Actions of locally convex Lie groups

In this section we provide the elementary notions of actions of infinite-dimensional Lie groups on locally convex manifolds.

Definition A.3.1 (Smooth actions of Lie groups). If G is a locally convex Lie group and M is a locally convex manifold, then a smooth map λ:G×M →M, (g, s)7→g.s is called a smooth action ofGonM if the map λg:M →M, s7→g.sis a diffeomorphism for eachg∈G andG3g7→λg∈Diff(M) is a homomorphism of abstract groups.

If, moreover,M =H is a locally convex Lie group and eachλgis an element of Aut(H), then we call the action asmooth automorphic action. Furthermore, ifH =Y is a locally convex space and eachλg is an element of GL(Y), then we call the action a smooth linear action. Finally, if Y =kis a locally convex Lie algebra and eachλ_gis an element of Aut(k), then we call the action also asmooth automorphic action.

Remark A.3.2 (Adjoint action). A locally convex Lie groupGacts in a natural way on its Lie algebragby

Ad :G×g→g, Ad(g).x=T c_g(x),

wherec_g:G→Gdenotes the conjugation maph7→g·h·g⁻¹andgis identified withT_eG. This action is in particular smooth and automorphic.

Lemma A.3.3 (Actions need only be smooth on unit neighbourhoods). LetGbe a lo-cally convex Lie group, M be a locally convex manifold and λ:G×M →M be an abstract action, i.e., λ_g∈Diff(M) for all g∈Gand G3g7→λ_g∈Diff(M)is a homomorphism of ab-stract groups. Then λ is smooth if and only if there exists an open unit neighbourhood U ⊆G such that λ|_U×M is smooth.

Proof. For eachg∈G, letU_gbe an open neighbourhood ofgsuch thatg⁻¹·x∈Ufor allx∈U_g. Thenλ(x, v) =λ(g, λ(g⁻¹·x, v)) implies that λ|_U

g×M =λ_g◦λ◦(λ_g−1×id_M) is smooth.

Lemma A.3.4 (Smoothness criterion for automorphic actions). LetG andH be locally convex Lie groups, and λ:G×H→H be an automorphic action of abstract groups., i.e., λg∈Aut(H) for all g∈G and G3g7→λg∈Aut(H) is a homomorphism of abstract groups.

Thenλis smooth if and only if the orbit maps

G3g7→λ(g, h)∈H

are smooth for eachh∈H and there exists an open unit neighbourhoodU ⊆H such that λ|_G×U is smooth.

Proof. For each h∈H, let Uh be an open neighbourhood of g such that h⁻¹·x∈U for all x∈Uh. Thenλ(g, h⁰) =λ(g, h)·λ(g, h⁻¹·h⁰)) implies that λ|_G×U

h is smooth, because λ(g, h) depends smoothly ong for fixedh.

Proposition A.3.5 (Automorphic actions on central extensions of Lie algebras).

Let H be a locally convex Lie group and bgω be a central extension, given by some cocycle ω:g×g→z as in Remark A.2.2. If H×z→z, (h, z)7→h.z is a linear action, H×g→g, (h, x)7→h.xis an automorphic action and R:H×g→zis a map, then

H×z×g→z×g, h.(z, x) = (h.z+R(h, x), h.x) (A.9) defines a smooth automorphic action ofH onz⊕ωgif and only if

h.R(h⁰, x) +R(h, h⁰.x) =R(h·h⁰, x), (A.10) for all h, h⁰∈H and x∈g, the restriction of R to U×(z⊕g) is smooth for some open unit neighbourhood U ⊆H and

ω(h.x, h.x⁰) =h.ω(x, x⁰) +R(h,[x, x⁰]) (A.11) for allh∈H andx, x⁰∈g.

Proof. A direct computation yields that (A.10) is equivalent to the condition that (A.9) defines an abstract action. Then the smoothness of the action follows from Lemma A.3.3 and the assertion follows from Lemma A.2.3.

