Connections on distributional bundles

Connections on Distributional Bundles.

DANIEL CANARUTTO(*)

ABSTRACT- A general approach to the geometry of distributional bundles is pre- sented. In particular, the notion of connection on these bundles is studied. A few examples, relevant to quantum field theory, are discussed.

Introduction.

The notion of smoothness introduced by Frölicher [Fr] provides a general setting for calculus in functional spaces [FK, KM] and differen- tial geometry in functional bundles [JM, KM, CK, MK]. An important aspect of that approach is that the essential results can be formulated in terms of finite-dimensional spaces and maps, without heavy involvement in infinite-dimensional topology and other intricated questions. In par- ticular, the notion of a smooth connection on a functional bundle has been applied in the context of the «covariant quantization» approach to Quantum Mechanics [JM, CJM].

In a previous paper [C00a] I applied these ideas to the differential geometry of certain bundles whose fibres are distributional spaces, more specifically scalar-valued generalized half-densities. The main purpose of the present paper is to extend those results to the general case of the bundle of generalized «tube» sections of a 2-fibred «classical» (i.e. finite dimensional) bundle; basic notions of standard differential geometry – such as tangent space, jet space, connection and curvature – are intro- (*) Indirizzo dell’A.: Dipartimento di Matematica Applicata «G. Sansone», Via S. Marta 3, 50139 Firenze, Italia.

E-mail: canaruttoHdma.unifi.it; http://www.dma.unifi.it/^Acanarutto

duced for this case; adjoint connections and tensor product connections are shown to exist. Furthermore, a suitable connection on the underlying classical bundle is shown to yield a connection on the corresponding dis- tributional bundle; some particularly important cases are the vertical bundle and its tensor algebra, which turn out to be closely related to the notion of adjoint connection. Finally, I consider a few examples which are relevant in view of applications to quantum field theory: the «Dirac connection» on the bundle of 1-electron states for a given observer, and the connections induced on the phase-distributional bundles describing electron and photon fields.

1. Generalized sections.

Let p:YKY be a real or complex classical vector bundle, namely a finite-dimensional vector bundle over the Hausdorff paracompact smooth real manifold Y. Moreover assume that Y is oriented, let n»4dimY, and denote the positive component of RⁿTY by VY»4

»4(RⁿTY)¹. Then VYKY is a semi-vector bundle [C98, C00a, C00b, CJM], as well as its dual bundle V*Yf(RⁿT*Y)¹KYwhich is called the bundle of positive densities on Y.

Let

Y Y Y

0f

D D D

0(Y,V*Y7

Y Y* ) be the vector space of all smooth sec- tions YKV*Y7

YY* which have compact support. A topology on this space can be introduced by a standard procedure [Sc]; its topological du- al will be denoted as

Y Y Y

^f

D D D

^(Y,Y) and called the space of generalized sections, or distribution-sections of the given classical bundle, while

Y Y Y

0

is called the space oftest sections. In particular, a sufficiently regular or- dinary section s:YKY can be seen as a generalized section by the rule

as,ub4

Y

as(y),u(y)b, u

Y Y Y

0.

On turn,

Y Y Y

has a natural topology [Sc], and its subspace

Y Y Y

^*0f

D D D

0(Y,Y) of all smooth sections with compact support is dense in it. Some particu- lar cases of generalized sections are that ofr-currents (YfR^rT*Y,r N) and that of half-densities(¹) (Yf(V*Y)^{1 /2}).

(¹) The «square root» bundle (V* )^{1 /2}is characterized, up to isomorphism, by (V* )^{1 /2}7(V* )^{1 /2}`V*.

The topological dual of

Y Y Y

^*0 is

Y Y Y

^*f

D D D

^(Y,V*Y7

Y Y* ), that is the space ofgeneralized Y*-valued densitiesonY, or the adjoint space of

Y Y Y

^.

REMARK. If u

Y Y Y

^and f

Y Y Y

* then, possibly, the contraction au,fb may be defined even if neither one is a test section.

Generalized sections can be naturally restricted to any open subset Y^a%Y of the base manifold, namely there is a natural linear projection

Y

Y Y

_K

Y Y Y

a

f

D D D

^(Y

a

,Y^a), where Y^a»4p²¹(Y^a). Accordingly, if (b_i) is a local frame ofY, a generalized sectionz

Y Y Y

has the local expression z4zⁱb_i with zⁱ

D D D

^(Y

a

,C).

There is no inclusion

Y Y Y

a

%K

Y Y Y

, since elements in

Y Y Y

a

cannot be natural- ly extended to generalized sections onY(such extension may not exist at all). However, a gluing property holds: if ]Y_i( is a covering of Y and ]ui

Y Y Y

i( is a family of generalized sections such thatui andujcoincide onY_iOY_j, then there exists a uniqueu

Y Y Y

whose restriction to Y_icoin- cides with ui(i.

