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Differential Geometry, Proceedings, Peniscola 1988,

arXiv:math/9202207v1 [math.DG] 1 Feb 1992Differential Geometry, Proceedings, Peniscola 1988,Springer Lecture Notes in Mathematics, Vol. 1410 (1989), 249–261GRADED DERIVATIONSOF THE ALGEBRA OF DIFFERENTIAL FORMSASSOCIATED WITH A CONNECTIONPeter W. MichorInstitut f¨ur Mathematik, Universit¨at Wien,Strudlhofgasse 4, A-1090 Wien, AustriaIntroductionThe central part of calculus on manifolds is usually the calculus of differential formsand the best known operators are exterior derivative, Lie derivatives, pullback and inser-tion operators.

Differential forms are a graded commutative algebra and one may ask forthe space of graded derivations of it. It was described by Fr¨olicher and Nijenhuis in [1],who found that any such derivation is the sum of a Lie derivation Θ(K) and an insertionoperator i(L) for tangent bundle valued differential forms K, L ∈Ωk(M; TM).

The Liederivations give rise to the famous Fr¨olicher-Nijenhuis bracket, an extension of the Liebracket for vector fields to a graded Lie algebra structure on the space Ω(M; TM) ofvector valued differential forms. The space of graded derivations is a graded Lie algebrawith the graded commutator as bracket, and this is the natural living ground for eventhe usual formulas of calculus of differential forms.

In [8] derivations of even degree wereintegrated to local flows of automorphisms of the algebra of differential forms.In [6] we have investigated the space of all graded derivations of the graded Ω(M)-module Ω(M; E) of all vector bundle valued differential forms. We found that any suchderivation, if a covariant derivative ∇is fixed, may uniquely be written as Θ∇(K) +i(L) + µ(Ξ) and that this space of derivations is a very convenient setup for covariantderivatives, curvature etc.

and that one can get the characteristic classes of the vectorbundle in a very straightforward and simple manner. But the question arose of how allthese nice formulas may be lifted to the linear frame bundle of the vector bundle.

Thispaper gives an answer.In [7] we have shown that differential geometry of principal bundles carries over nicelyto principal bundles with structure group the diffeomorphism group of a fixed manifold S,and that it may be expressed completely in terms of finite dimensional manifolds, namelyas (generalized) connections on fiber bundles with standard fiber S, where the structuregroup is the whole diffeomorphism group. But some of the properties of connectionsremain true for still more general situations: in the main part of this paper a connectionwill be just a fiber projection onto a (not necessarily integrable) distribution or sub vector1991 Mathematics Subject Classification.

53C05, 58A10.Key words and phrases. connections, graded derivations, Fr¨olicher-Nijenhuis bracket.Typeset by AMS-TEX1

2PETER W. MICHORbundle of the tangent bundle. Here curvature is complemented by cocurvature and theBianchi identity still holds.

In this situation we determine the graded Lie algebra of allgraded derivations over the horizontal projection of a connection and we determine theircommutation relations. Finally, for a principal connection on a principal bundle and theinduced connection on an associated bundle we show how one may pass from one to theother.

The final results relate derivations on vector bundle valued forms and derivationsover the horizontal projection of the algebra of forms on the principal bundle with valuesin the standard vector space.We use [6] and [7] as standard references: even the notation is the same. Some formulasof [6, section 1] can also be found in the original paper [1].1.

Connections1.1. Let (E, p, M, S) be a fiber bundle: So E, M and S are smooth manifolds, p : E →Mis smooth and each x ∈M has an open neighborhood U such that p−1(x) =: E|U is fiberrespectingly diffeomorphic to U × S. S is called the standard fiber.

The vertical bundle,called V E or V (E), is the kernel of Tp : TE →TM.It is a sub vector bundle ofTE →E.A connection on (E, p, M, S) is just a fiber linear projection φ : TE →V E; so φ is a1-form on E with values in the vertical bundle, φ ∈Ω1(E; V E). The kernel of φ is calledthe horizontal subspace kerφ.

Clearly for each u ∈E the mapping Tup : kerφ →Tp(u)Mis a linear isomorphism, so (Tp, πE) : kerφ →TM ×M E is a diffeomorphism, fiberlinear in the first coordinate, whose inverse χ : TM ×M E →kerφ →TE is calledthe horizontal lifting. Clearly the connection φ can equivalently be described by givinga horizontal sub vector bundle of TE →E or by specifying the horizontal lifting χsatisfying (Tp, πE) ◦χ = IdT M×M E.The notion of connection described here is thoroughly treated in [7].