Remark A.3.6 (Cocycle for group actions). LetGandH be locally convex Lie groups and λ:G×H →H be a smooth action. A crossed homomorphism or 1-cocycle is a smooth map f :G→N with

f(g·h) =f(g)·g.f(h) for all g, h∈H,

which is equivalent to (f,id_G) :G→HoG being a group homomorphism. We note that in view of Lemma A.3.3 this implies, in particular, that for a 1-cocycle, smoothness on an identity neighbourhood is equivalent to global smoothness.

Definition A.3.7 (Derived action). IfGis a locally convex Lie group,Y is a locally convex space andλ:G×Y →Y is a smooth action, then

λ˙ :g×Y →Y, x.y=dλ(e, y)(x,0)

is called the derived action. In the special case of the adjoint action of G on g, we get λ(x, y) = ad(x, y) = [x, y] .˙

Definition A.3.8 (Left logarithmic derivative). IfM is a locally convex manifold with cor-ners, G is a locally convex Lie group and f ∈C^∞(M, G), then the left logarithmic derivative δ^l(f)∈Ω¹(M,g) off is defined to be

δ^l(f).Xm:=T λ_f⁻¹_(m)(T f(Xm)).

Is is simply the pull-back f^∗κG of the Maurer–Cartan form κG:T G→TeG, Xg 7→T λg⁻¹(Xg) toM alongf.

Lemma A.3.9 (Product rule for left logarithmic derivative). If M is a locally convex manifold with corners,Gis a locally convex Lie group andf, g∈C^∞(M, G), then

δ^l(f ·g) =δ^l(g) + Ad(g)⁻¹.δ^l(f) and in particular δ^l(f⁻¹) =−Ad(f).δ^l(f).

Proof. This follows from the definition and an elementary calculation.

Lemma A.3.10 (Product rule for pointwise action). Let M be a smooth locally convex manifold with corners, G be a locally convex Lie group and λ:G×Y →Y be a smooth lin-ear action on the locally convex space Y. If h:M →G and f :M →Y are smooth, then we have

d(λ(h).f).X_m=λ(h).(df.X_m) + ˙λ Ad(h).δ^l(h).X_m

.(λ(h(m)).f(m)) (A.12) withλ(h⁻¹).f :M →E,m7→λ h(m)⁻¹

.f(m). Ifλ= Adis the adjoint action ofGong, then we have

d(Ad(h).f).Xm= Ad(h).(df.Xm) + Ad(h).

δ^l(h).Xm, f(m)

Proof. We write λ(h, f) instead of λ(h).f, interpret it as a function of two variables, suppress the dependence onmand calculate

d(λ(h, f)) (X_m, X_m) =d(λ(h, f)) (0_m, X_m) + (X_m,0_m)

=d2(λ(h, f)) (Xm) +d1(λ(h).f) (Xm)

=λ(h, df(X_m)) +dλ(·, f).T h(X_m)

=λ(h).(df(Xm)) +dλ(·, f).T(λh◦λ_h⁻¹◦λh◦λ_h⁻¹◦h)(Xm)

=λ(h).(df(X_m) +d(λ(·, f)◦λ_G(h)).Ad(h).δ^l(h)(X_m)

=λ(h).(df(Xm)) + ˙λ Ad(h).δ^l(h)(Xm), λ(h(m), f(m)) ,

d2) denotes the differential of λwith respect to the first (respectively second) variable, keeping constant the second (respectively first) variable.

Notions of bundle theory

B.1 Vector- and Principal Bundles

In this section we provide the basic concepts of continuous and smooth vector bundles. In particular, we focus on a description of principal bundles in terms of transition functions (or cocycles), because this is the picture of principal bundles we mostly use.

Throughout the thesis, we always assume that the base spaces of the bundles under consid-eration areconnected.

Definition B.1.1 (Continuous vector bundle). Let X be a topological space and Y be a locally convex space. Acontinuous vector bundle over X with fibreY (or shortly acontinuous vector bundle) is a topological space Etogether with continuous mapξ:E→X such that each fibre E_x:=ξ⁻¹(x) is a locally convex space and that for each point inX there exists an open neighbourhoodU, called atrivialising neighbourhood, and a homeomorphism

Θ :ξ⁻¹(U)→U×Y,

calledlocal trivialisation, such that pr₁◦Θ = ξ|_ξ−1(U)and that Θ|_E

xis an isomorphism of topo-logical vector spaces fromExto{x} ×Y ∼=Y for eachx∈U. We often refer to a vector bundle as a tuple (Y, ξ:E→X) with the calligraphic letterE. IfE andE⁰ are two vector bundles, then amorphism of vector bundles is a continuous mapf :E→E⁰ such thatf(E_ξ(e))⊆E_ξ(f(e)) and

f|_E

ξ(e) is a continuous linear map for eache∈E.

