
,
r
in
(
x
,
v
)
,
d
φ
is
C
∞
and
(
δ
f
(
x
)
)
(
v
)
is
C
s
,
r
-
1
in
(
x
,
v
)
by the exponential law [3, Proposition 4.5].
Thus
ψ
∧
is indeed
C
s
,
r
-
1
, by Chain Rule 2 [3, Lemma 3.18].
∎
Proposition 3.28
Let 𝑁 be a locally convex manifold, 𝑀 a connected locally compact smooth manifold (both possibly with rough boundary); let
r
∈
N
0
∪
{
∞
}
,
s
∈
N
∪
{
∞
}
, and let 𝐺 be a
C
s
-
1
-regular Lie group.
Let
m
0
∈
M
.
Then a function
f
:
N
×
M
→
G
is
C
r
,
s
if and only if
f
m
0
:
N
→
G
,
n
↦
f
(
n
,
m
0
)
is
C
r
, and
the functions
f
n
:
M
→
G
,
m
↦
f
(
n
,
m
)
are
C
s
and
F
:
N
→
Ω
C
s
-
1
1
(
M
,
g
)
,
n
↦
δ
(
f
n
)
is
C
r
.
Proof
If 𝑓 is a
C
r
,
s
-map, then the map
f
m
0
is
C
r
and each
f
n
is
C
s
.
Since
Ω
C
s
-
1
1
(
M
,
g
)
is a closed vector subspace of
C
s
-
1
(
T
M
,
g
)
, it only remains to show that the map
F
:
N
→
C
s
-
1
(
T
M
,
g
)
is
C
r
.
By [3, Proposition 4.5], it suffices to show that
F
∧
:
N
×
T
M
→
g
,
(
n
,
v
)
↦
δ
(
f
n
)
v
=
κ
G
(
T
(
f
n
)
v
)
is
C
r
,
s
-
1
.
Now the Maurer–Cartan form
κ
G
is a smooth map
T
G
→
g
, and the map
N
×
T
M
→
T
G
,
(
n
,
v
)
↦
T
(
f
n
)
(
v
)
is a
C
r
,
s
-
1
-map (cf. [3, Lemma 3.11]).
In view of [3, Lemma 3.18], the assertion follows.
Conversely, assume that (1) and (2) hold.
We first show that
f
m
is
C
r
for each
m
∈
M
.
Pick a smooth path
γ
:
[
0
,
1
]
→
M
with
γ
(
0
)
=
m
0
and
γ
(
1
)
=
m
.
Then
f
m
(
n
)
=
f
n
(
m
)
=
f
n
(
m
0
)
evol
G
(
δ
(
f
n
∘
γ
)
)
=
f
n
(
m
0
)
evol
G
(
γ
*
δ
(
f
n
)
)
=
f
m
0
(
n
)
evol
G
(
γ
*
F
(
n
)
)
.
Since
f
m
0
and 𝐹 are
C
r
, the smoothness of
evol
G
and the smoothness of
γ
*
:
Ω
C
s
-
1
1
(
M
,
g
)
→
C
s
-
1
(
[
0
,
1
]
,
g
)
(see Lemma 3.26) imply that
f
m
is
C
r
.
Now we show that 𝑓 is
C
r
,
s
.
Because 𝑀 can be covered with the interiors of well-located compact manifolds
L
⊆
M
with rough boundary, and the pullbacks
Ω
C
s
-
1
1
(
M
,
g
)
→
Ω
C
s
-
1
1
(
M
,
g
)
induced by inclusion are continuous linear, we may assume that 𝑀 is compact for the remainder of the proof.
Let
i
:
C
r
(
M
,
G
)
→
C
(
M
,
G
)
be the inclusion map.
By Proposition 3.14, 𝑓 will be
C
r
,
s
if
f
∨
:
N
→
C
s
(
M
,
G
)
,
n
↦
f
n
is
C
r
.
By Lemma 3.27,
f
∨
will be
C
r
if
δ
∘
f
∨
=
F
is
C
r
(which holds by hypothesis) and
i
∘
f
∨
:
N
→
C
(
M
,
G
)
is
C
r
.
By Proposition 3.14,
i
∘
f
∨
will be
C
r
if we can show that 𝑓 is
C
r
,
0
.
Let
m
∈
M
, and choose a chart
(
ϕ
,
U
)
of 𝑀 for which
ϕ
(
U
)
is convex with
ϕ
(
m
)
=
0
.
We have to show that the map
h
:
N
×
ϕ
(
U
)
→
G
,
(
n
,
x
)
↦
f
(
n
,
ϕ
-
1
(
x
)
)
is
C
r
,
0
.
For
γ
x
(
t
)
:=
t
x
,
0
≤
t
≤
1
, we have
h
(
n
,
x
)
=
h
(
n
,
γ
x
(
1
)
)
=
h
(
n
,
0
)
evol
G
(
δ
(
f
n
∘
ϕ
-
1
∘
γ
x
)
)
=
f
m
(
n
)
evol
G
(
γ
x
*
(
ϕ
-
1
)
*
F
(
n
)
)
.
Since
f
m
and 𝐹 are
C
r
-maps and
evol
G
is smooth and
(
ϕ
-
1
)
*
:
Ω
C
s
-
1
1
(
U
,
g
)
→
Ω
C
s
-
1
1
(
ϕ
(
U
)
,
g
)
is a topological linear isomorphism, in view of Chain Rule 1 [3, Lemma 3.17], it suffices to show that the map
Ω
C
s
-
1
1
(
ϕ
(
U
)
,
g
)
×
ϕ
(
U
)
→
C
s
-
1
(
[
0
,
1
]
,
g
)
,
(
α
,
x
)
↦
γ
x
*
α
is
C
r
,
0
.
As the map is linear in 𝛼, it suffices to show that it is
C
0
,
0
, i.e., continuous.
The latter holds since
T
ϕ
(
U
)
is locally compact, entailing that the evaluation map
C
s
-
1
(
T
ϕ
(
U
)
,
g
)
×
T
ϕ
(
U
)
→
g
is continuous.
∎
Lemma 3.29
Let 𝑀 be a connected, locally compact smooth manifold (possibly with rough boundary), let
k
∈
N
0
∪
{
∞
}
, and let 𝐺 be a
C
k
-regular Lie group with Lie algebra 𝔤.
If
γ
:
[
0
,
1
]
→
M
is a piecewise smooth curve, then the map
Ω
C
k
1
(
M
,
g
)
→
G
,
α
↦
evol
G
(
γ
*
α
)
is smooth.
Let
(
φ
,
U
)
be a chart of 𝑀 for which
φ
(
U
)
is a convex 0-neighbourhood and
γ
x
(
t
)
:=
φ
-
1
(
t
φ
(
x
)
)
for
x
∈
U
,
t
∈
[
0
,
1
]
.
Let
l
∈
N
0
∪
{
∞
}
.
Then the map
Ω
C
k
+
l
1
(
M
,
g
)
×
U
→
G
,
(
α
,
x
)
↦
evol
G
(
γ
x
*
α
)
is
C
∞
,
l
.
Proof
(1) This follows from the smoothness of
evol
G
and the fact that, for each smooth path
η
:
[
0
,
1
]
→
M
, the map
Ω
C
k
1
(
M
,
g
)
→
C
k
(
[
0
,
1
]
,
g
)
,
α
↦
η
*
α
=
α
∘
η
′
is smooth (see Lemma 3.26).
(2) We may assume that
M
=
U
=
ϕ
(
U
)
and
ϕ
=
id
U
.