There one canfind parallel transport, which is not globally defined in general, holonomy groups, holo-nomy Lie algebras, a method for recognizing G-connections on associated fiber bundles,classifying spaces for fiber bundles with fixed standard fiber S and universal connectionsfor this. Here we want to treat a more general concept of connection.1.2.

Let M be a smooth manifold and let F be a sub vector bundle of its tangent bundleTM. Bear in mind the vertical bundle over the total space of a principal fiber bundle.Definition.

A connection for F is just a smooth fiber projection φ : TM →F which weview as a 1-form φ ∈Ω1(M; TM).So φx : TxM →Fx is a linear projection for all x ∈M. kerφ =: H is a sub vectorbundle of constant rank of TM.

We call F the vertical bundle and H the horizontalbundle.h := IdT M −φ is then the complementary projection, called the horizontalprojection. A connection φ as defined here has been called an almost product structureby Guggenheim and Spencer.Let Ωhor(M) denote the space of all horizontal differential forms {ω : iXω = 0 for X ∈F}.

Despite its name, this space depends only on F, not on the choice of the connectionφ. Likewise, let Ωver(M) be the space of vertical differential forms {ω : iXω = 0 for X ∈H}.

This space depends on the connection.1.3. Curvature.

Let[,] : Ωk(M; TM) × Ωl(M; TM) −→Ωk+l(M; TM)

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS3be the Fr¨olicher-Nijenhuis-bracket as explained in [6] or in the original [1]. It inducesa graded Lie algebra structure on Ω(M; TM) := L Ωk(M; TM) with one-dimensionalcenter generated by Id ∈Ω1(M; TM).

For K, L ∈Ω1(M; TM) we have (see [1] or [6,1.9])[K, L](X, Y ) = [K(X), L(Y )] −[K(Y ), L(X)]−L([K(X), Y ] −[K(Y ), X])−K([L(X), Y ] −[L(Y ), X])+ (L ◦K + K ◦L)([X, Y ]).From this formula it follows immediately that[φ, φ] = [h, h] = −[φ, h] = 2(R + ¯R),where R ∈Ω2hor(M; TM) and ¯R ∈Ω2ver(M; TM) are given by R(X, Y ) = φ[hX, hY ] and¯R(X, Y ) = h[φX, φY ], respectively. Thus R is the obstruction against integrability ofthe horizontal bundle H; R is called the curvature of the connection φ.

Likewise ¯R is theobstruction against integrability of the vertical bundle F; we call ¯R the cocurvature of φ. [φ, φ] = 2(R + ¯R) has been called the torsion of φ by Fr¨olicher and Nijenhuis.1.4.

Lemma. (Bianchi-Identity) [R + ¯R, φ] = 0 and [R, φ] = i(R) ¯R + i( ¯R)RProof:.

We have 2[R + ¯R, φ] = [[φ, φ], φ] = 0 by the graded Jacobi identity. For thesecond equation we use the Fr¨olicher-Nijenhuis operators as explained in [6,section1].2R = φ.

[φ, φ] = i([φ, φ])φ, and by [6,1.10.2] we have i([φ, φ])[φ, φ] = [i([φ, φ])φ, φ] −[φ, i([φ, φ])φ] + i([φ, [φ, φ]])φ + i([φ, [φ, φ]])φ = 2[i([φ, φ])φ, φ] = 4[R, φ].So [R, φ] =14i([φ, φ])[φ, φ] = i(R + ¯R)(R + ¯R) = i(R) ¯R + i( ¯R)R, since R has vertical values and killsvertical values, so i(R)R = 0; likewise for ¯R.□2. Graded derivations for a connection2.1.. We begin with some algebraic preliminaries.

Let A = Lk∈Z Ak be a graded com-mutative algebra and let I : A →A be an idempotent homomorphism of graded al-gebras, so we have I(Ak) ⊂Ak, I(a.b) = I(a).I(b) and I ◦I = I. A linear mappingD : A →A is called a graded derivation over I of degree k, if D : Aq →Aq+k andD(a.b) = D(a).I(b) + (−1)k.|a|I(a).D(b), where | a | denotes the degree of a.Lemma.

If Dk and Dl are derivations over I of degree k and l respectively, and iffurthermore Dk and Dl both commute with I, then the graded commutator [Dk, Dl] :=Dk ◦Dl −(−1)k.lDl ◦Dk is again a derivation over I of degree k + l.The space DerI(A) = L DerIk(A) of derivations over I which commute with I is agraded Lie subalgebra of (End(A), [,] ).The proof is a straightforward computation.2.2.. Let M be a smooth manifold and let F be a sub vector bundle of TM as consideredin section 1. Let φ be a connection for F and consider its horizontal projection h : TM →H.