A continuous section of E is a continuous map σ:X →E, which satisfies ξ◦σ= idX

and we denote by Sc(E) the space of all continuous sections. If U ⊆X is a subset, then EU = (Y, ξ|_ξ−1(U):ξ⁻¹(U)→U) denotes the restricted vector bundle overU andSc(EU) is cor-respondingly the space of sections defined onU.

Remark B.1.2 (Transition functions in vector bundles). IfE is a continuous vector bun-dle with fibre Y and U and U⁰ with U∩U⁰6=∅ are two trivialising neighbourhoods, then we have for eachx∈U ∩U⁰an isomorphismϕ_x:= Θ⁰ Θ⁻¹(x,·)

∈GL(Y) induced from the home-omorphism

(U∩U⁰)×Y →(U∩U⁰)×Y (x, v)7→Θ⁰ Θ⁻¹(x, v) .

Since we have in general no nice topology on GL(Y) if Y fails to be a Banach space, it does not make sense to put any requirements on the continuity of the mapx7→ϕx.

Furthermore, if K is a topological group acting continuously on Y (i.e., K acts on Y as an abstract group andK×Y →Y, (k, y)7→k.y is continuous), thenE is avector K-bundle if the local trivialisations can be chosen such that for each pair of trivialising neighbourhoods U and U⁰, there exists a continuous mapping

k_{U U}⁰ :U ∩U⁰→K 104

withϕ_x(y) =k_{U U}⁰(x).y for ally∈Y andx∈U∩U⁰.

Definition B.1.3 (Smooth vector bundle). IfE and M are manifolds with corners, then a continuous vector bundleξ:E→M with fibreY is asmooth vector bundle if all local triviali-sations can be chosen to be diffeomorphisms. IfK is a Lie group acting smoothly onY, then a continuousK-vector bundle ξis a smooth vector K-bundle if thekU U⁰ from Remark B.1.2 can be chosen to be smooth.

Definition B.1.4 (Continuous principal bundle). LetK be a topological group. IfX is a topological space, then acontinuous principalK-bundle overX(or shortly acontinuous principal K-bundle) is a topological space P together with a continuous right action ρ:P×K→P, (p, k)7→p·kand a mapπ:P →Xsuch that for eachx∈X there exists an open neighbourhood U, called atrivialising neighbourhood, such that there exists a homeomorphism

Θ :π⁻¹(U)→U×K, (B.1)

called local trivialisation, satisfying pr₁◦Θ = π|_π−1(Ui) and Θ(p·k) = Θ(p)·k, where K acts on U ×K by right multiplication in the second factor. An arbitrary subset A⊆X is called trivialisingif it has a neighbourhood which is trivialising. We often refer to a continuous principal bundle as a tuple (K, π:P →X) by the calligraphic letterP, where we assume the action ofK on the domain of π to be given implicitly. If confusion with homotopy groups could occur, we denote the bundle projection byη instead ofπ.

Amorphism of continuous principalK-bundlesor acontinuous bundle mapbetween two con-tinuous principalK-bundlesP andP⁰ is a continuous mapf :P →P⁰ satisfyingρ⁰_k◦f =f◦ρk, whereρkandρ⁰_k are the right actions ofk∈KonPandP⁰. Since the above definition implies in particularX ∼=P/K andX⁰∼=P⁰/K, we obtain an induced mapf_X:X ∼=P/K→X⁰∼=P⁰/K given byf_X(p·K) :=f(p)·K. Furthermore, ifX =X⁰, then we callf abundle equivalence if it is an isomorphism andf_X = id_X.