Since 𝐺 is
C
k
-regular, we only need to show that the map
Ω
C
k
+
l
1
(
U
,
g
)
×
U
→
C
k
(
[
0
,
1
]
,
g
)
,
(
α
,
x
)
↦
γ
x
*
α
is
C
∞
,
l
.
By linearity in the first argument, we only need to show that the map is
C
0
,
l
.
By the exponential law for
C
α
maps [2, Proposition 4.3], we only need to show that
Ω
C
k
+
l
1
(
U
,
g
)
×
U
×
[
0
,
1
]
→
g
,
(
α
,
x
,
t
)
↦
(
γ
x
*
α
)
t
=
α
γ
x
(
t
)
γ
x
′
(
t
)
is
C
∞
,
l
,
k
as a function of 3 variables, which holds if it is
C
∞
,
l
+
k
as a function of the 2 variables
(
α
,
(
x
,
t
)
)
.
But
α
γ
x
(
t
)
γ
x
′
(
t
)
=
α
(
x
t
,
x
)
=
ε
(
α
,
(
x
t
,
x
)
)
is
C
∞
,
l
+
k
, like
ε
:
C
l
+
k
(
T
U
,
g
)
×
T
U
→
g
.
∎
4 Lie group structure
In this section, we study Lie group structures on groups of the form
C
k
(
M
,
K
)
, where 𝑀 is a non-compact smooth manifold and 𝐾 is a, possibly infinite-dimensional, Lie group.
Proposition 4.1
Let 𝑀 be a locally compact smooth manifold with rough boundary and 𝐾 a regular Lie group, and let
k
∈
N
0
∪
{
∞
}
.
Assume that the group
G
:=
C
k
(
M
,
K
)
carries a Lie group structure which is compatible with evaluations in the sense that
g
:=
C
k
(
M
,
k
)
is the corresponding Lie algebra and all point evaluations
ev
m
:
G
→
K
,
m
∈
M
, are smooth with
L
(
ev
m
)
=
ev
m
:
g
→
k
.
Then the following holds.
The evaluation map
ev
:
G
×
M
→
K
,
(
f
,
m
)
↦
f
(
m
)
is
C
∞
,
k
.
If 𝑁 is a locally convex
C
r
-manifold and the map
f
:
N
→
G
is
C
r
for some
r
∈
N
0
∪
{
∞
}
, then
f
∧
:
N
×
M
→
K
is
C
r
,
k
.
If, in addition, 𝐺 is
C
r
-
3
-regular, where
r
≥
3
and 𝑀 is connected, then a map
f
:
N
→
G
is
C
r
if and only if the corresponding map
f
∧
:
N
×
M
→
K
is
C
r
,
k
.
Proof
(1) Let
N
⊆
M
be a compact smooth manifold with rough boundary which is well located in 𝑀.
Then
C
k
(
N
,
K
)
carries the structure of a regular Lie group (see [10]).
Let
q
G
:
G
~
0
→
G
0
denote the universal covering of the identity component
G
0
of 𝐺.
Consider the continuous homomorphism of Lie algebras
ψ
:
L
(
G
)
=
C
k
(
M
,
k
)
→
C
k
(
N
,
k
)
,
f
↦
f
|
N
.
In view of the regularity of
C
k
(
N
,
K
)
, there exists a unique morphism of Lie groups
φ
~
:
G
~
0
→
C
k
(
N
,
K
)
with
L
(
φ
~
)
=
ψ
.
Then, for each
n
∈
N
, the homomorphism
ev
n
∘
φ
~
:
G
~
0
→
K
is smooth with differential
L
(
ev
n
∘
φ
~
)
=
ev
n
, so that
ev
n
∘
φ
~
=
ev
n
∘
q
G
.
We conclude that
ker
q
G
⊆
ker
φ
~
and hence that
φ
~
factors to the restriction map
ρ
:
C
k
(
M
,
K
)
0
→
C
k
(
N
,
K
)
, i.e.,
φ
~
=
ρ
∘
q
G
.
In particular, the restriction map
C
k
(
M
,
K
)
→
C
k
(
N
,
K
)
is a smooth homomorphism of Lie groups.
Since
ε
:
C
k
(
N
,
K
)
×
N
→
K
is a
C
∞
,
k
-map, by Theorem 3.12, (1) follows.
(2) If 𝑓 is
C
r
, then
f
∧
=
ev
∘
(
f
×
id
M
)
is
C
r
,
k
, using that
ev
is
C
∞
,
k
by (1).
(3) We may without loss of generality assume that 𝑁 is 1-connected.
If
f
∧
is
C
r
,
k
, we define
β
∈
Ω
C
r
-
1
1
(
N
,
g
)
by
β
(
ν
)
(
m
)
=
κ
K
(
T
(
f
∧
(
⋅
,
m
)
)
(
ν
)
)
,
which is a
C
r
-map as
β
∧
:
T
N
×
M
→
k
is
C
r
-
1
,
k
.
We claim that 𝛽 satisfies the Maurer–Cartan equation.
Since the evaluation map
ev
m
:
g
→
k
is a continuous homomorphism of Lie algebras, and the corresponding maps
(
ev
m
)
*
:
Ω
C
r
-
2
2
(
N
,
g
)
→
Ω
C
r
-
2
2
(
N
,
k
)
,
ω
↦
ev
m
∘
ω
separate the points, for
m
∈
M
, it follows that 𝛽 satisfies the Maurer–Cartan equation, using that
β
(
ν
)
(
m
)
=
δ
f
∧
(
⋅
,
m
)
(
ν
)
.
Fix a point
n
0
∈
N
.
The fundamental theorem (Theorem 3.24) implies the existence of a unique
C
r
-map
h
:
N
→
G
with
h
(
n
0
)
=
f
(
n
0
)
and
δ
(
h
)
=
β
.
Then
δ
(
ev
m
∘
h
)
=
ev
m
∘
δ
(
h
)
=
ev
m
∘
β
=
δ
(
ev
m
∘
f
)
, so Lemma 3.5, applied to 𝐾-valued functions, yields
ev
m
∘
h
=
ev
m
∘
f
for each 𝑚, which leads to
h
=
f
.
This proves that 𝑓 is a
C
r
-map.
∎
Example 4.2
If 𝑀 is a compact manifold (possibly with boundary), then the ordinary Lie group structure on
G
:=
C
k
(
M
,
K
)
is compatible with evaluations.
To identify
T
1
(
G
)
with
C
k
(
M
,
k
)
, pick a chart
φ
:
U
→
V
⊆
k
of 𝐾 around 1 such that
φ
(
1
)
=
0
and
d
φ
|
k
=
id
k
.
Then
ψ
:=
d
(
φ
*
)
|
T
1
G
:
T
1
G
→
C
k
(
M
,
k
)
is a suitable isomorphism (cf. [7]).
Note that
ψ
-
1
is the map
C
k
(
M
,
k
)
→
T
1
G
,
γ
↦
d
d
t
|
t
=
0
(
φ
-
1
∘
t
γ
)
.
If 𝐾 has a smooth exponential function, then
ψ
-
1
coincides with the map
γ
↦
d
d
t
|
t
=
0
(
exp
K
∘
(
t
γ
)
)
because the smooth map
(
φ
∘
exp
K
)
*
:
C
k
(
M
,
k
)
→
C
k
(
M
,
k
)
,
γ
↦
φ
∘
exp
K
∘
γ
satisfies
d
(
φ
∘
exp
K
)
*
(
0
,
∙
)
=
id
and thus
d
d
t
|
t
=
0
(
exp
K
∘
(
t
γ
)
)
=
d
d
t
|
t
=
0
(
φ
-
1
∘
φ
∘
exp
K
∘
(
t
γ
)
)
=
d
(
φ
-
1
)
*
(
0
,
d
(
φ
∘
exp
K
)
*
(
γ
)
)
=
ψ
-
1
.