We define h∗: Ω(M) →Ωhor(M) by(h∗ω)(X1, . .

. , Xp) := ω(hX1, .

. .

, hXp).Then h∗is a surjective graded algebra homomorphism and h∗|Ωhor(M) = Id.Thush∗: Ω(M) →Ωhor(M) →Ω(M) is an idempotent graded algebra homomorphism.

4PETER W. MICHOR2.3. Let now D be a derivation over h∗of Ω(M) of degree k. Then D|Ω0(M) might benonzero, e.g.

h∗◦d. We call D algebraic, if D|Ω0(M) = 0.

Then D(f.ω) = 0 + f.D(ω),so D is of tensorial character.Since D is a derivation, it is uniquely determined byD|Ω1(M). This is given by a vector bundle homomorphism T ∗M →Λk+1T ∗M, whichwe view as K ∈Γ(Λk+1 ⊗TM) = Ωk+1(M; TM).

We write D = ih(K) to express thisdependence.Lemma. 1.

We have(ih(K)ω)(X1, . .

. , Xk+p) ==1(k+1)!

(p−1)!Xσε(σ).ω(K(Xσ1, . .

. , Xσ(k+1)), hXσ(k+2), .

. .

, hXσ(k+p),and for any K ∈Ωk+1(M; TM) this formula defines a derivation over h∗.2. We have [ih(K), h∗] = ih(K)◦h∗−h∗◦ih(K) = 0 if and only if h◦K = K ◦Λk+1h :Λk+1TM →TM.

We write Ωk+1(M; TM)h for the space of all h-equivariant forms likethat.3. For K ∈Ωk+1(M; TM)h and L ∈Ωl+1(M; TM)h we have: [K, L]∧,h := ih(K)L −(−1)kl ih(L)K is an element of Ωk+l+1(M; TM)h and this bracket is a graded Lie algebrastructure on Ω∗+1(M; TM)h such that ih([K, L]∧,h) = [ih(K), ih(L)] in DerhΩ(M).4.

For K ∈Ωk+1(M; TM) and the usual insertion operator [6, 1.2] we have:h∗◦i(K) = ih(K ◦Λk+1h) = h∗◦ih(K)i(K) ◦h∗= ih(h ◦K) = ih(K) ◦h∗[i(K), h∗] = [ih(K), h∗] = ih(h ◦K −K ◦Λk+1h)h∗◦i(K) ◦h∗= h∗◦ih(K) ◦h∗= ih(h ◦K ◦Λk+1h).Proof. 1.

We first need the following assertion: For ωj ∈Ω1(M) we have(1)ih(K)(ω1 ∧. .

. ∧ωp) =pXj=1(−1)(j+1).k(ω1 ◦h) ∧.

. .

∧(ωj ◦K) ∧. .

. ∧(ωp ◦h).This follows by induction on p from the derivation property.

From (1) we get(2)ih(K)(ω1 ∧. .

. ∧ωp)(X1, .

. .

, Xk+p) ==pXj=1(−1)(j−1)k(ω1 ◦h) ∧. .

. ∧(ωj ◦K) ∧.

. .

∧(ωp ◦h)(X1, . .

. , Xk+p) ==Xj(−1)(j−1)k1(k+1)!Xσε(σ)ω1(hXσ1) .

. .ωjK(Xσj, .

. .

, Xσ(j+k)). .

.. . .

ωp(hXσ(k+p)).Now we consider the following expression:(3)Xπ∈Sk+pε(π). (ω1 ∧.

. .

∧ωp)K(Xπ1, . .

. , Xπ(k+1)), hXπ(k+2), .

. .

, hXπ(k+p)==Xπε(π)Xρ∈Spε(ρ)ωρ1K(Xπ1, . .

. , Xπ(k+1))ωρ2(hXπ(k+2)) .

. .ωρp(hXπ(k+p)),

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS5where we sum over π ∈Sk+p and ρ ∈Sp.Reshuffling these permutations one maycheck that1(k+1)! (p−1)!

times expression (3) equals expression (2). By linearity assertion1 follows.2.For ω ∈Ω1(M) we have ω ◦h ◦K = ih(K)h∗ω = h∗ih(K)ω = h∗(ω ◦K) =ω ◦K ◦Λk+1h.3.

We have [ih(K), ih(L)] = ih([K, L]∧,h) for a unique [K, L]∧,h ∈Ωk+l+1(M; TM)hby Lemma 2.1. For ω ∈Ω1(M) we get thenω ◦[K, L]∧,h = ih([K, L]∧,h)ω = [ih(K), ih(L)]ω == ih(K)ih(L)ω −(−1)klih(L)ih(K)ω = ih(K)(ω ◦L) −(−1)klih(L)(ω ◦K).So by 1 we getih(K)(ω ◦L)(X1, .

