of Fr´echet manifolds
Mansour Aghasi and Ali Suri
Abstract.In this paper for a vector bundle (v.b.) (E, p, M), we show that there are two splitting theorems forT E at the presence of a connection. In spite of natural difficulties with non-Banach modeled v.b.’s, we gener-alize these theorems for a wide class of Fr´echet v.b.’s i.e. those which can be considered as projective limits of Banach v.b.’s. Afterward using the concept of parallelism we propose an alternative way of studying ordinary differential equations on Banach v.b.’s as well as a suitable basis for fur-ther steps. Notwithstanding of the lack of a general solvability-uniqueness theorem for differential equations on non-Banach modeled v.b.’s, we will prove an interesting result for the category of our discussing v.b.’s. For the case of the tangent bundle of a projective system of manifolds as a corollary we observe that according to [1], the connection may be replaced with an equivalence structure like a dissection, Christoffel structure, spray or a Hessian Structure. These splitting theorems have applications in the study of the geometry of bundle of accelerations.
M.S.C. 2010: 58B25, 58A05.
Key words: Banach vector bundles; Fr´echet vector bundles; tensorial splitting; Dom-browski splitting; connection; parallelism; projective limit; ordinary differential equa-tions.
1
Introduction
The study of Fr´echet manifolds was the subject of investigation for many authors due to interaction of this field with modern differential geometry and theoretical physics. (see for example [2], [3], [7], [12], [16], [17], and [20].) In the study of Mathemat-ical Physics one often encounters with manifolds and spaces which are obtained as projective limits. Some of these fields are the loop quantization of Gauge theories as quantum gravity, the 2D Yang-Mills theory [4], [5], and string theory [21]. Moreover projective systems of manifolds arise in other areas of differential geometry like the
∗
Balkan Journal of Geometry and Its Applications, Vol.15, No.2, 2010, pp. 1-13 (electronic version); pp. 1-13 (printed version).
c
study of group of diffeomorphisms and the geometry of infinite jets ([18], [23], [22], [25]).
At the second section we introduce our notation for v.b.’s and manifolds. All the maps, for the sake of simplicity, are assumed to be smooth but less degrees of differentiability may be assumed. In section 3 we state two main theorems for a Banach v.b. π:E−→M which propose two ways of slitting ofT Eat the presence of a connection. We used the nominating of tensorial splitting and Dombrowski splitting due to the fact that in the case ofE=T Msimilar theorems are proved in [11] and [15]. Afterward the notion of parallelism on v.b.’s is introduced and the local equations are determined. In fact we propose an alternative way of studying ordinary differential equations on Banach v.b.’s which serves as a suitable basis for the further steps i.e. ordinary differential equations on non-Banach v.b.’s.
Section 4 is devoted to the study of non-Banach v.b.’s and manifolds. There are two main problems with these geometric objects. The first is the pathological structure of GL(F) i.e. the general linear group of a Fr´echet space. In fact if F is
non-Banach,GL(F) does not admit a reasonable topological group structure (see [12].
[14]). As we know when we are engaged with v.b.’s with fibres of typeF, the transition
functions take their values in GL(F) and we fail to have the important geometric
objects like v.b’s, tangent bundles and frame bundles [10]. The other obstacle is the lack of a general solvability-uniqueness theorem for ordinary differential equations on non-Banach manifolds and even spaces. As we will see, both of these overcome if we restrict ourselves to the category of those Fr´echet v.b.’s which may be considered as projective limits of Banach v.b.’s.
As a corollary we prove that for the second order tangent bundle, T T M of a projective limit manifold M there are two different splittings at the presence of a linear connection. Afterward using [1] we prove that the linear connection can be substituted with a Christoffel structure, spray, dissection or a Hessian structure. (In [15] these splitting theorems are proved for Banach manifolds and at the presence of a spray.) Finally flat connection on the Banach and Fr´echet model spaces, and the canonical direct connection on Banach and Fr´echet Lie groups are used to construct our introduced splittings on their double tangent bundles.