Remark B.1.5 (Sections define local trivialisations). LetP = (K, π:P →M) be a con-tinuous principal bundle. IfU ⊆X is open or closed, then a continuous map σ:U →P with π◦σ= idU is acontinuous section. In particular, ifU ⊆X is a trivialising neighbourhood, then the corresponding trivialisation Θ :π⁻¹(U)→U×K determines a continuous section

σΘ:U →P, σ(x) = Θ⁻¹(x, e).

Conversely, if σ:U →P is a continuous section of π, then this defines a local trivialisation as follows. For eachp∈π⁻¹(U) we can writep=σ(π(p))·kσ(p) for somekσ(p)∈K. This defines a continuous mapk_σ:π⁻¹(U)→K, becausek_σ(p) = pr₂(Θ(σ(p)))⁻¹·pr₂(Θ(p)). We thus have a local trivialisation

Θ_σ:π⁻¹(U)→U×K, p7→(π(p), k_σ(p)).

Since Θ_σΘ= Θ andσ_Θσ =σ, we have a one-to-one correspondence between local trivialisations and continuous local sections ofπ.

Definition B.1.6 (Trivialising system). LetP = (K, π:P→X) be a continuous principal K-bundle. If (Ui)_i∈I is an open cover ofX by trivialising neighbourhoods and (σi:Ui→P)_i∈I is a collection of continuous sections, then the collectionU = (Ui, σi)i∈I is called ancontinuous open trivialising system ofP.

If (Ui)i∈I is a closed cover of X by trivialising sets and (σi:Ui→P)i∈I is a collection of continuous sections, then the collectionU = (U_i, σ_i)_i∈I is called acontinuous closed trivialising system ofP.

If U = (U_i, σ_i)_i∈I and V = (V_j, τ_j)_j∈J are two continuous open trivialising systems of P, then V is a refinement of U if there exists a map J 3j7→i(j)∈I such that V_j⊆U_i(j) and

τ_j =σ_i(j)

Vj, i.e., (V_j)_j∈J is a refinement of (U_i)_i∈I and the sections τ_j are obtained from the sectionσ_i by restrictions.

IfU = (U_i, σ_i)_i∈I is a continuous open trivialising system andV = (V_j, τ_j)_j∈Jis a continuous closed trivialising system, thenV is arefinementofU if there exists a mapJ 3j7→i(j)∈Isuch thatVj ⊆U_i(J) andτj =σ_i(j)

_V

j and vice versa.

Remark B.1.7 (Principal bundles and Cocycles). IfP is a continuous principalK-bundle over X, and U and U⁰ are open trivialising neighbourhoods with U∩U⁰6=∅, then the corre-sponding local trivialisations, given by sectionsσU :U →P andσU⁰ :U⁰→P, define continuous mappingskU U⁰ :U∩U⁰→Kby

kU U⁰(x) =kσ_U(σU⁰(x)) or equivalently σU(x)·kU U⁰(x) =σU⁰(x), (B.2) calledtransition functions. They satisfy thecocycle condition

kU U(x) =eforx∈U and kU U⁰(x)·kU⁰U⁰⁰(x)·kU⁰⁰U(x) =eforx∈U∩U⁰∩U⁰⁰, (B.3) for any third continuous section σ⁰⁰:U⁰⁰→P with open U⁰⁰⊆X. If U = (Ui, σi)i∈I is a con-tinuous open trivialising system, we thus have a collection KP := (kij:Ui∩Uj→K)i,j∈I of continuous functions satisfying (B.3).

On the other hand, if (Ui)i∈I is an open cover of X, then each collection K= (k_ij :U_i∩U_j→K)_i,j∈I of continuous maps satisfying (B.3) is called acontinuous cocycle.

It defines a continuous principalK-bundlePK overX if we set PK=[

i∈I

{i} ×Ui×K/∼ with ((i, x, k)∼(j, x⁰, k⁰)) :⇔(x=x⁰ andkji(x)·k=k⁰).

Then a bundle projection π:P_K→X is given by [i, x, k]7→x, (U_i)_i∈I is a cover by trivialising open sets with local trivialisations given by

Θi: [

x∈Ui

[(i, x, k)]→Ui×K, [(i, x, k)]7→(x, k)

and theK-action is given by ([(i, x, k)], k⁰)7→[(i, x, kk⁰)]. Thus (U_i, τ_i)_i∈I withτ_i(x) = [(i, x, e)]

is a continuous open trivialising system of P_K. Since P_KP is equivalent to P by the equiva-lence [(U, x, k)]7→σU(x)·k, each principal K-bundle may equivalently be described by such a collection of continuous functionsK.