Remark 4.3
If 𝐾 is regular and 𝑀 as in Proposition 4.1, then a Lie group structure on
G
:=
C
k
(
M
,
K
)
compatible with evaluations is unique whenever it exists.
In fact, assume that there is another structure
G
~
.
Let
f
:
G
~
→
G
and
g
:
G
→
G
be the maps
x
↦
x
.
Because 𝑔 is smooth, the map
f
∧
=
g
∧
is
C
∞
,
k
by Proposition 4.1 (2), and hence 𝑓 is smooth by Proposition 4.1 (3).
Likewise,
f
-
1
is smooth, and thus
G
~
=
G
.
Proposition 4.4
If 𝐾 is a
C
k
-
1
-regular Lie group with
k
≥
2
, 𝑀 a connected finite-dimensional smooth manifold with rough boundary and
m
0
∈
M
, then the map
δ
:
C
*
k
(
M
,
K
)
→
Ω
C
k
-
1
1
(
M
,
k
)
is a topological embedding.
Let
Evol
K
:=
δ
-
1
:
im
(
δ
)
→
C
*
k
(
M
,
K
)
denote its inverse.
Then 𝛿 is an isomorphism of topological groups if we endow
im
(
δ
)
with the group structure defined by
(4.1)
α
*
β
:=
β
+
Ad
(
Evol
K
(
β
)
)
-
1
⋅
α
,
(4.2)
α
-
1
:=
-
Ad
(
Evol
K
(
α
)
)
⋅
α
.
Proof
By definition of the topology on
C
k
(
M
,
K
)
, the tangent map induces a continuous group homomorphism
T
:
C
k
(
M
,
K
)
→
C
k
-
1
(
T
M
,
T
K
)
,
f
↦
T
(
f
)
.
Let
κ
K
:
T
K
↦
k
denote the (left) Maurer–Cartan form of 𝐾.
Since
δ
(
f
)
=
f
*
κ
K
=
κ
K
∘
T
(
f
)
,
it follows that the composition
C
k
(
M
,
K
)
→
C
k
-
1
(
T
(
M
)
,
T
(
K
)
)
→
C
k
-
1
(
T
(
M
)
,
k
)
,
f
↦
T
(
f
)
↦
δ
(
f
)
is continuous.
Next we show that 𝛿 is an embedding.
Consider
α
=
δ
(
f
)
with
f
∈
C
*
k
(
M
,
K
)
, i.e.,
f
(
m
0
)
=
1
holds for the base point
m
0
∈
M
.
To reconstruct 𝑓 from 𝛼, since 𝑀 is connected, we can find for
m
∈
M
a smooth path
γ
:
[
0
,
1
]
→
M
with
γ
(
0
)
=
m
0
and
γ
(
1
)
=
m
.
Then
δ
(
f
∘
γ
)
=
γ
*
δ
(
f
)
=
γ
*
α
implies
f
(
m
)
=
evol
K
(
γ
*
α
)
.
We now choose an open neighbourhood 𝑈 of 𝑚 and a chart
(
φ
,
U
)
of 𝑀 such that
φ
(
U
)
is convex with
φ
(
m
)
=
0
.
For each
x
∈
U
, define
γ
x
:
[
0
,
1
]
→
U
,
γ
x
(
t
)
:=
φ
-
1
(
t
φ
(
x
)
)
.
Then
δ
(
f
(
m
)
-
1
(
f
∘
γ
x
)
)
=
δ
(
f
∘
γ
x
)
=
γ
x
*
δ
f
=
γ
x
*
α
implies that
f
(
m
)
-
1
f
(
x
)
=
evol
K
(
γ
x
*
α
)
, and hence
f
(
x
)
=
f
(
m
)
.
evol
K
(
γ
x
*
α
)
.
From Lemma 3.29, we immediately derive that the map
Ω
C
k
-
1
1
(
M
,
k
)
×
U
→
K
,
(
α
,
x
)
↦
evol
K
(
γ
*
α
)
⋅
evol
K
(
γ
x
*
α
)
is continuous so that the corresponding map
Ω
C
k
-
1
1
(
M
,
k
)
↦
C
0
(
U
,
K
)
is continuous.
We conclude that the map
δ
(
C
*
k
(
M
,
k
)
)
→
C
0
(
U
,
K
)
,
δ
(
f
)
↦
f
|
U
is continuous.
We finally observe that, for each open covering
M
=
⋃
j
∈
J
U
j
, the restriction maps to
U
j
lead to a topological embedding
C
*
0
(
M
,
K
)
↪
∏
j
∈
J
C
0
(
U
j
,
K
)
.
Hence
δ
(
C
*
k
(
M
,
k
)
)
→
C
0
(
M
,
K
)
,
δ
(
f
)
↦
f
is continuous.
Now, we show by induction that
θ
j
:
δ
(
C
*
k
(
M
,
k
)
)
→
C
j
(
M
,
K
)
,
δ
(
f
)
↦
f
is continuous for
j
=
0
,
…
,
k
.
The topology on
C
j
(
M
,
K
)
is initial with respect to the inclusion
C
j
(
M
,
K
)
→
C
0
(
M
,
K
)
and the map
T
:
C
j
(
M
,
K
)
→
C
j
-
1
(
T
M
,
T
K
)
.
Because
incl
∘
θ
j
=
θ
0
is continuous, the map
θ
j
will be continuous if we can show that
T
∘
θ
j
is also continuous.
Let 𝑚 be the continuous group multiplication of
C
j
-
1
(
T
M
,
T
K
)
.
We have
T
f
=
f
⋅
δ
f
=
f
⋅
α
for
α
=
δ
f
and thus
T
θ
j
(
α
)
=
θ
j
-
1
(
α
)
⋅
α
inside
C
j
-
1
(
T
U
,
T
K
)
.
Because the inclusion
Ω
C
k
-
1
1
(
U
,
k
)
↪
C
j
-
1
(
T
U
,
T
K
)
is continuous, so
T
∘
θ
j
=
m
∘
(
θ
j
-
1
,
incl
)
is continuous (since
θ
j
-
1
is continuous by induction).
∎
Theorem 4.5
Let
s
,
k
∈
N
0
∪
{
∞
}
with
k
≥
s
+
1
, and let 𝑀 be a connected locally compact smooth manifold (with rough boundary),
m
0
∈
M
and 𝐾 a
C
s
-regular Lie group.
Assume that the subset
δ
(
C
*
k
(
M
,
K
)
)
is a smooth submanifold of
Ω
C
k
-
1
1
(
M
,
k
)
.
Endow
C
*
k
(
M
,
K
)
with the smooth manifold structure for which
δ
:
C
*
k
(
M
,
K
)
→
im
(
δ
)
is a diffeomorphism, and endow
C
k
(
M
,
K
)
≅
K
⋉
C
*
k
(
M
,
K
)
with the product manifold structure.
Assume that
L
j
for
j
∈
J
are compact smooth manifolds with rough boundary which are well located in 𝑀 and whose interiors
L
j
∘
cover 𝑀 such that
m
0
∈
L
j
and such that
δ
j
:
C
*
k
(
L
j
,
K
)
→
Ω
C
k
-
1
1
(
L
j
,
k
)
is an embedding of smooth manifolds onto a submanifold of
Ω
C
k
-
1
1
(
L
j
,
k
)
.
Then the following assertions hold.