2
Preliminaries
Letp:E −→M be a smooth Banach vector bundle with fibres of typeE andB as
the model space of M (both Banach spaces). According to [27] a connection on a smooth vector bundlep:E−→M is a smooth splitting of the short exact sequence of bundles onE;
(2.1) 0−→V E−→i T E−→p˜ p∗
T M −→0
whereV E is the vertical sub-bundle ofE,iis the inclusion map,p∗
(T M)⊂E×T M
is the pullback bundle induced by the projectionp:E−→M and ˜pis defined by the tangent mapT p:T E−→T M.
SinceV E is isomorphic top∗
The connection mapD is fibre preserving for both vector bundle structures ofT E, i.e. πE : T E −→E and T P :T E −→ T M, and fibre linear for the first mentioned
structure. IfDis also fibre linear for the second indicated structure, thenDis called a linear connection onE.
Let Φ = (φ,φ¯) :E|U −→φ(U)×Ebe a local trivialization where (U,Φ) is a chart
ofM and assume that Ψ = ((ψ,ψ¯), V) is another local trivialization withU∩V 6=¡❢. Then Ψ◦Φ−1(x, v) = ((ψ◦φ−1)(x), G
ΨΦ(x)v) where GΨΦ :U ∩V −→ GL(E,E) is
smooth and for anyx∈ U∩V, GΨΦ(x) is a toplinear isomorphism. The canonical
induced trivialization forT E takes the form
T(Ψ◦Φ−1)(x, ξ, y, η) = ¡
(ψ◦φ−1)(x), G
ΨΦ(x)ξ, T(ψ◦φ−1)(x)y,
(2.2)
GΨΦ(x)η+T GΨΦ(x)(y, ξ)¢.
for (x, ξ, y, η)∈φ(U)×E×B×E. By these means on (Φ, U),D has the form
DΦ(x, ξ, y, η) = (x, η+ωΦ(x, y)ξ)
whereDΦ= Φ◦D◦TΨ−1andωΦ:φ(U)×E−→L(B,E) is smooth and is called the
local form of the connectionD.
IfD is a linear connection on π : E −→ M, then for the local forms we have ωΦ :
φ(U) :−→L(E, L(B;E)). Using (2.2) and the factDΨ◦T(Ψ◦Φ−1) = (Ψ◦Φ−1)◦D Φ
we see that
DΨ◦T(Ψ◦Φ−1)(x, ξ, y, η) = ¡(ψ◦φ−1)(x), GΨΦ(x)η+T GΨΦ(x)(y, ξ)
+ ωΨ((ψ◦φ−1)(x))[T(ψ◦φ−1)(x)y, GΨΦ(x)ξ]¢.
On the other hand,
(Ψ◦Φ−1)◦D
Φ(x, ξ, y, η) =¡(ψ◦φ−1)(x), GΨΦ(η+ωΦ(x)[y, ξ])¢.
Comparing the second components of the above equalities for different charts ofM
yields the following well known local compatibility condition (2.3)
GΨΦ(ωΦ(x)[y, ξ]) =T GΨΦ(x)(y, ξ) +ωΨ((ψ◦φ−1)(x))[T(ψ◦φ−1)(x)y, GΨΦ(x)ξ].
For the manifold letπM :T M −→M be its tangent bundle andT M =Sx∈MTxM
whereTxM consists of all equivalent classes of the form [c, x] such that
c∈Cx={c: (−ǫ, ǫ)−→M; ǫ >0, csmooth andc(0) =x},
with the equivalence relation c1 ∼x c2 iff c′1(0) = c′2(0) for c1, c2 ∈ Cx. If A =
{(Uα, φα);α∈I}is an atlas forM, then consider the atlasesB={(π
−1
M (Uα),Φα); α∈
I} andC={(π−1
T M(π
−1(U
α)),Φ˜α); α∈I} forT M andT(T M) respectively.
A connection on a manifold M is a connection on its tangent bundle i.e. a v.b. morphismD:T(T M)−→T M with the local representation;
Dα:ψα(Uα)×E×E×E −→ ψα(Uα)×E
whereDα=ϕα◦ϕ˜−α1 and ωα :ψα(Uα)×E−→ L(E,E),α∈I, are the local forms
of D. The connection D is linear if {ωα}α∈I are linear with respect to the second
variable. Moreover we can fully determine a linear connection by the family of its Christoffel symbols defined by;
Γα:ψα(Uα)−→ L(E,L(E,E)); α∈I.