Lemma B.1.8 (Forcing transition functions into open covers). Let X be a compact space,K be topological group and (O`)<sub>`∈L</sub> be an open cover of K. IfP is a continuous principal K-bundle over X, then for each continuous open trivialising system U = (Ui, σi)i=1,...,n there exists a refinement V= (Vs, τs)s=1,...,r such that for each transition function kst:Vs∩Vt→K of V we have kst(Vs∩Vt)⊆O` for some`∈L.

Proof. Letκij:Ui∩Uj→Kbe the transition functions ofU. Furthermore, letV₁⁰, . . . , V_m⁰ be an open cover ofX such that for eachq∈ {1, . . . , m}we haveV⁰q⊆U_i(q)for somei(q)∈ {1, . . . , n}.

By replacingU by the refinement (U_i(q), σ_i(q))q=1,...,m we may thus assumeV⁰i⊆Ui.

For each pair (i, j) with 1≤i, j,≤m, the open cover (O`)<sub>`∈L</sub> pulls back to an open cover (Oe<sub>`</sub><sup>(i,j)</sup>)`∈L of Ui∩Uj, i.e., Oe<sub>`</sub><sup>(i,j)</sup>:=κ<sup>−1</sup><sub>ij</sub> (O`). Then eachx∈V⁰i∩V⁰j has an open neighbour-hood Ux^(i,j) such that Ux^(i,j)⊆V_q⁰ for some q∈ {1, . . . , m} and Ux^(i,j)⊆Oe^(i,j)<sub>`</sub> for some `∈L.

Then

V_(i,j):= (V₁⁰\(V⁰i∩V⁰j), . . . , V_m⁰\(V⁰i∩V⁰j),(U_x^(i,j))_x∈V0 i∩V⁰j) is an open cover ofX and each set of this cover is contained in someV_q⁰.

Now take a common refinementV₁, . . . , V_rof all the open coversV_(i,j)for 1≤i, j≤m. That means, that for each (i, j) and each s∈ {1, . . . , r} we have that V_s is contained in one of the open sets of the cover V_(i,j). Note that this is possible since for each two covers (Q_s)_s∈S and (R_t)_t∈T we have (Q_s∩R_t)_(s,t)∈S×T as a common refinement. SinceX is compact there exists a finite subcoverV1, . . . , Vr of the common refinement of allV_(i,j).

Now for each s∈ {1, . . . , r} we have that V_s is contained in some V_i(s)⁰ for some i(s)∈ {1, . . . , m} and we thus haveV_s∩V_t⊆V_i(s)⁰ ∩V_i(t)⁰ . We claim thatV_s∩V_tis contained in oneU(i(s),i(t))

x ifVs∩Vt6=∅. First, recall thatVsis contained in one of the open sets ofV(i(s),i(t)), and the same holds forV_t. The claim is trivially true ifV_sorV_tare contained in oneU(i(s),i(t))

x , so

assumeV_s⊆V_q⁰\(V⁰_i(s)∩V⁰_i(t)) andV_t⊆V_q⁰0\(V⁰_i(s)∩V⁰_i(t)) for someq, q⁰∈ {1, . . . , m}. Then V_s∩V_t⊆(V_q⁰∩V_q⁰0)\(V⁰i(s)∩V⁰_i(t)) and V_s∩V_t⊆V_i(s)⁰ ∩V_i(t)⁰ ⊆V⁰_i(s)∩V⁰_i(t) implyVs∩Vt=∅ and the claim is established.

We now set τs:= σ_{i(s) V}

s for s∈ {1, . . . , r}. Then V:= (Vs, τs)s=1,...,r is a continuous open trivialising system of P, which is a refinement of U. Denote the transition functions of V by kst:Us∩Ut→K. Since the sections of V are given by restricting the sections of U and the sections determine the transition functions by σ_s·k_st=σ_t, we have k_st=κ_i(s)i(t)

_V

s∩Vt. We have seen before that ifV_s∩V_t6=∅, thenV_s∩V_t⊆U(i(s),i(t))

x for somex∈V⁰_i(s)∩V⁰_i(t). Since U(i(s),i(t))

x ⊆Oe(i(s),i(t))

` for some `∈Lwe thus have k_st(V_s∩V_t)⊆k_st(Oe(i(s),i(t))

` ) =κ_i(s)i(t)(Oe(i(s),i(t))

` )⊆O<sub>`</sub>.