A map
f
:
N
×
M
→
K
is
C
r
,
k
for each
r
∈
N
0
∪
{
∞
}
and locally convex
C
r
-manifold 𝑁
if and only if the map
f
n
:
M
→
K
,
m
↦
f
(
n
,
m
)
is
C
k
for all
n
∈
N
, and the corresponding map
f
∨
:
N
→
C
k
(
M
,
K
)
,
n
↦
f
n
is
C
r
.
𝐾 acts smoothly by conjugation on
C
*
k
(
M
,
K
)
, and the above manifold structure on
C
k
(
M
,
K
)
yields a
C
s
-regular Lie group structure compatible with evaluations.
Proof
(1) Let
m
0
be the base point of 𝑀.
According to Proposition 3.28, we have that
f
:
N
×
M
→
K
is
C
r
,
k
if and only if
f
m
0
is
C
r
, all the maps
f
n
are
C
k
and
δ
∘
f
∨
:
N
→
Ω
C
k
-
1
1
(
M
,
k
)
is
C
r
.
In view of our definition of the manifold structure on
C
*
k
(
M
,
K
)
, the latter condition is equivalent to the
C
r
-property for the map
N
→
C
*
k
(
M
,
K
)
,
n
↦
f
n
(
m
0
)
-
1
f
n
=
f
m
0
(
n
)
-
1
f
n
.
Since the evaluation in
m
0
coincides with the projection
C
k
(
M
,
K
)
≅
K
⋉
C
*
k
(
M
,
K
)
→
K
, we see that 𝑓 is
C
r
,
k
if and only if all the maps
f
n
are
C
k
and
f
∨
is
C
r
.
(2) For the evaluation map
f
=
ev
:
C
k
(
M
,
K
)
×
M
→
K
, we have
ev
∨
=
id
G
with
G
:=
C
k
(
M
,
K
)
, and
ev
g
=
g
for each
g
∈
G
.
Hence (1) implies that
ev
is
C
∞
,
k
.
In view of Proposition 4.4, 𝛿 is an isomorphism of topological groups if
im
(
δ
)
is endowed with the group structure (4.1).
We now show that the operations (4.1) and (4.2) are smooth with respect to the submanifold structure on
im
(
δ
)
.
The Lie group structure. To see that
C
*
k
(
M
,
K
)
is a Lie group, it suffices to show that the map
(
α
,
β
)
↦
α
*
β
-
1
=
Ad
(
Evol
K
(
β
)
)
(
α
-
β
)
is smooth, which holds if the map
θ
:
im
(
δ
)
×
im
(
δ
)
→
Ω
C
k
-
1
1
(
M
,
k
)
,
(
α
,
β
)
↦
Ad
(
Evol
K
(
α
)
)
⋅
β
is smooth.
Let
(
L
j
)
j
∈
J
be a family of compact manifolds (with rough boundary), as described in the theorem.
Then
Ω
C
k
-
1
1
(
M
,
k
)
→
∏
j
∈
J
Ω
C
k
-
1
1
(
L
j
,
k
)
,
α
↦
(
α
|
T
L
j
)
j
∈
J
is linear and a topological embedding with closed image.
Let
ρ
j
:
Ω
C
k
-
1
1
(
M
,
k
)
→
Ω
C
k
-
1
1
(
L
j
,
k
)
,
α
↦
α
|
T
L
j
be the restriction map.
Then 𝜃 will be smooth if we can show that
ρ
j
∘
θ
is smooth for each
j
∈
J
.
Now, by the assumption and using the Lie group structure on
C
*
k
(
L
j
,
K
)
, the map
θ
j
:
im
(
δ
j
)
×
im
(
δ
j
)
→
Ω
C
k
-
1
1
(
L
j
,
k
)
analogous to 𝜃 is smooth.
Consider the commutative diagram, in which 𝜓 is the restriction map,
In the above diagram,
ρ
j
∘
θ
=
θ
j
∘
ψ
is smooth; thus 𝜃 is smooth.
To see that
C
k
(
M
,
K
)
=
K
⋉
C
*
k
(
M
,
K
)
is a Lie group, it remains to show that the action
σ
:
K
×
C
*
k
(
M
,
K
)
→
C
*
k
(
M
,
K
)
,
(
g
,
γ
)
↦
g
γ
g
-
1
is smooth.
This holds if and only if
δ
∘
σ
is smooth.
Now, for
g
∈
K
and
γ
∈
C
*
k
(
M
,
K
)
,
δ
(
σ
(
γ
)
)
=
δ
(
γ
g
-
1
)
=
Ad
(
g
-
1
)
-
1
δ
(
γ
)
+
δ
(
g
-
1
)
⏟
=
0
=
Ad
(
g
)
δ
(
γ
)
(considering 𝑔 as a constant path in
C
k
(
M
,
K
)
).
Equivalently, writing
δ
(
γ
)
=
α
, we thus have to show that
K
×
im
(
δ
)
→
im
(
δ
)
,
(
g
,
α
)
↦
Ad
(
g
)
∘
α
is smooth.
This follows if
τ
:
K
×
C
k
-
1
(
T
M
,
k
)
→
C
k
-
1
(
T
M
,
k
)
,
(
g
,
γ
)
↦
Ad
(
g
)
∘
γ
is smooth.
Now we have that
τ
∧
(
g
,
γ
,
v
)
=
Ad
(
g
)
γ
(
v
)
=
Ad
(
g
)
ε
(
γ
,
v
)
is
C
∞
,
k
-
1
in
(
(
g
,
γ
)
,
v
)
, by the chain rule, with evaluation
ε
:
C
k
-
1
(
T
M
,
k
)
×
T
M
→
k
which is
C
∞
,
k
-
1
(see Proposition 3.11).
Hence
τ
∧
is
C
∞
,
k
-
1
in
(
(
g
,
γ
)
,
v
)
, and thus
τ
=
(
τ
∧
)
∨
is
C
∞
indeed.
If 𝑀 is compact, then the Lie group structure on
C
*
k
(
M
,
K
)
coincides with the ordinary one.
Indeed, write
C
*
k
(
M
,
K
)
ord
for the latter.
Also, write
f
:
C
*
k
(
M
,
K
)
ord
→
C
*
k
(
M
,
K
)
,
g
:
C
*
k
(
M
,
K
)
ord
→
C
*
k
(
M
,
K
)
ord
and
h
:
C
*
k
(
M
,
K
)
→
C
*
k
(
M
,
K
)
for the maps given by
γ
↦
γ
.
Since ℎ is smooth,
h
∧
=
f
∧
is
C
∞
,
k
by (1).
Hence 𝑓 is smooth by (1).
Likewise, 𝑔 is smooth; whence
g
∧
=
(
f
-
1
)
∧
is
C
∞
,
k
(see Proposition 3.14).
Hence
f
-
1
is smooth by Proposition 3.14.
Thus 𝑓 is an isomorphism, and thus
C
*
k
(
M
,
K
)
ord
=
C
*
k
(
M
,
K
)
.
To emphasize the dependence on 𝑀, we occasionally write
δ
M
instead of 𝛿.
If
M
1
with base point
m
1
∈
M
1
has properties analogous to 𝑀 and
f
:
M
1
→
M
is a smooth map with
f
(
m
1
)
=
m
0
, then
f
*
:
C
*
k
(
M
,
K
)
→
C
*
k
(
M
1
,
K
)
,
γ
↦
γ
∘
f
is a smooth homomorphism of Lie groups and the diagram
commutes, where we also use the continuous linear (and hence smooth) map
f
*
:
Ω
C
k
-
1
1
(
M
,
k
)
→
Ω
C
k
-
1
1
(
M
1
,
k
)
,
ω
↦
f
*
ω
.