. .

, Xk+l+1) ==1(k+1)! l!Xσ∈Sk+l+1ε(σ)(ω ◦L)K(Xσ1, .

. .

, Xσ(k+l+1)), hXσ(k+2), . .

. , hXσ(k+l+1))and similarly for the second term.4.

All these mappings are derivations over h∗of Ω(M). So it suffices to check that theycoincide on Ω1(M) which is easy for the first two assertions.

The latter two assertionsare formal consequences thereof.□2.4. For K ∈Ωk(M; TM) we have the Lie derivation Θ(K) = [i(K), d] ∈DerkΩ(M),where d denotes exterior derivative.

See [6, 1.3].Proposition. 1.

Let D be a derivation over h∗of Ω(M) of degree k. Then there areunique elements K ∈Ωk(M; TM) and L ∈Ωk+1(M; TM) such thatD = Θ(K) ◦h∗+ ih(L).D is algebraic if and only if K = 0.2. If D is in DerhkΩ(M) (so [D, h∗] = 0) thenD = h∗◦Θ(K) ◦h∗+ ih(˜L)for unique K ∈Ωkhor(M; TM) and ˜L ∈Ωk+1(M; TM)h. K is the same as in 1.Define Θh(K) := h∗◦Θ(K) ◦h∗∈DerhkΩ(M), then in 2 we can write D = Θh(K) +ih(˜L).Proof.

Let Xj ∈X(M) be vector fields and consider the mapping ev(X1,... ,Xk) ◦D :C∞(M) = Ω0(M) →Ωk(M) →Ω0(M) = C∞(M), given by f 7→(Df)(X1, . .

. , Xk).This map is a derivation of the algebra C∞(M), since we have D(f.g)(X1, .

. .

, Xk) =(Df.g + f.Dg)(X1, . .

. , Xk) = (Df)(X1, .

. .

, Xk).g + f.(Dg)(X1, . .

. , Xk).

So it is givenby the action of a unique vector field K(X1, . .

. , Xk), which clearly is an alternatingand C∞(M)-multilinear expression of the Xj; we may thus view K as an element ofΩk(M; TM).The defining equation for K is Df = df ◦K for f ∈C∞(M).Nowwe consider D −[i(K), d] ◦h∗.

This is a derivation over h∗and vanishes on Ω0(M),

6PETER W. MICHORsince [i(K), d]h∗f = [i(K), d]f = df ◦K = Df.It is algebraic and by 2.3 we haveD −[i(K), d] ◦h∗= ih(L) for some unique L ∈Ωk+1(M; TM).Now suppose that D commutes with h∗, i. e. D ∈Derhk(M). Then for all f ∈C∞(M)we have df ◦K ◦Λkh = h∗(df ◦K) = h∗Df = Dh∗f = Df = df ◦K, so K ◦Λkh = Kor h∗K = K and K is in Ωkhor(M; TM).Now let us consider D −h∗[i(K), d]h∗∈DerhkΩ(M), which is algebraic, since h∗[i(K), d]h∗f = h∗(df ◦K) = df ◦(h∗K) =df ◦K = Df.

So D −h∗[i(K), d]h∗= ih(˜L) for some unique ˜L ∈Ωk+1(M; TM)h by 2.3again.□2.5.. For the connection φ (respectively. the horizontal projection H) we define theclassical covariant derivative Dh := h∗◦d.

Then clearly Dh : Ω(M) →Ωhor(M) andDh is a derivation over h∗, but Dh does not commute with h∗, so it is not an elementof Derh1Ω(M). Therefore and guided by 2.4 we define the (new) covariant derivative asdh := h∗◦d◦h∗= Dh ◦h∗= Θh(Id).

We will consider dh as the most important elementin Derh1Ω(M).Proposition.1. dh −Dh = ih(R), where R is the curvature.2.

[d, h∗] = Θ(φ) ◦h∗+ ih(R + ¯R).3. d ◦h∗−dh = Θ(φ) ◦h∗+ ih( ¯R).4. Dh ◦Dh = ih(R) ◦d.5.

[dh, dh] = 2dh ◦dh = 2ih(R) ◦d ◦h∗= 2h∗◦i(R) ◦d ◦h∗.6. Dh|Ωhor(M) = dh|Ωhor(M).Assertion 6 shows that not a lot of differential geometry, on principal fiber bundlese.g., is changed if we use dh instead of Dh.