Using (2.3) we see that the compatibility condition for Christoffel symbols are
(2.4) σ′
αβ(x)(Γβ(x)[v, z]) = Γα(σαβ(x))[σαβ′ (x)(v), σ
′
αβ(x)(z)] +σ
′′
αβ(x)(v, z),
where (x, v, z)∈ψα(Uα)×E×E.
3
Splitting theorems for Banach manifolds
Suppose thatp:E−→M is a Banach vector bundle and (U,Φ) a local trivialization on it i.e. E|U ≃φ(U)×E. Then there is a canonical diffeomorphism betweenT E|U
andφ(U)×E×B×E. Now define the map (κ1, κ2) :T E−→p∗
T M⊕p∗
E locally by
κ1Φ : (φ(U)×E)×B×E −→ (φ(U)×E)×B
(x, ξ, y, η) 7−→ (x, ξ, y)
and
κ2Φ: (φ(U)×E)×B×E −→ (φ(U)×E)×E
(x, ξ, y, η) 7−→ (x, ξ, η+ωΦ(x)[y, ξ]).
Theorem 3.1. Let p:E −→M be a Banach v.b. and D a linear connection on it. Then(κ1, κ2) :T E−→π∗T M⊕π∗E (Whitney sum) defines a v.b.-isomorphism.
Proof. Clearlyκis a vector bundle morphism. Moreover (κ1, κ2) is smooth and linear
on fibres and locally is a bijection. Let (U,Φ) and (V,Ψ) be local trivializations ofp
withU∩V 6=∅. Then
(κ1Ψ, κ2Ψ)◦T(Ψ◦Φ−1)(x, ξ, y, η)
= (κ1Ψ, κ2Ψ)
³
(ψ◦φ−1)(x), G
ΨΦ(x)ξ, T(ψ◦φ−1)(x)y
, GΨΦ(x)η+T GΨΦ(x)(y, ξ)
´
= ³(ψ◦φ−1)(x), G
ΨΦ(x)ξ, T(ψ◦φ−1)(x)y
´
⊕³(ψ◦φ−1)(x), G ΨΦ(x)ξ
, GΨΦ(x)η+T GΨΦ(x)(y, ξ) +ωΨ((ψ◦φ−1)(x))[T(ψ◦φ−1)(x)y, GΨΦ(x)ξ]
´
∗
= ³(ψ◦φ−1)(
x), GΨΦ(x)ξ, T(ψ◦φ−1)(x)y
´
⊕³(ψ◦φ−1)(
x), GΨΦ(x)ξ
, GΨΦ(x)η+GΨΦ(ωΦ(x)[y, ξ])
´
= (Ψ◦Φ−1)◦(κ
1Φ, κ2Φ)(x, ξ, y, η)
The next useful theorem proposes another splitting for T E. Note that in the case of E = T M this theorem splits T T M to three copies of T M. In fact this is a generalization of Dombrowski’s splitting theorem [11] for Banach vector bundles. In [11] the theorem is proved for E = T M with M finite dimensional and this is developed by Lang for Banach manifolds [15].
Theorem 3.2. Let p:E −→M be a Banach vector bundle with a connectionD on it. Then the map
(πE, T p, κ2) :T E−→E⊕T M⊕E
locally defined by(x, ξ, y, η)7−→(x, ξ)⊕(x, y)⊕(x, η+ω(x)(y, ξ)), is a fibre bundle isomorphism.
Proof. It can be easily checked that the map is a fibre bundle morphism and a local bijection. Moreover the compatibility condition ,similar to the previous theorem, yields that these local isomorphisms are compatible and consequently the bundles are
isomorphic. ¤
3.1
Parallelism and ordinary differential equations on Banach
vector bundles
If A is a section of π : E −→ M and X a vector field on M, then the covariant derivativeDXAis defined to beD◦T A◦X. Letα: (−ǫ, ǫ)−→M. Denote the basic
section ofTRby∂:t7−→(t,1) and putα′(
t) =T α(∂(t)).