Indeed, (4.3) commutes because
f
*
(
δ
M
(
γ
)
)
=
f
*
(
γ
*
(
κ
K
)
)
=
(
γ
∘
f
)
*
(
κ
K
)
=
(
f
*
(
γ
)
)
*
(
κ
K
)
=
δ
M
1
(
f
*
(
γ
)
)
using the Maurer–Cartan form
κ
K
on 𝐾.
Since
δ
M
and
δ
M
1
are isomorphisms onto their images, and
f
*
on the left-hand side of (4.3) is a group homomorphism, also the smooth map
f
*
:
im
(
δ
M
)
→
im
(
δ
M
1
)
is a homomorphism of groups.
As a consequence,
(4.4)
f
*
:
C
k
(
M
,
K
)
→
C
k
(
M
1
,
K
)
is also a smooth group homomorphism.
The Lie algebra. We first determine the Lie algebra of
G
*
:=
C
*
k
(
M
,
K
)
in the special case
M
=
[
0
,
1
]
.
We know that
δ
:
G
*
→
C
k
-
1
(
[
0
,
1
]
,
k
)
,
γ
↦
δ
γ
is an isomorphism of Lie groups.
Also, it is known that
L
(
δ
)
∘
ψ
-
1
=
d
δ
|
T
1
G
∘
ψ
-
1
is the map
γ
↦
γ
′
, where
ψ
:=
d
(
φ
*
)
|
T
1
(
G
*
)
is the usual isomorphism between
T
1
G
and the Lie algebra
C
*
k
(
[
0
,
1
]
,
k
)
(see [9]).
Hence
L
(
δ
)
∘
ψ
-
1
:
C
*
k
(
[
0
,
1
]
,
k
)
→
C
k
-
1
(
[
0
,
1
]
,
k
)
,
γ
↦
γ
′
is an isomorphism of topological Lie algebras if the pointwise Lie bracket is used on the left-hand side.
General case: For general 𝑀, we first determine the tangent space
T
0
(
im
(
δ
)
)
to see the Lie algebra of this group.
Let
η
:
I
→
im
(
δ
)
be a
C
k
-curve with
η
(
0
)
=
0
and
β
:=
η
′
(
0
)
.
Then
1
=
per
η
(
t
)
m
0
(
γ
)
=
evol
K
(
γ
*
η
(
t
)
)
for each smooth loop 𝛾 in
m
0
and each
t
∈
I
.
Taking the derivative in
t
=
0
, we get (see [9])
0
=
T
0
(
evol
K
)
(
γ
*
β
)
=
∫
0
1
γ
*
β
=
∫
γ
β
.
Hence all periods of 𝛽 vanish so that 𝛽 is exact.
If, conversely,
β
∈
Ω
C
k
-
1
1
(
M
,
k
)
is an exact 1-form, then
β
=
d
f
for some
f
∈
C
*
k
(
M
,
k
)
.
We show that the curve
α
:
[
0
,
1
]
→
im
(
δ
)
,
t
↦
δ
(
exp
K
∘
(
t
f
)
)
satisfies
α
′
(
0
)
=
β
.
For
x
∈
M
and
v
∈
T
x
M
, choose a smooth path 𝛾 in 𝑀 from
m
0
to 𝑥 such that
γ
′
(
1
)
=
v
.
Then
α
′
(
0
)
(
v
)
=
d
d
t
|
t
=
0
δ
(
exp
K
∘
(
t
f
)
)
(
v
)
=
d
d
t
|
t
=
0
δ
(
exp
K
∘
(
t
f
)
)
(
γ
′
(
1
)
)
=
d
d
t
|
t
=
0
γ
*
(
δ
(
exp
K
∘
t
f
)
)
(
1
)
=
d
d
t
|
t
=
0
δ
[
0
,
1
]
(
exp
K
∘
t
(
f
∘
γ
)
)
(
1
)
=
d
δ
[
0
,
1
]
(
d
d
t
|
t
=
0
(
exp
K
∘
t
(
f
∘
γ
)
)
)
(
1
)
=
d
δ
[
0
,
1
]
(
ψ
-
1
(
f
∘
γ
)
)
(
1
)
=
(
f
∘
γ
)
′
(
1
)
=
d
f
(
γ
′
(
1
)
)
=
d
f
(
v
)
=
β
(
v
)
;
thus
α
′
(
0
)
=
β
.
This shows that
T
0
(
im
(
δ
)
)
=
d
C
*
k
(
M
,
k
)
≅
C
*
k
(
M
,
k
)
as a topological vector space (apply Proposition 4.4 to the Lie group
(
k
,
+
)
)
.
By the preceding, the map
d
:
C
*
k
(
M
,
k
)
→
T
0
(
im
(
δ
)
)
is an isomorphism of topological vector spaces.
We now show that
d
is a homomorphism (hence an isomorphism) of Lie algebras if
C
*
k
(
M
,
k
)
is endowed with the pointwise Lie bracket.
We already know this if
M
=
[
0
,
1
]
.
In the general case, note that the maps
γ
*
:
T
0
(
im
(
δ
)
)
→
C
k
-
1
(
[
0
,
1
]
,
k
)
,
ω
↦
γ
*
(
ω
)
separate points (for 𝛾 ranging through the set of all smooth paths in 𝑀 starting in
m
0
).
Moreover,
γ
*
is a Lie algebra homomorphism, as it is the tangent map at 0 of the analogous smooth group homomorphism
γ
*
:
im
(
δ
)
→
C
k
-
1
(
[
0
,
1
]
,
k
)
.
It therefore suffices to show that
γ
*
∘
d
is a Lie algebra homomorphism for each 𝛾.
But
(
γ
*
∘
d
)
(
f
)
=
γ
*
(
d
f
)
=
d
f
∘
γ
′
=
(
f
∘
γ
)
′
=
(
γ
*
(
f
)
)
′
for
f
∈
C
*
k
(
M
,
k
)
, where
γ
*
:
C
*
k
(
M
,
k
)
→
C
*
k
(
[
0
,
1
]
,
k
)
is a Lie algebra homomorphism and so is
C
*
k
(
[
0
,
1
]
,
k
)
→
C
k
-
1
(
[
0
,
1
]
,
k
)
,
f
↦
f
′
,
by the special case of
[
0
,
1
]
.
Hence
γ
*
∘
d
is a Lie algebra homomorphism.
Consider the map
Ψ
:
C
*
k
(
M
,
k
)
→
T
1
(
G
*
)
,
f
↦
d
d
t
|
t
=
0
(
exp
K
∘
(
t
f
)
)
.
By the chain rule and the preceding, we have
T
1
δ
Ψ
(
f
)
=
T
1
δ
d
d
t
|
t
=
0
(
exp
K
∘
t
f
)
=
d
d
t
|
t
=
0
δ
(
t
↦
exp
K
∘
t
f
)
=
d
f
for
f
∈
C
*
k
(
M
,
k
)
, i.e.,
T
1
(
δ
)
∘
Ψ
=
d
.
Since
T
1
(
δ
)
and
d
are isomorphisms of topological Lie algebras, also Ψ is an isomorphism of topological Lie algebras.
We now show that the maps
L
(
ev
x
)
, for
ev
x
:
C
*
k
(
M
,
K
)
→
K
,
f
↦
f
(
x
)
, separate points on
L
(
G
*
)
.
It suffices to show that the maps
L
(
ev
x
)
∘
Ψ
:
C
*
k
(
M
,
k
)
→
k
separate points on
C
*
k
(
M
,
k
)
.
This follows if we can establish
(4.5)
L
(
ev
x
)
∘
Ψ
=
ε
x
with
ε
x
:
C
*
k
(
M
,
k
)
→
k
,
f
↦
f
(
x
)
.