Let p8:Y8 KY8 be another classical vector bundle and W: YKY8 a smooth fibred isomorphism over the diffeomorphismW:YKY8; namely, p8ⁱW4Wip. Clearly,Wdetermines a natural isomorphism between the spaces of ordinary sections of the two bundles; one easily sees that this restricts to an isomorphism of the corresponding spaces of test sections, and extends to an isomorphism W*:

Y Y Y

_K

Y Y Y

₈. One also has the adjoint construction. It is not difficult to see thatW

*turns out to be a continuous isomorphism (the proof is essentially the same as given in [C00a] for the particular case of scalar-valued half-densities).

2. F-smoothness in distributional spaces.

Let I%R be an open interval. A curve a: IK

Y Y Y

is said to be F- smooth (Frölicher-smooth) if the map

aa,ub:IKC:tOaa(t),ub

is smooth for every u

Y Y Y

0. Accordingly, a function f:

Y Y Y

KC is called F-smooth iffia:IKC is smooth for all F-smooth curve a, and a map F:

Y Y Y

_K

W W W

between any two distributional spaces is called F-smooth if fiFia is smooth for all F-smooth a:IK

Y Y Y

^and f:

W W W

_K^C.

It can be proved [Bo] that a functionf : MKR, whereMis a classical manifold, is smooth (in the standard sense) iff the composition fic is a smooth function of one variable for any smooth curvec: IKM. Thus one has a unique notion of smoothness based on smooth curves, including both classical manifolds and distributional spaces. This is convenient for dealing with smoothness relatively to product spaces such as M3

Y Y Y

^; moreover, one has a natural notion of smoothness for mapsMK

Y Y Y

^and

Y

Y Y

_KM. Hence, one may simply write smooth for F-smooth.

Let

C C C

YYYbe the set of all F-smooth curves in

Y Y Y

^{; take anyiN}N]0( and consider the following binary relation in R3

C C C

YYY:

(t,a)A

i

(s,b) ` D^kaa,ub(t)4D^kab,ub(s) (u

Y Y Y

0, k40 ,R,i. Then clearly A

i

is an equivalence relation; the quotient Tⁱ

Y Y Y

^»₄

C C C

YYYO A

i

will be called thetangent space of order iof

Y Y Y

. The equivalence class of (t,a)

C C C

YYYwill be denoted by¯ⁱa(t). Obviously, Tⁱ

Y Y Y

is a fibred set over

Y

Y Y

; the fibre over some l

Y Y Y

will be denoted by Tⁱ_l

Y Y Y

. In particular T⁰

Y Y Y

₄

Y Y Y

^.

The set T

Y Y Y

^»₄^T1

Y Y Y

is called simply the tangent space of

Y Y Y

^{, and}

¯a(t)»4¯¹a(t) is called the tangent vector of aat a(t).Any element in T

Y Y Y

can be represented as ¯a( 0 ), for a suitable curve a defined on a neighbourhood I of 0. It is not difficult to see that there is a natural isomorphism

Y

Y Y

₃

Y Y Y

_K^T

Y Y Y

^{: (l,m)}O¯[l1tm]_t40.

PROPOSITION 2.1. Let

A A A

^and

B B B

be smooth spaces (each one is either a classical manifold or a distributional space) and F:

A A A

_K

B B B

^a smooth map. Then there exists a unique smooth map TF: T

A A A

_K^T

B B B

^, called the tangent prolongation of F, such that for every smooth curve a:IK

A A A

^{one has}

¯[Fia](t)4TFi¯a(t), tI.

The proof of this non-trivial statement is omitted because it is essen- tially similar to that of the particular case considered in [C00a]. It is not difficult to see that tangent prolongations behave naturally in terms of any compositions.

3. Distributional bundles.

The basic classical geometric setting underlying distributional bun- dles is the following. One considers a classical 2-fibred bundle

VK

q EK

q

B,

where q:VKE is a complex (or real) vector bundle, and the fibres of the bundle EKB are smoothly oriented. Moreover, one assumes that qiq: VKB is also a bundle (not a vector bundle in general), and that for any sufficiently small open subset X%B there are bundle trivializa- tions

(q,y) :E_XKX3Y, (qiq,y) :V_XKX3Y

(hereE_X»4q²¹(X) and the like) with the following projectability prop- erty: there exists a surjective submersion p: YKY such that the diagram

VX q

I

E_X K

(qiq,y)

K_(q,y)

X3Y

I

^1X3p

X3Y

commutes; this implies that YKY is a vector bundle, not trivial in general.

The above conditions are easily checked to hold in many cases which are relevant for physical applications (as in the cases considered in § 11 and § 12). In particular, the above conditions hold if V4E3_B W where WKBis a vector bundle, ifV4VE(the vertical bundle ofEKB) and if V is any component of the tensor algebra of VEKE.