For a curvecin E we say thatcis parallel alongαifπ◦c=αandD◦c′= 0. If (U,Φ) is a local trivialization forpthenD◦c′= 0 locally means that
DU¡(φ◦α)(t),( ¯φ◦c)(t),(φ◦α)′(t),( ¯φ◦c)′(t) =
¡
(φ◦α)(t),( ¯φ◦c)′
(t) +ωφ((φ◦α)(t))[(φ◦α)′(t),( ¯φ◦c)(t)] = 0
(3.1)
wherec(t) is considered as c(t) = (α(t),c(t)).
In the case ofE=T M where we consider the canonical lift of the curveα, the above equation takes the familiar form
(φ◦α)′′
(t) +ωφ((φ◦α)(t))[(φ◦α)′(t),(φ◦α)′(t)] = 0
which is the autoparallel (geodesic whenD is a metric connection) equation onM. Here we state the following theorem which is a direct consequence of the existence and uniqueness theorem for solution of ordinary differential equations on Banach manifolds.
Theorem 3.3. Let p:E −→M be a Banach vector bundle and D be a connection on it. Then for a curveα inM andξ∈Eα(0) there exists a unique parallel curve c
(inE) alongαwithc(0) =ξ.
4
The Fr´
echet case
that{(Mi, ϕji)}i,j∈Nis a projective system of Banach manifolds with the limitM =
lim
←−Mi such thatMi is modeled on the Banach spaceBiand{Bi, ρji}i∈Nalso form a
projective system of Banach spaces.
Furthermore let for eachx= (x)i∈N∈M there exists a projective system of local
charts{(Ui, ψi)}i∈Nsuch thatxi∈Ui andU = lim
←−Ui is open inM.
The vector bundle structure ofT M for a Fr´echet manifoldM links to pathological structure of general linear groupGL(F) and this causes troubles. It is shown in [12]
that by considering the generalized topological Lie group
H0(F) ={(li)i∈N∈
Y
i∈I
GL(Ei) : lim
←−l
i
exists}
rather thanGL(F) this obstacle can be solved. Moreover as we will see in the rest
of this paper, the problems related to the lack of a general solvability for differential equations on Fr´echet manifolds will overcome with the introduced technique.
Suppose that for anyi∈N, (Ei, pi, Mi) is a Banach v.b. onMi with the fibres of
typeEi where {Ei, λji}i,j∈N forms also a projective system of Banach spaces. With
these notations we state the following definition;
Definition 4.1. The system {(Ei, fji)}i∈N is called a strong projective system of
Banach v.b.’s on{(Mi, ϕji)}i∈Nif for any (xi)i∈N, there exists a projective system of
trivializations (Ui, τi) (whereτi:pi−1(Ui)−→Ui×Ei are local diffeomorphisms) of
(Ei, pi, Mi), such thatxi∈Mi,U = lim←−Uiis open inM and (ϕji×λji)◦τj=τi◦fji
for alli, j∈Nwithj≥i.
Remark 4.2. If {Mi, ϕji}
i∈N is a projective system of Banach manifolds, then
{T Mi,}i∈N,{T T Mi}i∈Nand{T2Mi}i∈Nform strong systems of Banach v.b.’s. (see
[9] and [12]). Hence we have concrete examples for our definition.
Remark 4.3. Galanis in [14] defined the strong system of Banach v.b.’s in a same way. The difference is in the base manifolds. More precisely in [14] the base manifold in all v.b.’s is a fixed Banach manifoldM. Note that in the case that one needs to study the natural bundles over a projective system of manifolds, as we will see, the base manifold is not fixed and hence we stated our definition with a slight modification. A similar theorem is proved in [14] but with a fixed base manifold.
Proposition 4.4. Let{(Ei, pi, Mi)}
i∈Nbe a strong projective system of Banach vec-tor bundles. Then(lim←−Ei,lim
←−pi,lim←−Mi)is a Fr´echet vector bundle.
Proof. We claim thatE = lim←−Eiis a differentiable manifold modeled on lim←−Bi×lim
←−Ei
where{(Bi, ρji)}i∈N is the projective system of model spaces of the base manifolds.