But indeed, using the chain rule twice,
L
(
ev
x
)
Ψ
(
f
)
=
L
(
ev
x
)
d
d
t
|
t
=
0
(
exp
K
∘
t
f
)
=
d
d
t
|
t
=
0
ev
x
(
exp
K
∘
t
f
)
=
d
d
t
|
t
=
0
exp
K
(
t
f
(
x
)
)
=
T
0
exp
K
f
(
x
)
=
f
(
x
)
=
ε
x
(
f
)
.
Let
i
:
K
→
C
k
(
M
,
K
)
be the map taking
g
∈
K
to the constant function
x
↦
g
.
Then
G
=
C
k
(
M
,
K
)
=
C
*
k
(
M
,
K
)
⋉
i
(
K
)
internally, entailing that
L
(
C
k
(
M
,
K
)
)
=
L
(
C
*
k
(
M
,
K
)
)
⋉
L
(
i
(
K
)
)
internally.
Hence
H
:
C
k
(
M
,
k
)
=
C
*
k
(
M
,
k
)
⋉
k
→
L
(
C
k
(
M
,
K
)
)
,
η
=
γ
+
v
↦
Ψ
(
γ
)
+
L
(
i
)
(
v
)
(for
γ
=
η
-
η
(
m
0
)
∈
C
*
k
(
M
,
k
)
,
v
=
η
(
m
0
)
∈
k
) is an isomorphism of topological vector spaces.
Consider the evaluation maps
ev
x
:
C
k
(
M
,
K
)
→
K
.
Then the maps
L
(
ev
x
)
separate points on
L
(
G
)
.
Indeed, we have
ker
(
L
(
ev
m
0
)
)
=
L
(
C
*
k
(
M
,
K
)
)
because
G
=
G
*
⋊
K
with
ev
m
0
the projection onto 𝐾.
It therefore only remains to check that the maps
L
(
ev
x
)
separate points on
L
(
C
*
k
(
M
,
K
)
)
.
But this has already been checked.
Since each
L
(
ev
x
)
is a Lie algebra homomorphism, 𝐻 will be a Lie algebra homomorphism (hence an isomorphism) if we can show that
L
(
ev
x
)
∘
H
is a Lie algebra homomorphism for each
x
∈
M
.
The restriction of this map to
C
*
k
(
M
,
k
)
is
L
(
ev
x
)
∘
Ψ
, hence a Lie algebra homomorphism.
Moreover, the restriction to the constant functions corresponds to
L
(
ev
x
)
∘
L
(
i
)
on 𝔨, and hence is a Lie algebra homomorphism.
Because
C
k
(
M
,
k
)
=
C
*
k
(
M
,
k
)
⋊
k
, it only remains to show that
(4.6)
L
(
ev
x
)
H
(
[
γ
,
v
]
)
=
[
L
(
ev
x
)
H
(
γ
)
,
L
(
ev
x
)
H
(
v
)
]
for
γ
∈
C
*
k
(
M
,
k
)
,
v
∈
k
.
The left-hand side of (4.6) is (using (4.5))
L
(
ev
x
)
Ψ
(
[
γ
,
v
]
)
=
ε
x
(
[
γ
,
v
]
)
=
[
γ
(
x
)
,
v
]
.
The right-hand side of (4.6) is
[
L
(
ev
x
)
Ψ
(
γ
)
,
L
(
ev
x
)
L
(
i
)
(
v
)
]
=
[
ε
x
(
γ
)
,
L
(
ev
x
∘
i
⏟
=
id
)
(
v
)
]
=
[
γ
(
x
)
,
v
]
as well.
Hence 𝐻 is an isomorphism of topological Lie algebras.
Identifying
C
k
(
M
,
k
)
with
L
(
G
)
via 𝐻, the map
L
(
ev
x
)
corresponds to the point evaluation
δ
x
:
C
k
(
M
,
k
)
→
k
,
f
↦
f
(
x
)
,
i.e.,
(4.7)
L
(
ev
x
)
∘
H
=
δ
x
.
In fact, it suffices to show that both sides of (4.7) coincide on both
C
*
k
(
M
,
k
)
and 𝔨.
For
γ
∈
C
*
k
(
M
,
k
)
, we have
L
(
ev
x
)
H
(
γ
)
=
L
(
ev
x
)
Ψ
(
γ
)
=
γ
(
x
)
indeed.
For
v
∈
k
, we have
L
(
ev
x
)
H
(
v
)
=
L
(
ev
x
)
L
(
i
)
(
v
)
=
v
as well.
Remark 4.6
(1) The restriction maps
ρ
n
:
C
k
(
M
,
K
)
→
C
k
(
L
n
,
K
)
,
γ
↦
γ
|
L
n
are smooth since
ρ
n
=
f
*
as in (4.4) for the inclusion map
L
n
→
M
.
Now a map
f
:
N
→
G
from a manifold 𝑁 to 𝐺 is smooth if and only if
ρ
n
∘
f
:
N
→
C
k
(
L
n
,
K
)
is smooth for each 𝑛.
In fact, assume that
ρ
n
∘
f
is smooth.
Then
N
∋
x
↦
f
(
x
)
(
m
0
)
=
(
ρ
n
∘
f
)
(
x
)
(
m
0
)
is smooth, and after replacing 𝑓 with
x
↦
f
(
x
)
f
(
x
)
-
1
(
m
0
)
, we may assume that
im
(
f
)
⊆
G
*
.
Now
δ
n
∘
ρ
n
∘
f
=
i
n
*
∘
δ
∘
f
is smooth, where
i
n
:
L
n
→
M
is the inclusion map and
i
n
*
:
Ω
C
k
-
1
1
(
M
,
k
)
→
Ω
C
k
-
1
1
(
L
n
,
k
)
.
Since
Ω
C
k
-
1
1
(
M
,
k
)
=
lim
←
Ω
C
k
-
1
1
(
L
n
,
k
)
with the limit maps
i
n
*
, it follows that
δ
∘
f
is smooth as a map to
Ω
C
k
-
1
1
(
M
,
k
)
, hence also smooth as a map to the submanifold
im
(
δ
)
.
Hence
f
=
δ
-
1
∘
(
δ
∘
f
)
is smooth as well.
Consequently,
C
k
(
M
,
g
)
:
C
k
(
M
,
k
)
→
C
k
(
M
,
K
)
,
γ
↦
g
∘
γ
is smooth for each smooth map
g
:
k
→
K
because
C
k
(
L
n
,
g
)
is smooth (cf. [7]) and
ρ
n
∘
C
k
(
M
,
g
)
=
C
k
(
L
n
,
g
)
∘
ρ
n
.
(2) If
l
∈
N
0
∪
{
∞
}
and a map
θ
:
M
→
C
l
(
I
,
K
)
is
C
k
, then
θ
*
:
I
→
C
k
(
M
,
K
)
,
θ
*
(
t
)
(
x
)
=
θ
(
x
)
(
t
)
is
C
l
.
Because the point evaluation
ev
t
:
C
l
(
I
,
K
)
→
K
,
γ
↦
γ
(
t
)
is smooth, we have
θ
*
(
t
)
=
ev
t
∘
θ
∈
C
k
(
M
,
K
)
for each
t
∈
I
.
By (1),
θ
*
will be
C
l
if we can show that
ρ
n
∘
θ
*
is
C
l
for each 𝑛.
But
(
ρ
n
∘
θ
*
)
=
θ
*
(
t
)
|
L
n
=
(
θ
|
L
n
)
*
, where
(
θ
|
L
n
)
*
:
I
→
C
k
(
L
n
,
K
)
is
C
l
, as follows by applying Proposition 3.14 twice.