Let n be the dimension of the fibres of EKB. The orientation re- quirement implies thatRⁿVEKEis a trivializable bundle with smooth- ly oriented fibres, and one has the smooth bundleV*E»4(RⁿV*E)¹K KE. Then for eachxBone may consider the distributional space

V V V

x»4 4

D D D

^(Ex,V_x), and obtains the fibred set

[:

V V V

^f

D D D

B(E,V)»4

I

₂

I

xB

V V V

xKB.

For any two classical local bundle trivializations (q,y) and (qiq,y) as

above, let

Y:

V V V

Xf[²¹(X)K

Y Y Y

^f

D D D

^{(Y,Y) ,} Y_x»4(y_x)

*, xX.

Then ([,Y) :

V V V

XKX3

Y Y Y

is a local bundle trivialization of

V V V

_K^B.

Moreover, if (q,y8) :E_X8KX8 3Y8and (qiq,y8) :V_X8KX8 3Y8are two other classical bundle trivializations related by the same projectability property, then ([,Y8)i([,Y)²¹:XOX8 3

Y Y Y

_K^XOX8 3

Y Y Y

₈ ^{is F-} smooth and linear. Hence, suitable classical bundle atlases onVKB and EKB determine a linear F-smooth bundle atlas on

V V V

_KB, which is said to be anF-smooth distributional bundle(²). Clearly,

V V V

turns out to be an F-smooth space in a natural way: a curvea: IK

V V V

is defined to be F-smooth if ([,Y)ia is such for any local F-smooth trivialization; in general, the F-smoothness of any map from or to

V V V

is equivalent to the F-smoothness of its local trivialized expressions.

If a is F-smooth then it is natural to set

T

(

([,Y)ia

)

₄

(

T([ⁱa), T(Yia)

)

:I3RKTX3T

Y Y Y

^.

One says that two F-smooth curves are first-order equivalent at some point if their trivialized expressions are such; in this way one obtains the definition of the tangent space T

V V V

. Obviously, this is a fibred set over

V

V V

; a local bundle trivialization ([,Y) of

V V V

yields the local bundle trivialization

T([,Y) : T

V V V

XKT(X3

Y Y Y

^)fTX₃^T

Y Y Y

^,

and the transition maps between two induced trivializations are F- smooth and linear. Hencep_VVV: T

V V V

K

V V V

, the tangent bundle of

V V V

^{, is an} F-smooth vector bundle. One has another F-smooth bundle with the same total F-smooth space, namely

T[: T

V V V

_K^TB: ¯aO¯(q_ia) . Moreover one has the vertical subbundle

V

V V V

^»₄Ker T[%T

V V V

, the natural identification V

V V V

₄

V V V

₃

B

V V V

and the exact sequence over

V V V

0KV

V V V

KT

V V V

K

V V V

3_B TBK0 .

(²) Not every trivialization of a distributional bundle derives from trivializa- tions of the underlying classical 2-fibred bundle.

The subbundle of T*B7

V V V

T

V V V

which projects over the identity of TB is called thefirst jet bundle, denoted by J

V V V

_K

V V V

. This is an affine bun- dle over

V V V

, with «derived» vector bundle T*B7

V V V

V

V V V

. The restric- tion of T*[7T([,Y) is a local bundle trivialization which is denoted by

J([,Y) : J

V V V

XKJ(X3

Y Y Y

^)`

Y Y Y

3(T*X7

Y Y Y

^).

If xf(x^a) :XKR^m is a coordinate chart then one has the fibred charts

(x,Y) :

V V V

_KR^m3

Y Y Y

^, (x^a,Y,x.

a,Y.

)»4T(x,Y) : T

V V V

_K^Rm₃

Y Y Y

₃^Rm₃

Y Y Y

^, (x^a,Y,Ya)»4J(x,Y) : J

V V V

KR^m3

Y Y Y

3(R^m7

Y Y Y

^{) .}

Tangent prolongations of F-smooth maps involving

V V V

can be expressed through local trivializations; in particular, if s: BK

V V V

is an F-smooth section, then Ts: TBKT

V V V

projects over the identity of TB, so that it can be viewed as a section js:BKJ

V V V

. Settings^Y»4Yis:BK

Y Y Y

one has

(x^a,Y,x._a ,Y.

)iTs4Ts^Y4(x^a,s^Y,x .a, x._a

¯_as^Y) , (x^a,Y,Y_a)ijs4Js^Y4(x^a,s^Y,¯_as^Y) .

For maps f :

V V V

KR one introduces the notation

¯_Yf»4Vfi

(

1_VVV3([,Y)²¹

)

:

V V V

₃

Y Y Y

_K^R, and obtains the local coordinate expression

df»4pr₁iTf4¯_a fdx^a1(¯_Yf)idY. REMARK. IfY

a

%Yis an open subset such thatY

a

»4p²¹(Y

a

) is trivial- izable, and (yⁱ,y^A) :Y

a

KRⁿ3R^p is a linear bundle chart, thens^Y has a coordinate expression whose components are scalar-valued distributions s^A

D D D

X(Y

a

,R).