Lete= (ei)i∈N∈E thenx= (xi=pi(ei))i∈Nbelongs toM = lim
←−Mi since forj≥i;
ϕji(xj) =ϕji◦pj(ej) =pi◦fji(ej) =pi(ei) =xi.
Since the projective system is strong then forx∈M there exists a projective system of trivializations{(Ui, τi)}i∈Nsuch thatxi∈Ui andϕji×λji◦τj=τi◦fjiforj≥i.
Define the projective system of charts{(Vi,Φi)}
i∈Nfor{Ei}i∈Nto beVi =pi
−1
(Ui)
It may be checked that these charts form a projective system with the limit (V = lim
←−Vi,Φ = lim←−Φi).
The equalityϕji◦pj=pi◦fji yields thatp= lim←−pi can be defined andplocally is the projection on the first component i.e. pr1◦Φ =p. Also the last equality yields that for anyx∈M, Φ|p−1(x)= lim
←−Φ|pi−1(xi)exists and is a linear isomorphism. Due
to the pathological structure ofGL(E) we restrict ourselves to an appropriate subset
ofGL(E) say it H0(E). More preciselyH0(E) is the projective limit of Banach Lie
and consequentlyH0(E) becomes a generalized Fr´echet Lie group [13].
To see the differentiability of transition functions consider the projective limit trivializations (U, τ) and (V,¯τ) of p: E −→ E with x ∈U ∩V. Then we see that
Now we state a slight modified version of Proposition 2.1 of [14] to introduce projective limit of connections.
Proposition 4.5. Let (lim←−Ei = E,lim
←−pi = p,lim←−Mi = M) be an strong P.L.B.
v.b.’s. If {Di}
i∈N is a projective system of connections (possibly nonlinear) on this system, thenlim←−Di is a connection on p:E −→M with the local component {ω
U}
given by ωU(x) = lim←−ωiUi(xi) where x = (xi)i∈N ∈ U = lim
←−Ui and ωUii is the corresponding local components ofDi for anyi∈N.
Proof. The proof follows from the previous theorem and section 2 of [14]. ¤
Lemma 4.6.Let{(Ei, pi, Mi)}i∈Nbe a strong system of Banach v.b.’s, then{pi
∗
T Mi⊕
pi∗Ei} also forms a strong system of Banach v.b.’s with the limit isomorphic to p∗
Ej. (We will use these notations during
the proof alternatively.) Clearly gik◦gji =gjk for j ≥ i ≥ k, and this ensures us
that {pi∗
Ei, gji}
trivializations for the system{(Ei, pi, Mi)}i∈N. Then the corresponding systems for the trivializations onpi:Ei−→Mi with domains contained in domains of charts of
Mi and the corresponding natural induced maps.)
Lete∈E. As it is shown in prop. 4.4, (xi) = (pi(ei))∈M and consequently there
(strong) system of v.b.’s with the limitT M isomorphic to lim←−T Mi. Using the first part of this lemma with a modification (as stated above),{(pi∗T Mi, pi∗ , Ei)}i∈Nis
also a strong system of v.b.’s with the limit (p∗
T M, p∗(
πM), E).
After a direct calculation and just writing the steps of the proof forp∗
Eandp∗
T M
together, we see that{pi∗Ei⊕pi∗T Mi}i∈N also satisfies the conditions of prop 4.4
and hence the proof is complete. ¤
To be more familiar with projective limit manifolds techniques, we state a direct proof for the case ofE=T M in appendix.
Theorem 4.7. Under the assumptions as in the previous lemma, let {Di}
i∈N be a projective system of linear connections with the limit D = lim←−Di. Then the map
(κ1, κ2) = lim←−(κi1, κi2) :T E−→p
Since (κ1, κ2) is the projective limit of diffeomorphisms, it is a generalized
One can consider the previous assumptions and prove the following theorem for the case of projective limit v.b.’s.
Theorem 4.8. The projective limit map (πE, T p, κ2) :T E −→E⊕T M ⊕E exists
and is a fibre bundle diffeomorphism.