Regularity. To verify the
C
s
-regularity of 𝐺, we show first that each
γ
∈
C
s
(
I
,
g
)
has an evolution
Evol
G
(
γ
)
.
Identifying
C
s
(
I
,
C
k
(
M
,
k
)
)
with
C
s
(
I
,
g
)
via the isomorphism
C
s
(
I
,
H
)
, we consider 𝛾, as a
C
s
-map
I
→
C
k
(
M
,
k
)
.
Then
γ
*
:
M
→
C
s
(
I
,
k
)
,
γ
*
(
x
)
(
t
)
:=
γ
(
t
)
(
x
)
is
C
k
, using the exponential law [3, Theorem 4.6] twice.
Hence
Evol
K
∘
γ
*
:
M
→
C
s
+
1
(
I
,
K
)
is
C
k
, and therefore
η
:=
(
Evol
K
∘
γ
*
)
*
:
I
→
C
k
(
M
,
K
)
,
η
(
t
)
(
x
)
:=
(
Evol
K
∘
γ
*
)
(
x
)
(
t
)
is
C
s
+
1
(see Remark 4.6 (2)).
We claim that 𝜂 is the evolution of 𝛾.
Indeed,
η
(
0
)
(
x
)
=
Evol
K
(
γ
*
(
x
)
)
(
0
)
=
1
for all
x
∈
M
; whence
η
(
0
)
=
1
.
To see that
δ
η
=
γ
, we only need to show that
(
δ
η
)
(
t
)
(
x
)
=
γ
(
t
)
(
x
)
=
γ
*
(
x
)
(
t
)
for all
x
∈
M
and
t
∈
[
0
,
1
]
, i.e.,
ev
x
∘
δ
η
=
γ
*
(
x
)
.
However, recalling that
L
(
ev
x
)
=
ev
x
,
ev
x
∘
δ
η
=
Lev
x
∘
δ
η
=
δ
(
ev
x
∘
η
)
=
δ
(
t
↦
Evol
K
(
γ
*
(
x
)
)
(
t
)
)
=
δ
(
Evol
K
(
γ
*
(
x
)
)
)
=
γ
*
(
x
)
.
Thus
Evol
G
(
γ
)
=
η
.
In particular,
evol
G
(
γ
)
=
Evol
G
(
γ
)
(
1
)
=
η
(
1
)
is the map
M
→
K
taking 𝑥 to
Evol
K
(
γ
*
(
x
)
)
(
1
)
=
evol
K
(
γ
*
(
x
)
)
.
Thus
evol
G
(
γ
)
=
evol
K
∘
γ
*
,
i.e.,
evol
G
=
C
k
(
M
,
evol
K
)
∘
Φ
,
where
C
k
(
M
,
evol
K
)
is smooth by Remark 4.6 (1) and
Φ
:
C
s
(
I
,
C
k
(
M
,
k
)
)
→
C
k
(
M
,
C
s
(
I
,
k
)
)
,
γ
↦
γ
*
is an isomorphism of topological vector spaces by the exponential law [3, Theorem 4.6].
Thus
evol
G
is smooth, which completes the proof.
∎
Corollary 4.7
If 𝑀 is a paracompact, one-dimensional 1-connected real manifold (with boundary),
k
,
s
∈
N
0
∪
{
∞
}
with
k
≥
s
+
1
and 𝐾 a
C
s
-regular Lie group, then the group
C
*
k
(
M
,
K
)
carries a unique
C
s
-regular Lie group structure for which
δ
:
C
*
k
(
M
,
K
)
→
Ω
C
k
-
1
1
(
M
,
k
)
≅
C
*
k
(
M
,
k
)
is a
C
∞
-diffeomorphism.
Also,
C
k
(
M
,
K
)
≅
C
*
k
(
M
,
K
)
⋊
K
carries the structure of a
C
s
-regular Lie group compatible with evaluations and the compact-open
C
k
-topology.
Proof
We may assume that
M
=
R
,
M
=
[
0
,
1
]
or
M
=
[
0
,
1
[
.
Take
L
n
=
[
-
n
,
n
]
,
L
n
=
[
0
,
1
]
and
L
n
=
[
0
,
1
-
1
n
]
, respectively.
Then
im
(
δ
n
)
=
Ω
C
k
-
1
1
(
L
n
,
k
)
, and Theorem 4.5 applies.
∎
5 Iterative constructions
Lemma 5.1
Let
(
G
n
)
n
∈
N
be a sequence of Lie groups,
ϕ
n
m
:
G
m
→
G
n
morphisms of Lie groups defining an inverse system,
G
:=
lim
←
G
n
the corresponding topological projective limit group and
ϕ
n
:
G
→
G
n
the canonical maps.
Let
r
∈
N
0
∪
{
∞
}
, and assume that 𝐺 carries a Lie group structure with the following properties.
A map
f
:
M
→
G
of a smooth manifold 𝑀 (which may have a rough boundary) with values in 𝐺 is
C
r
if and only if all the maps
f
n
:=
ϕ
n
∘
f
are
C
r
.
L
(
G
)
≅
lim
←
L
(
G
n
)
as topological Lie algebras, with respect to the projective system defined by the morphisms
L
(
ϕ
n
m
)
:
L
(
G
m
)
→
L
(
G
n
)
.
Then the map
Ψ
:
C
r
(
M
,
G
)
→
lim
←
C
r
(
M
,
G
n
)
,
f
↦
(
f
n
)
n
∈
N
is an isomorphism of topological groups.
Proof
First we note that our assumptions imply that
T
G
≅
L
(
G
)
⋊
G
≅
lim
←
(
L
(
G
n
)
⋊
G
n
)
≅
lim
←
T
(
G
n
)
as topological groups.
Moreover, writing
|
L
(
G
)
|
for the topological vector space underlying
L
(
G
)
, considered as an abelian Lie algebra, we have
L
(
T
G
)
≅
|
L
(
G
)
|
⋊
L
(
G
)
≅
lim
←
(
|
L
(
G
n
)
|
⋊
L
(
G
n
)
)
≅
lim
←
L
(
T
G
n
)
so that the Lie group
T
G
is a projective limit and satisfies the analogue of (2).
Hence we may iterate this argument to obtain
T
k
G
≅
lim
←
T
k
G
n
for each 𝑘.
We thus have homeomorphisms
C
(
T
k
M
,
T
k
G
)
c
.
o
.
≅
lim
←
C
(
T
k
M
,
T
k
G
n
)
c
.
o
.
,
which lead to a topological embedding
C
r
(
M
,
G
)
↪
∏
k
∈
N
0
C
(
T
k
M
,
T
k
G
)
c
.
o
.
≅
∏
k
∈
N
0
lim
←
C
(
T
k
M
,
T
k
G
n
)
c
.
o
.
≅
lim
←
∏
k
∈
N
0
C
(
T
k
M
,
T
k
G
n
)
c
.
o
.
,
entailing that Ψ is a topological isomorphism.
∎
For compact manifolds 𝑁 and 𝑀, a Lie group
C
k
,
r
(
N
×
M
,
K
)
can be defined similarly to the classical construction of
C
k
(
N
,
K
)
.
Lemma 5.2
If 𝐾 is a locally convex Lie group and 𝑁 and 𝑀 are compact manifolds (possibly with rough boundary), then the map
Φ
:
C
r
(
N
,
C
k
(
M
,
K
)
)
→
C
r
,
k
(
N
×
M
,
K
)
,
f
↦
f
∧
is an isomorphism of Lie groups.