4. F-smooth fibred morphisms.

Let V8 KE8 KB8another 2-fibred bundle with the same properties, and [8:

V V V

_{8 K}^B₈ the induced distributional bundle. Let moreover

F:

V V V

_K

V V V

₈be a fibred F-smooth map over the smooth mapf:BKB8. Then, similarly to the classical case, the tangent prolongation

TF: T

V V V

KT

V V V

8

is a linear fibred morphism over F and a fibred morphism over Tf: TBKTB8. setting F^Y8»4Y8ⁱF:

V V V

_K

Y Y Y

₈ ^{one has (3)}

(x8,Y8,x. 8,Y.

8)_iTF4

g

f^a8,F^Y8,x.

a¯_af^x8,x.

a¯_aF^Y81¯_YF^Y8iY.

h

^.

If moreoverf is a diffeomorphism, then the restriction off*7TF de- termines a fibred morphism JF: J

V V V

KJ

V V V

8 over F.

If F is linear over f, then one writes

F^Y8_Y»4¯_YF^Y84F^Y8i([,Y)²¹:XKLin (

Y Y Y

^,

Y Y Y

8) ,

which is analogous to the matrix expression of a linear morphism in fi- nite-dimensional case.

Let nowW:VKV8be a classical linear isomorphism over the fibred diffeomorphismW:EKE8, which on turn is projectable over the diffeo- morphism f:BKB8. Then one has the induced linear isomorphism F»4W

*:

V V V

K

V V V

8over B. In the domain of a local coordinate chart one has (⁴)

(Fl)^A84(F^Y8_Yl^Y)^A84(W^A8_Al^A)i^JW, l

V V V

^,

(¯_aF^Y8Yl^Y)^A84(¯_aW^A8AiW^J)(l^Ai^JW)1[¯_i(W^A8_Al^A)i^JW]¯_a8W^Ji(¯_af^a8if^J) , where back pointing arrows indicate the inverse maps. By using these formulas one can write down the coordinate expressions of TF and JF.

As a special case, one also gets the transformation formulas in T

V V V

^and J

V V V

between any two charts induced by classical charts; a detailed treat- ment of these aspects lies outside the scope of a short paper and will be exposed in a future survey paper.

When V4VE, V8 4VE8 and W is a fibred diffeomorphism over f, then one has the special caseW4VW, which extends to any component of the tensor algebra of VEKE. In particular, one is interested in the bun- (³) These partial derivatives are naturally defined as a consequence of propo- sition 2.1.

(⁴) The proof of the second formula is not difficult but somewhat delicate, as one must take carefully into account the various involved compositions.

dles of scalar q-densities, where q is a rational number, namely in the distributional bundles

D D D

B(E,C7V^2qE) where V^2qEf(V*E)^q and the like. One gets

¯_aF^Y8(l)4(¯_il^YiW^J)¯_a8W^Ji(¯_af^a8if^J)NVW^JN^q1

1q(l^YiW^J)QNVW^JN^q(¯_iWⁱ⁸i^JWW)¯_a8¯_i8^JiW(¯_af^a8i^Jf) , where NVW^JN denotes the vertical Jacobian determinant of W^J.

5. Distributional connections.

Similarly to the standard finite-dimensional case, aconnectionon the distributional bundle

V V V

is defined to be an F-smooth section

÷:

V V V

KJ

V V V

^.

In the domainX%Bof a local bundle chart (x,Y) :

V V V

XKR^m3

Y Y Y

^{one has} the local expression

÷aY»4Y_ai÷:

V V V

K

Y Y Y

^.

The existence of global connections then follows from standard argu- ments, using the paracompactness of B.

Basically, one deals with linear connections, that is connections ÷ which are linear morphisms over B. Then one writes

÷aY

4÷aY

YiY, ÷aY

Y: XKEnd (

Y Y Y

^{) .} If÷aY8

Y8is the expression of÷in a different fibred chart (x8,Y8) over the same domain X, then

÷a8

Y8Y84¯_a8^Jk^aQ(¯a

K

^Y8Y1

K

^Y8Yi÷aY

Y)i

K

^YY8, where

K

^f(k,

K

^Y8Y)»4(x8,Y8)i(x,Y)²¹:R^m3

Y Y Y

_KR^m3

Y Y Y

₈ denotes the transition map.