4.1
Double tangent bundles
Let {Mi}
i∈N be a projective system of manifolds with the limit M = lim
←−Mi and
D= lim←−Di be a linear connection on it. Then we have the following generalizations
of tensorial and Dombrowski’s splitting theorems for Fr´echet manifolds.
Theorem 4.9. Two morphisms (κ1, κ2) : T T M −→ πM∗ T M ⊕ π
∗
MT M and
(πT M, T πM, κ2) : T T M −→ T M ⊕T M ⊕T M exist and are v.b. and fibre bundle
diffeomorphisms respectively.
Remark 4.10.According to [1], the linear connection can be replaced with a Christof-fel structure, spray, Hessian structure or a dissection. Note that these structures must be locally associated with bilinear symmetric maps. (For a detailed study in the case of second order structures on Fr´echet manifolds see [1]).
4.2
Parallelism and ordinary differential equations on Fr´
echet
vector bundles
Let α = lim←−αi be a curve in M such that for any i ∈ N, αi : (−ǫ, ǫ) :−→ Mi is
a smooth curve. In spite of the lack of a general solvability-uniqueness theorem for non-Banach manifolds and even spaces we have the following Theorem.
Theorem 4.11. Let (Ei, pi, Mi)i∈N be a strong system of Banach v.b.’s and D =
lim
←−Di be a projective limit connection on it. Then for the curve α in M and ξ =
(ξi)i∈N ∈ Eα(0) there exists a unique curve c along α in E which is parallel with respect toD andc(0) =ξ.
Proof. For any i∈N,Di is a connection on pi :Ei−→Mi andαi is a curve inMi.
According to 3.3, forξi ∈Eαi(0) there exists a unique curveci in Ei withci(0) =ξi
and
(4.1) ( ¯φi◦ci)′
(t) +ωUii((φi◦αi(t)))[(φi◦αi)
′
(t),( ¯φi◦ci)(t)] = 0
for any local trivialization (Ui,Φi). For j ≥ i, we claim that fji◦cj = ci and
consequentlyc= lim←−ci exists as a curve inE.
( ¯φi◦fji◦cj)′
(t) +ωiUi((φ
i◦
αi)(t))[(φi◦αi)′
(t),( ¯φi◦fji◦cj)(t)] = (λji◦φ¯j◦cj)′
(t) +ωiUi((φi◦αi)(t))[(φi◦αi)
′
(t),(λji◦φ¯j◦cj)(t)] =λji◦( ¯φj◦cj)′
(t) +ωiUi((φ
i◦
αi)(t))[(φi◦αi)′
(t), λji◦( ¯φj◦cj)(t)] =λji◦¡
( ¯φj◦cj)′
(t) +ωUjj((φ
j◦αj)(t))[(φj◦αj)′
(t),( ¯φj◦cj)(t)]¢
= 0.
Furthermorefji◦cj(0) =fji(ξj) =ξi. Using 3.3 and the fact that forαithere exists
a unique curveci withci(0) =ξi, we conclude thatfji◦cj =ci. Hence{ci}
a projective system of curves andc= lim←−ci is a curve in E = lim←−Ei. On the other handp◦c= (pi◦ci)i∈N= (α
i)
i∈N=αi.e. c is parallel alongα.