Proof
The bijectivity of Φ follows from Proposition 3.14.
To see that Φ is an isomorphism of Lie groups, let
(
ϕ
,
U
)
be a 𝔨-chart of 𝐾 with
ϕ
(
1
)
=
0
.
Then
C
k
(
M
,
U
)
is an open identity neighbourhood so that
C
r
(
N
,
C
k
(
M
,
U
)
)
is an open identity neighbourhood and so is
C
r
,
k
(
N
×
M
,
U
)
.
That Φ restricts to a diffeomorphism
C
r
(
N
,
C
k
(
M
,
U
)
)
→
C
r
,
k
(
N
×
M
,
U
)
now follows from [3, Proposition 4.5] which asserts that
C
r
(
N
,
C
k
(
M
,
k
)
)
→
C
r
,
k
(
N
×
M
,
k
)
,
f
↦
f
∧
is an isomorphism of topological vector spaces, hence restricts to diffeomorphisms on open subsets.
∎
A Lie group structure on
C
r
,
k
(
N
×
M
,
K
)
compatible with evaluations is defined analogously to the case of
C
r
(
N
,
K
)
.
Theorem 5.3
Let 𝐾 be a Lie group and 𝑁 and 𝑀 smooth manifolds with rough boundary which are locally compact.
We assume that
G
:=
C
k
(
M
,
K
)
carries a
C
s
-regular Lie group structure compatible with evaluations and the compact-open
C
k
-topology, for some
s
∈
N
0
∪
{
∞
}
.
We assume that 𝑀 admits an exhaustion
M
=
⋃
n
∈
N
M
n
by compact smooth manifolds
M
n
with rough boundary which are well located in 𝑀 such that
M
n
⊆
M
n
+
1
0
for all
n
∈
N
; likewise, we assume an exhaustion
N
=
⋃
n
∈
N
N
n
.
Let
r
∈
N
0
∪
{
∞
}
with
r
-
3
≥
s
.
If
C
r
(
N
,
G
)
carries a
C
s
-regular Lie group structure compatible with evaluations and the compact-open
C
r
-topology, then
C
r
,
k
(
N
×
M
,
K
)
carries a
C
s
-regular Lie group structure compatible with evaluations and the compact-open
C
r
,
k
-topology.
Moreover, the canonical map
Φ
:
C
r
,
k
(
N
×
M
,
K
)
→
C
r
(
N
,
G
)
,
f
↦
f
∨
is an isomorphism of Lie groups.
Proof
In view of Proposition 4.1, the map Φ is a bijective group homomorphism.
First we show that it is an isomorphism of topological groups.
Let
M
=
⋃
n
M
n
be an exhaustion of 𝑀 by compact submanifolds
M
n
with boundary satisfying
M
n
⊆
M
n
+
1
0
.
Then our definition of the group topology implies that
G
=
C
k
(
M
,
K
)
≅
lim
←
C
k
(
M
n
,
K
)
as a topological group.
Put
G
n
:=
C
k
(
M
n
,
K
)
, and recall from Proposition 3.12 that it carries a Lie group structure compatible with evaluations.
We also have the isomorphism of topological Lie algebras
L
(
G
)
=
C
k
(
M
,
k
)
≅
lim
←
L
(
G
n
)
≅
lim
←
C
k
(
M
n
,
k
)
.
Now let
(
N
m
)
m
∈
N
be an exhaustion of 𝑁 by compact submanifolds with boundary.
Then Lemmas 5.1 and 5.2 lead to the following isomorphisms of topological groups:
C
r
(
N
,
G
)
≅
lim
←
C
r
(
N
,
G
n
)
=
lim
←
C
r
(
N
,
C
k
(
M
n
,
K
)
)
≅
lim
←
n
lim
←
m
C
r
(
N
m
,
C
k
(
M
n
,
K
)
)
≅
lim
←
n
lim
←
m
C
r
,
k
(
N
m
×
M
n
,
K
)
≅
C
r
,
k
(
N
×
M
,
K
)
.
The preceding isomorphism leads to a
C
s
-regular Lie group structure on the topological group
C
r
,
k
(
N
×
M
,
K
)
.
To see that this Lie group structure is compatible with evaluations, we first observe that
ev
(
n
,
m
)
=
ev
m
∘
ev
n
∘
Φ
, where
ev
m
∘
ev
n
:
C
r
(
N
,
C
k
(
M
,
K
)
)
→
K
is smooth.
Now
L
(
C
r
,
k
(
N
×
M
,
K
)
)
→
≅
L
(
Φ
)
L
(
C
r
(
N
,
G
)
)
≅
C
r
(
N
,
L
(
G
)
)
≅
C
r
(
N
,
C
k
(
M
,
k
)
)
≅
C
r
,
k
(
N
×
M
,
k
)
.
The map
L
(
ev
n
)
corresponds to
ev
n
:
C
r
(
N
,
L
(
G
)
)
→
L
(
G
)
.
Also, identifying
L
(
G
)
with
C
k
(
M
,
k
)
,
L
(
ev
m
)
corresponds to
ev
m
:
C
k
(
M
,
k
)
→
k
.
Thus we have
L
(
ev
m
∘
ev
n
)
=
ev
m
∘
ev
n
on
C
r
(
N
,
C
k
(
M
,
k
)
)
, which corresponds to
ev
(
n
,
m
)
on
C
r
,
k
(
N
×
M
,
k
)
.
∎
Example 5.4
Let
k
,
r
,
s
∈
N
0
∪
{
∞
}
with
k
≥
s
+
1
and
r
≥
s
+
3
.
If 𝐾 is a
C
s
-regular Lie group, then
C
r
,
k
(
R
×
R
,
K
)
admits a
C
s
-regular Lie group structure compatible with evaluation and the compact-open
C
r
,
k
-topology.
In fact,
G
:=
C
k
(
R
,
K
)
admits a
C
s
-regular Lie group structure compatible with evaluations and the compact-open
C
k
-topology, by Corollary 4.7.
Hence
C
r
(
R
,
G
)
admits a
C
s
-regular Lie group structure compatible with the evaluations and the compact-open
C
r
-topology, by Corollary 4.7.
The assertion now follows from Theorem 5.3.
Example 5.5
Let
k
,
r
,
s
∈
N
0
∪
{
∞
}
with
r
≥
s
+
3
, and let 𝑀 be a compact smooth manifold with rough boundary.
Then
C
r
,
k
(
R
×
M
,
K
)
admits a
C
s
-regular Lie group structure compatible with evaluations and the compact open
C
r
,
k
-topology.
In fact,
C
k
(
M
,
K
)
has a
C
s
-regular Lie group structure compatible with evaluations and the compact-open
C
k
-topology.
By Corollary 4.7, we have that
C
r
(
R
,
C
k
(
M
,
K
)
) admits a
C
s
-regular Lie group structure compatible with evaluations and the compact open
C
r
-topology.
The desired structure on
C
r
,
k
(
R
×
M
,
K
)
≅
C
r
(
R
,
C
k
(
M
,
K
)
)
is provided by Theorem 5.3.
Remark 5.6
Continuing by induction, given a
C
s
-regular Lie group 𝐾 and compact smooth manifold 𝑀 with rough boundary, we expect that a
C
s
-regular Lie group structure can be constructed on
C
α
(
R
n
×
M
,
K
)
for all
n
∈
N
and
α
=
(
α
1
,
…
,
α
n
+
1
)
with
α
1
,
…
,
α
n
sufficiently large.
The following problem remains: Can
C
k
(
R
n
×
M
,
K
)
be made a Lie group if
n
≥
1
and
dim
(
R
n
×
M
)
≥
2
?