As in the finite-dimensional case, a connection yields a number of structures (whose assignment is actually equivalent to that of the con- nection itself). First, ÷ can be viewed as a linear map

V V V

₃

BTBKT

V V V

^,

and (p_VVV, T[)i÷ is the identity of

V V V

₃

B TB. The image H_÷

V V V

^»₄÷(

V V V

₃

BTB)

is a vector subbundle of T

V V V

_K

V V V

, with m-dimensional fibres; the re- striction of ÷i(p_VVV, T[) is the identity of H_÷

V V V

^{. If v: B}KTB is a smooth vector field, then ÷v:

V V V

_K^T

V V V

is an F-smooth vector field, called its horizontal lift, with coordinate expression

x.

ai÷v4v^a, Y.

i÷v4v^a÷aY. One also has the complementary map

V»412÷: T

V V V

_K^V

V V V

^f

V V V

₃

B

V V V

^,

so that the map (÷i(p_VVV, T[),V) determines the decomposition T

V V V

4H_÷

V V V

5

VVV

V

V V V

^.

Lets: BK

V V V

be an F-smooth section. Thecovariant derivativeofs is defined to be the linear morphsim over B

˜sf˜[÷]s»4pr2iViTs: TBK

V V V

^.

Ifv:BKTBis a vector field, then one also writes˜_vs»4˜siv. The lo- cal coordinate expression of the covariant derivative is

(˜s)^Y»4Yi˜s4x._a

(¯_as^Y2÷aY is) .

The curvature tensor of a linear connection ÷ can be defined, as in the finite-dimensional case, as the section D:BKR²T*B7

B End (

V V V

⁾ given by

D(u,v)s»4˜_u˜_vs2˜_v˜_us2˜_[u,v]s, u,v:BKTB,s:BK

V V V

^, which has the local chart expression

D^Y_Y4D_ab^Y_Ydx^aRdx^b42(¯_b÷aY Y1÷aY

Yi÷bY

Y) dx^aRdx^b.

A more general definition of curvature, valid also in the non-linear case, can be given in terms of the Frölicher-Nijenhuis bracket [FN, MK, MM, KMS]. First, one must define the Lie bracket of any two F-smooth vector fieldsW,Z:

V V V

_K^T

V V V

. Using the canonical involutions: TT

V V V

_K

KTT

V V V

^{, and TZ}iW2s(TWiZ) :

V V V

_K^VT

V V V

^`T

V V V

₃

V V V

T

V V V

^{, one sets} [W,Z]»4pr₂(TZiW2s(TWiZ) ) :

V V V

_K^T

V V V

^,

which has the local expression

[W,Z]^a4W^b¯_bZ^a2Z^b¯_bW^a1¯_YZ^aiW^Y2¯_YW^aiZ^Y, [W,Z]^Y4W^b¯_bZ^Y2Z^b¯_bW^Y1¯_YZ^YiW^Y2¯_YW^YiZ^Y. The Frölicher-Nijenhuis bracket of F-smooth tangent-valued forms

V

V V

_KRT*B7

V V V

T

V V V

can now be introduced by a straightforward exten- sion of the standard definition, namely by the requirement that for de- composable forms one has

[a7W,7Z]4aR7[W,Z]1aR(W.b)7Z2(Z.a)R7W1 1(21 )^r(ZNa)Rdb7W1(21 )^rda(WNb)7Z, where a:BKR^rT*B, b:BKR^sT*B, and W,Z:BKTB.

If ÷:

V V V

_K^J

V V V

is an F-smooth connection then its curvature is de- fined to be

D»4 2[÷,÷] :

V V V

_KR²T*B7

VVV

V

V V V

^.

6. Adjoint connections.

The distributional bundle

V V V

^*»₄

D D D

B(E,V*E7_E V* )KB is called the adjoint bundle of

V V V

_KB; its fibre type is

Y Y Y

*, the adjoint of

Y Y Y

(§ 1).

An endomorphismAEnd (

D D D

) of an arbitrary distributional space

D D D

determines a dual endomorphism A8End (

D D D

0) of the test space, de- fined byA8u»4uiA, that isaA8u,fb4au,Afb. Moreover it may hap- pen thatA8can be extended to an endomorphismA* of the distribution- al completion

D D D

^{* of}

D D D

0; this possible extension is called the adjointof A. This requirement is fulfilled, in particular, by the polynomial deriva- tion operators [C01].

PROPOSITION 6.1. Let the F-smooth connection ÷:

V V V

_K^J

V V V

^be such that, in every local F-smooth chart (x,Y) :

V V V

_K^X₃

Y Y Y

^{, the}

local expression ÷^Y: TBKEnd (

Y Y Y

⁾ admits an adjoint (÷^Y)* : TBK KEnd (

Y Y Y

^{* ).}

Then, there exists a unique F-smooth connection ÷* :

V V V

*KJ

V V V

* such thatJci(÷,÷* )40 , where c:

V V V

3_B

V V V

^*KB3C: (s,l)O([(s), alb,s). Its chart expression is

÷*_aY^Y4 2(÷aY Y)* , that is

(˜*_vl)_Y4v^a(¯_alY2÷*_aY^Y_ilY)4v^a(¯_alY1lYi÷aY Y) . Equivalently, ÷* is determined by the requirement that

v.al,sb4a˜_v*l,sb1al,˜_vsb

hold for all smooth sectionsl: BK

V V V

^{*and s:B}K

V V V

^,and for all vec- tor field v: BKTB, whenever all contractions are well-defined.