Remark B.1.9 (Equivalences of principal bundles and cocycles). Let K be a topologi-cal group. If X is a topological space and (U_i)_i∈I is an open cover of a X, then a collection K= (k_ij :U_i∩U_j→K)_i,j∈I of continuous maps satisfying (B.3) is called a K-valued cocycle on X. Two such cocycles K and K⁰ are said to be equivalent if there exists a common refine-ment (Vj)_i∈J of their open covers together with two functions f :J →I and f⁰:J →I⁰ such thatVj ⊆U_f(j)andVj ⊆U_f⁰0(j)for allj∈Jand a collectionG= (gj :Vj→K)_j∈Jof continuous functions satisfying

g⁻¹_j (x)·k_f(j)f(j0)(x)·g_j⁰(x) =k⁰_f0(j)f⁰(j⁰)(x)

for all x∈Vj∩Vj⁰. If P_K and P_K⁰ are the associated principal K-bundles over X, then this defines a continuous bundle equivalencegG betweenPK andPK⁰ by setting

g_G :P_K→P_K⁰, [(f(j), x, k)]7→[(f⁰(j), x, gj(x)·k)].

Conversely, ifP_KandP_K⁰ are two principalK-bundles overX, given by two cocyclesKand K⁰, then there exists an open cover (V_i)_i∈I which is a common refinement of the open covers (Ui)_i∈I and (U_i⁰0)i⁰∈I⁰ underlyingK andK⁰. In fact,

(U_i∩U_i⁰0)_(i,i0)∈I×I⁰

is such a cover and, we assign to it the functionsf = pr₁andf⁰= pr₂. Then a bundle equivalence g:PK→PK⁰ defines for each (i, i⁰)∈I×I⁰ a continuous map

g⁰_(i,i0):Ui∩U_i⁰0×K→Kbyg([(i, x, k)]) = [(i⁰, x, g⁰_(i,i0)(x, k))]. (B.4) Sine g is assumed to satisfy g(p·k) =g(p)·k, we have g⁰_(i,i0)(x, k) =g_(i,i⁰ 0)(x, e)·k. If we set g_(i,i0)(x) :=g⁰_(i,i0)(x, e), we obtain a collection of continuous maps

G_g:= (g_(i,i0):U_i∩U_i⁰0 →K)_(i,i0)∈I×I⁰

satisfying

kj⁰i⁰(x)·g_(i,i0)(x) =g_(j,j0)(x)·kji(x) for allx∈Ui∩Ui⁰∩Uj∩Uj⁰, (B.5) because [(i, x, k)] = [(j, x, kji(x)k)] has to be mapped to the same element of P_K by g. Since G=GgG and g=gG_g and since each principal K-bundle may equivalently be described by a cocycle, the set of equivalence classes of principalK-bundles overX is parametrised by

Bun(X, K) ={K:K is aK-valued cocycle onX}/∼, where∼is the equivalence of cocycles described above.

Definition B.1.10 (Smooth principal bundle). LetKbe a locally convex Lie group andM be a manifold with corners. A continuous principalK-bundle overM is called asmooth principal K-bundle over M if P is a manifold with corners and the local trivialisations from (B.1) can be chosen to be diffeomorphisms. A morphism of smooth principal bundles is a morphism of continuous bundles that is also smooth.

Remark B.1.11 (Continuous vs. smooth principal bundles). All the remarks on the equivalent description of sections and local trivialisations, principal bundles and cocycles and bundle equivalences remain valid in exactly the same way if one only substitutes the assump-tions of being continuous with those of being smooth. In particular, we have the same noassump-tions of trivialising subsets, smooth bundle equivalences and smooth sections defining smooth lo-cal trivialisations. Smooth lolo-cal sections in turn define smooth transition functions , cocycles kij:Ui∩Uj→K and bundle equivalences are defined by smooth mappingsgj :VJ→K.