Letc1 be another curve inE alongαwithD◦c′1= 0 andc1(0) =ξ. For anyi∈N,
ci1 = λi◦c1 satisfies the equation (4.1) and ci1(0) = ξi.(λi : lim←−Ei −→ Ei is the
canonical projection.) 3.3 yields thatci1=c1 and consequentlyc1= lim←−ci1= lim←−ci=
c. Note that the life-time of the solution maybe trivial i.e. just the origin. To avoid this we have to assume some Lipschitz condition for the local components as it is
stated in appendix of [1]. ¤
Example 4.12. Splitting of the second order tangent bundle of Lie groups using the direct connection. Suppose that G is a Banach Lie group with the model space G. Letµ : G×ð −→ T Gbe given by µ(m, v) = Teλm(v), where λm
is the left translation onG and ð is the Lie algebra of G. According to [26], there
exists a unique connection DG on Gwhich is (µ, id
G)−related to the canonical flat
connection on the trivial bundleE= (G×ð, pr1, G). Locally the Christoffel symbols
ΓG ofDG are given by
ΓφG(x)(y, ξ) =Dfφ(x)(y, fφ−1(m)(ξ)); x∈φ(U), y, ξ∈G
wherefφ is the local expression of the isomorphism Teλx :TeG−→TxGand (U, φ)
chart ofG. IfG= lim←−Giis obtained as projective limit of Banach Lie groups andDGi
is the direct connection onEi= (Gi×ði, pr1, Gi), thenDG= lim←−DGi is exactly the
direct connection onE= (lim←−Gi×lim←−ði, pr1,lim←−Gi) [13]. Theorems 3.7 and 3.8 show
thatDGdetermines the following diffeomorphisms (κ
1, κ1) :T T G−→π∗GT G⊕π
∗
GT G
and (πT G, T πG, κ1) :T T G−→T G⊕T G⊕T Gwhich locally on the chart (U, φ) have
the form
(κ1, κ1)(x, ξ, y.η) =¡(x, ξ, y)⊕(x, ξ, η+Dfφ(x)(y, fφ−1(m)(ξ)))¢
and
(πT G, T πG, κ1)(x, ξ, y, η) = (x, ξ)⊕(x, y)⊕(x, η+Dfφ(x)(y, fφ−1(m)(ξ))).
Example 4.13. Flat connections. LetM =Ewith the global chart (E, idE). The
canonical flat connectionDCon the trivial bundleE= (M×E, pr1, M) is locally given
by the global Christoffel symbol{ΓC}, where ΓC(x)(ξ) = 0, for any (x, ξ)∈E×E.
LetM =F = lim
←−Ei and consider it with the global chart (F, idF) = lim
←−(Ei, idEi).
Moreover consider the canonical flat connection DC = lim←−DCi given by the form ΓC= lim
←−ΓCi on (M×F, pr1, M). Then the diffeomorphisms of theorems 3.7 and 3.8
are given by
(κ1, κ1)(x, ξ, y.η) = ¡(x, ξ, y)⊕(x, ξ, η)¢,
(πT G, T πG, κ1)(x, ξ, y, η) = (x, ξ)⊕(x, y)⊕(x, η).
5
Appendix
Lemma 5.1. Let{Mi}
in∈N be a projective system of Banach manifolds and for any i∈N,πi:T Mi−→Miits tangent bundle. Then{π∗
iT Mi}i∈Nalso forms a projective system with the limit isomorphic toπ∗
T M ≡lim←−π∗
Proof. Fori,j ∈Nwithj≥i define
respectively. On the other handP is also surjective since for ,j≥i,
and similarly (ϕi◦β)′(0) = (
βi)′(0). Henceϕi◦α,ϕi◦β,αiandβi, respectively, are equivalent onMifor anyi∈Ni.e. P([α, x],[β, x]) = ([αi, xi],[βi, xi])
i∈N. These help
us to conclude thatP is the desired bijection. ¤
Here we state a theorem from [12] without proof.
Theorem 5.2. T M admits an Er´echet type vector bundle structure overM with the structure groupH0(E)where
H0(E) ={(fi)i∈N∈
Y
i∈N
GL(Ei); lim
←−fiexists}.
More preciselyH0(E) = lim
←−H0i(E)where for any i∈N
H0i(E) ={(f1, ..., fi)∈ i
Y
k=1
GL(Ei); ρjk◦fj =fk◦ρjk f or k≤j≤i}
is a Banach lie group. In an analogous manner{T T Mi}i∈Nforms a projective system of manifolds with the limitT T M = lim←−T T Mi and the structure groupH0(E×E)and
fibres of typeE×E.
Using the above theorem it can be checked thatπ∗
T M is a Fr´echet vector bundle onT M with the structure groupH0(E).
Study of other types of the connections like [6] may be the subject of interest in this category with generalized Finsler structures. Also in other branches like [8] and [19] we may find applications for our theory.
Acknowledgement. Ali Suri expresses his deep gratitude to S.B. Hajjar.
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Authors’ address:
Mansour Aghasi, Ali Suri Department of Mathematics,