PROOF. Let÷* :

V V V

*KJ

V V V

* beanylinear connection; denote byzf fpr2 the (trivial) fibre coordinate on B3CKB, and observe that

Jci(÷,÷* ) :

V V V

₃

B

V V V

^*_K^C₃^T*B has the chart expression

z_aiJci(÷,÷* )(s,l)4alY,÷aY

Y(s^Y)b1a÷*_aY^Y(l_Y),s^Yb, which holds for any (s,l)

V V V

₃

B

V V V

* whenever all contractions are well- defined. This expression vanishes iff÷*_aY^Y4 2(÷aY

Y)*. Ifs:BK

V V V

0*%

V V V

is a section of the subbundle of test maps in

V V V

^{, one has ˜}vs:BK

V V V

ⁱⁿ general. For every u: BK

V V V

0, the map

˜*_vu:

V V V

0*KC:sOv.as,ub2a˜_vs,ub

is linear continuous, hence ˜*_vu: BK

V V V

*. Its chart expression is as,˜*_vub4v^a¯_aas^Y,u_Yb2av^a¯_as^Y,u_Yb1av^a÷aY

Yis^Y,u_Yb4 4as^Y,v^a(¯_au_Y1÷aY

Y)*u_Yb. By continuity, the operation ˜_v* can be extended to all sectionsl:BK K

V V V

*, and is seen to define a covariant derivative. r

REMARK. The adjoint connection ÷* is not reducible to the sub- bundle

V V V

0KB.

REMARK. Similarly to the finite-dimensional case, a distributional connection ÷ determines connections on any tensor bundle over B con- structed from

V V V

_KB. Together with its possible adjoint ÷*, it deter- mines connections on the tensor algebra of

V V V

_K^B and its sub- spaces.

7. Connection induced by a classical connection.

In this section, I’ll show that a suitable underlying classical structure determines a connection on a distributional bundle (though not all distri- butional connections arise in this way).

Consider again the classical 2-fibred bundleVKEKBas before. By VVand JVone denotes the vertical and jet spaces ofVrelatively to base B, while vertical and jet spaces relatively to base E will be denoted by V_EV and J_EV.

A connectionG:VKJVis said to beprojectable if there is a connec- tion G:EKJE such that the diagram

V

q

I

E K^G

K_G JV

I

^Jq

JE

commutes; moreover,Gis said to belinearif it is a linear morphism overG.

Let (x^a,yⁱ,y^A) :VKR^m3Rⁿ3R^p be a local 2-fibred coordinate chart, linear over (x^a,yⁱ) : EKR^m3Rⁿ; the coordinate expression of a linear projectable connection is then

G4dx^a7(¯xa1Gai¯yi1GaA

By^B¯yA) , G4dx^a7(¯x_a1G_aⁱ¯y_i) ,

with Gia,GaA

B:EKR.

A smooth sections:EKVcan be viewed as a section of a functional bundle, whose fibre over eachxM is the space of all smooth sections E_xKV_x; in the case when one considers local sectionsEKV, these must be defined on a «tubelike» open subset of E. Moreover, this functional bundle can be viewed as a subbundle of

V V V

^»₄

D D D

B(E,V)KB.

Observe now that the above s can be viewed as the vertical-valued 0-form

(1V,s) :VKV3_EVfVEV%TV,

which has the same coordinate expressions4s^A¯y_A. One may also view G as a projectable tangent-valued 1-form

G:VKT*M7

V TV%T*V7

V TV, and consider the Frölicher-Nijenhuis bracket

[G,s] :VKT*V7_V TV.

Actually, [G,s] turns out to be a basic vertical-valued form VK KT*M7

V V_EV, as one immediately sees from its coordinate expression [G,s]4(¯as^A1Gia¯_is^A2GaA

Bs^B) dx^a7¯yA.

From this, it is clear that [G,s] can be extended to the case whens is a sectionBK

V V V

; moreover, it can be seen as the covariant derivative of a linear connection÷:

V V V

_K^J

V V V

, which in the considered chart has the ex- pression ÷aY

Y(s^Y)4GaA

Bs^B2Gai

¯_is^A, that is (÷aY

Y)^A_B4G_a^A_B2d^A_BGⁱ_a¯_i.

It is not difficult (just a somewhat intricated calculation) to check that the above expression transforms in the right way under the distribution- al bundle chart transformation induced by a classical chart transforma- tion.