Furthermore, if P is a smooth principal K-bundle over M, then asmooth open trivialising system U ofP consists of an open cover (Ui)i∈I and smooth sectionsσi :Ui→P. If eachUi is also a manifold with corners and the sectionσican be extended to smooth sectionsσi:Ui→P, thenU = (U_i, σ_i)_i∈I is called asmooth closed trivialising system ofP. In this case,U is called the trivialising systemunderlying U.

Remark B.1.12 (Smooth Structure on Smooth Principal Bundles). Let K be a Lie group and P be a continuous principal K-bundle over the manifold with corners M. If there exists a trivialising cover (Ui)_i∈I and trivialisations Θi:π⁻¹(Ui)→Ui×K such that the corre-sponding transition functionskij :Ui∩Uj→K are smooth, then we define onP the structure of a manifold with corners by requiring the local trivialisations

Θi :π⁻¹(Ui)→Ui×K

to be diffeomorphisms. This actually defines a smooth structure onP, because it is covered by (π⁻¹(Ui))_i∈I and since the coordinate changes

(Ui∩Uj)×K→(Ui∩Uj)×K, (x, k)7→Θj(Θ⁻¹_i (x, k)) = (x, kij(x)·k) are smooth.

Lemma B.1.13 (Existence of smooth trivialising systems). If P = (K, π:P→M) is a smoothK-principal bundle with finite-dimensional baseM, then there exists an open cover(Vi)_i∈I such that eachVi is trivialising and a manifold with corners. In particular, there exists a smooth closed trivialising system V= (Vi, σi)_i∈I, where σi is the restriction of some smooth section, defined on an open neighbourhood of Vi. If, moreover, M is compact then we may assume I to be finite.

Proof. For each m∈M there exists an open neighbourhood U and a chart ϕ:U →(Rⁿ)⁺ such that U is trivialising, i.e. there exists a smooth section σ:U →P. Then there exists an ε >0 such that (Rⁿ)⁺∩(ϕ(m) + [−ε, ε]ⁿ)⊆ϕ(U) is a manifold with corners and we set V_m:=ϕ⁻¹((Rⁿ)⁺∩(ϕ(m) + (−ε, ε)ⁿ)). Then (V_m)_m∈M has the desired properties and ifM is compact it has a finite subcover.

Definition B.1.14 (Associated bundles). Let P be a smooth principal K-bundle and λ:K×N →N be a smooth left action ofK on some smooth locally convex manifoldN. Then we define theassociated bundle λ(P) :=P ×_λN to consist of the topological space

(P×N)/K,

whereKacts onP×N from the right by (p, n)·k:= (p·k, λ(k⁻¹, n)) and the bundle projection π_λ(P):P×N →M, [p, n]7→π_P(p),

whereπ_P :P →M is the bundle projection ofP.

Remark B.1.15 (Local trivialisations in associated bundles). If λ(P) =P ×λN is an associated bundle, then it is in particular a locally trivial K-bundle overM with fibreN, i.e., we have for each m∈M an open neighbourhood U, called trivialising neighbourhood and a diffeomorphism

Θ :π_λ(P)⁻¹ (U)→U ×N

such that for two trivialising neighbourhoodsU and U⁰ with local trivialisations Θ and Θ⁰ we have

Θ⁰(Θ⁻¹(x, n)) =kU⁰U(x)⁻¹.n (B.6) for x∈U∩U⁰ and some smooth function kU U⁰ :U∩U⁰→K. In fact, if πP :P →M is the bundle projection ofP,U is a trivialising neighbourhood forP andσ:U → P a smooth section ofπ_P, then

π⁻¹_λ(P)(U) = (U×N)/K→U×N, (p, n)7→(π_P(p), k_σ(p).n)

defines such a diffeomorphism with inverse (x, n)7→[(σ(x), n)]. Furthermore, two such trivialis-ing neighbourhoods define by (B.2) a smooth mapkU U⁰ :U∩U⁰→Ksuch that (B.6) holds.

Im Dokument Infinite-Dimensional Lie Theory for Gauge Groups (Seite 107-115)