There is a natural relation between the curvature R of G and the curvature D of the induced distributional connection ÷. Actually one has R4dx^aRdx^b(R_abⁱ¯_i1R_ab^A_By^B¯_A) with

R_abⁱ4 2¯_aG_bⁱ1¯_bGⁱ_a2G^j_a¯_jGⁱ_b1G^j_b¯_jG_aⁱ, RabA

B42¯_aGbA

B1¯_bGaA

B2Gaj¯_jGbA

B1Gbj¯_jGaA B2GbA

CGaC B1GaA

CGbC B. A direct calculation then gives

D_ab^Y_Ys^Y4R_ab^A_Bs^B2R_abⁱ¯_is^A,

that is, simply, the Frölicher-Nijenhuis bracket D(s)4 2[R,s] .

8. Induced connection and horizontal transport.

In this section it will be showed that the notion of distributional con- nection induced by a classical connection arises in a natural and some- what more intuitive way in terms of the parallel (i.e. horizontal) trans- ports related to the two connections.

Let I%R be an open neighbourhood of 0, and c:IKB a smooth curve. For any v0Vc( 0 ) one has, locally, a unique G-horizontal curve C_v0: I_v0KV, withI_v0%I, such that C_v0( 0 )4v₀. MoreoverC_v0is linear pro- jectable over C_v0:I_v0KE, the horizontal G-lift of c starting from v₀fq(v₀).

IftIv₀, so that the horizontal transport ofv₀V_{c( 0 )}toV_c(t)is defined, then there is a neighbourhood U%V_{c( 0 )} of v0 such that the horizontal transport of every uU toVc(t) is defined too (this is a consequence of the continuity ofG). From a general result in the theory of ordinary dif- ferential equations, on the other hand, it follows that horizontal trans- port relatively to alinear connection on a vector bundle determines an isomorphism of any two fibres along any smooth curve connecting their base points. This is not the case of the presently considered setting, sinceVKB is not a vector bundle in general. But the whole fibreV_v0is linearlysent to the whole fibreV_vt, whereC_v0(t)fv_tE_c(t); namely hori- zontal transport determines an isomorphism between these two fi- bres.

Momentarily forgetting these locality issues, assume horizontal transport along c determines, for all tI, a fibred isomorphism C_t:V_{c( 0 )}KV_c(t) over a diffeomorphism C_t:E_{c( 0 )}KE_c(t). In other terms one has a 1-parameter familiy of fibred isomorphisms over a 1-parameter familiy of diffeomorphisms, denoted by

C:I3V_{c( 0 )}KV, C: I3E_{c( 0 )}KE. Let now l

V V V

c( 0 )f

D D D

^(Ec( 0 ), V_{c( 0 )}). Then

(Ct)

*l

V V V

c(t)f

D D D

^(Ec(t),Vc(t)) .

Namely, the classical horizontal transport locally determines a lift C*:I3

V V V

c( 0 )K

V V V

of the base curvec. It can be seen that this is exactly the horizontal lift of c relatively to the distributional connection ÷ induced by G, namely that

÷: TM3_M

V V V

KT

V V V

: (¯c( 0 ),l)O¯(C

*l)( 0 ) .

This result follows from a coordinate calculation; from the definition of a horizontal curve one has

¯

¯tCⁱ( 0 ,v₀)4c._a

( 0 )Gai(v₀) , ¯

¯tC^A_B( 0 ,v₀)4c._a

( 0 )GaA B(v₀) , while the induced horizontal curveC

*l:IK

V V V

can be written, by some abuse of language, as

(C*l)^A(t,y)4C^A_B(t,C^J(t,y) )l^B(C^J(t,y) ) . Calculating the tangent vector ¯(C

*l) :IKT

V V V

is now a straightfor- ward (though not immediate) task; using the relation between G and ÷ one gets the claimed result.

As already observed, in general this horizontal lift of c through ÷ may not exist for everyl

V V V

c( 0 ), but it can defined for the restriction of l to a suitable open subset. Furtherermore, the horizontal lift construc- tion can be done wheneverlhas compact supportK%E_{c( 0 )}, by the follow- ing argument. For everyeE_{c( 0 )}choose an open neighbourhood ofe,U%

%E_{c( 0 )}, such that the restriction ofltoUis horizontally transported overc up tot4tUD0; from this open covering ofK select a finite subcovering

U

U U

, and definetK»4min]tU,U

U U U

(. Then by a partition of unity sub- jected to

U U U

one has horizontal transport of l over c up to t4tK.

9. Induced connections and tensor products.

Consider another 2-fibred bundle V8 KE8 KB over the same lower base manifoldB. The fibred tensor product ofVandV8 is defined to be the 2-fibred bundle

W»4V7_F V8 KF»4E3_BE8 KB.

Let (x^a,yⁱ,y^A) and (x^a,yⁱ⁸,y^A8) be 2-fibred coordinate charts onV and V8; then one has induced coordinates (x^a,yⁱ,y^A,yⁱ⁸,y^A8,w^AA8) on W, where

w^AA8fy^A7y^A8 i.e. w^AA8i74y^Ay^A8, 7:V3_BV8 KW: (v,v8)Ov7v8.

The jet prolongation J7: JV3_B JV8 KJWis characterized by the re- quirement that the diagram

commutes for any two sectionss:BKV,s8:BKV8. Thus one finds the coordinate expression

waAA8

iJ74yaAy^A81y^AyaA8.

Let now G:VKJV and G8: V8 KJV8 be linear projectable connec- tions overG:EKJE andG8:E8 KJE8, respectively; then there exists a unique connection G7G8:WKJW such that the diagram

JV3_B JV8

(G,G8)

H

V3_BV8 K^J7

K₇ JW

H

^G7G8

W

commutes; moreover, G7G8 is linear projectable over (G,G8) :E3_BE8 KJE3_B JE8, and its ccordinate expression is

(yai,yaA,yai8,yaA8,waAA8)i(G7G8)4 4(Gⁱ_a,GaA

By^B,Gia8,GaA8

B8y^B8,GaA

By^By^A81y^AGaA8 B8y^B8) , where the components of G8 are recognized by primed indices.

The distributional bundle

W W W

^»₄

D D D

B(F,W)KB is easily seen to co-

incide with the fibred tensor product of

V V V

^and

V V V

₈^{, namely}

W

W

W

^»₄

D D D

M(F,W)4

D D D

M(E3_ME8,V 7

E3ME8

V8)4

4

D D D

M(E,V)7_M

D D D

M(E8,V8)f

V V V

7_M

V V V

₈^. Let÷:

V V V

_K^J

V V V

^and÷8:

V V V

_{8 K}^J

V V V

₈be the distributional connections in- duced byGand G8. These yield, exactly by the same argument which is valid in the finite-dimensional case, a linear connection ÷7÷8:

W W W

_K KJ

W W W

; it is not difficult to proof:

PROPOSITION 9.1. The tensor product connection ÷7÷8 is exactly the distributional connection associated with the classical connection G7G8. For v

W W W

^{one has}

(÷7÷8)_a^YY8_YY8v^YY84GaA

Bv^BA82Gia

¯_iv^AA81GaA8

B8v^AB82Gai8¯_i8v^AA8. IfE4E8then one also has the 2-fibred bundleV7

E V8 KEKB. The distributional bundle

D D D

M(E,V7_EV8) is different from

V V V

7_M

V V V

₈^{. If} G:VKJV and G8:V8 KJV8 are now linear projectable connections over thesameconnectionG:EKJE, then, besidesG7G8, they also de- termine a different kind of tensor connection, that is

G7G8:V7_EV8 KJ(V7EV8) , which is characterized by the commuting diagram

J(V3_B V8)fJV3_JEJV8

(G,G8)

H

V3_E V8

K^J7

K₇ J(V7

E V8)

H

^G7G8

V7

EV8 and has the coordinate expression

(y_aⁱ,y_a^A,y_a^A8,w_a^AA8)i(G7G8)4 4(G_aⁱ,GaA

By^B,GaA8

B8y^B8,GaA

By^By^A81y^AGaA8 B8y^B8) . The induced distributional connection

÷7÷8:

D D D

M(E,V7_E V8)KJ

D D D

M(E,V7_EV8)

has the cordinate chart expression (÷7÷8)_a^YY8_YY8v^YY84GaA

Bv^BA81GaA8

B8v^AB82Gai

¯_iv^AA8. 10. Induced connection: vertical bundle and adjoint case.

A linear projectable connectionG:VKJV, as considered in the pre- vious sections, determines a unique «dual» connection G* : V*KJV*;

this is again linear projectable over the same G, and is characterized by Jci(G,G* )40 ,

wherec:V3_E V*KE3C denotes the duality contraction; it has the co- ordinate expression

G*_aA^B4 2GaA B.

On turn, G* determines a connection on the distributional bundle

D

D D

B(E,V* ). In general, this isnotthe adjoint connection÷* of÷, which is actually a connection on a different distributional bundle. In order to study the relation between÷* and the classical connectionG one has to perform some further constructions.

The first step consists in the vertical extension ofG:EKJE. Recall- ing the natural isomorphism JVE`VJE, one gets the morphism

G

q

»4VG: VEKJVE,

which turns out to be a linear projectable connection overG. Its coordi- nate expression is

G

q

ai

j4¯_jGⁱa.

Its dual connection G^q* : V*EKJV*E has the coordinate expression (G^q* )_ajⁱ4 2G^qai

j4 2¯_jGⁱa.

Now one finds induced linear projectable connections over G in all tensor product bundles overEKBconstructed from VEand V*E. Most noticeably, one has projectable linear connections overG on the 2-fibre bundles

R^rV*EKEKB, rN, V*EKEKB,