In this paper we focus our attention on dh.Proof. 1 is 2 minus 3.

Both sides of 2 are derivations over h∗and it is straightforwardto check that they agree on Ω0(M) = C∞(M) and Ω1(M). So they agree on the wholeof Ω(M).

The same method proves equation 3. The rest is easy.□2.6.

Theorem. Let K be in Ωk(M; TM).

Then we have:1. If K ∈Ωk(M; TM)h, then Θh(K) = [ih(K), dh].2. h∗◦Θ(K) = h∗◦Θ(K ◦Λkh) + (−1)k−1ih(ih(R)K).3.

Θh(K) = Θh(h∗K) + (−1)k−1ih(ih(R)(h ◦K)).4. If K ∈Ωk(M; TM)h, then Θh(K) = Θh(h∗K) = Θh(h ◦K).Now let K ∈Ωkhor(M; TM).

Then we have:5. h∗◦Θ(K) −Θh(K) = ih(φ ◦[K, φ] ◦Λh).6. Θ(K) ◦h∗−Θh(K) = ih((−1)ki( ¯R)(h ◦K) −h ◦[K, φ]).7.

[h∗, Θ(K)] = ih(φ ◦[K, φ] ◦Λh + h ◦[K, φ] −(−1)ki( ¯R)(h ◦K)).8. Let Kj ∈Ωkjhor(M; TM), j = 1, 2.

Then[Θh(K1), Θh(K2)] = Θh([K1, K2])−−ihh ◦[K1, φ ◦[K2, φ]] ◦Λh −(−1)k1k2h ◦[K2, φ ◦[K1, φ]] ◦Λh.

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS7Note. The third term in 8 should play some rˆole in the study of deformations of gradedLie algebras.Proof: 1.

Plug in the definitions.2. Check that h∗Θ(K)f = h∗Θ(h∗K)f for f ∈C∞(M).

For ω ∈Ω1(M) we geth∗Θ(K)ω = h∗ih(h∗K)dω −(−1)k−1h∗di(K)ω, using 2.3.4; also we have h∗Θ(h∗K)ω =h∗ih(h∗K)dω −(−1)k−1h∗dh∗i(K)ω. Collecting terms and using 2.5.1 one obtains theresult.3 follows from 2 and 2.3.4.4 is easy.5.It is easily checked that the left hand side is an algebraic derivation over h∗.

So itsuffices to show that both sides coincide if applied to any ω ∈Ω1(M). This will be doneby induction on k. The case k = 0 is easy.

Suppose k ≥1 and choose X ∈X(M). Thenwe have in turn:i(X)h∗Θ(K)ω = ih(hX)Θ(K)ω = h∗i(hX)Θ(K)ω= h∗[i(hX), Θ(K)]ω + (−1)kh∗Θ(K)i(hX)ω= h∗Θ(i(hX)K)ω + (−1)kh∗i([hX, K])ω + (−1)kh∗Θ(K)(ω(hX))i(X)h∗Θ(K)h∗ω = i(X)h∗Θ(K)(ω ◦h)= h∗Θ(i(hX)K)(ω ◦h) + (−1)kh∗i([hX, K])(ω ◦h) + (−1)kh∗Θ(K)(ω(hX))i(X)h∗Θ(K) −h∗Θ(K)h∗ω=h∗Θ(i(hX)K) −h∗Θ(i(hX)K)h∗ω + (−1)kh∗i([hX, K])(ω ◦φ)= ihφ ◦[i(hX)K, φ] ◦Λhω + (−1)kh∗i([hX, K])(ω ◦φ)by induction on k.i(X)ihφ ◦[K, φ] ◦Λhω = i(X)ω ◦φ ◦[K, φ] ◦Λh= (ω ◦φ)i(hX)[K, φ]) ◦Λh= (ω ◦φ)[i(hX)K, φ] + (−1)k[K, i(hX)φ = 0] −(−1)ki([K, hX])φ++ (−1)k−1i([φ, hX])K◦Λh,by [6, 1.10.2]= ihφ ◦[i(hX)K, φ] ◦Λh+ (−1)kh∗i([hX, K])(ω ◦φ)++ (−1)kω ◦φ ◦(i([φ, hX])K) ◦Λh.Now [φ, hX](Y ) = [φY, hX] −φ([Y, hX]) by [6,1.9].Therefore (i([φ, hX])K) ◦Λh =i(h◦[φ, hX])K ◦Λh = 0, and i(X)h∗◦Θ(K)−h∗◦Θ(K)◦h∗ω = i(X)ih(φ◦[K, φ]◦Λh)ωfor all X ∈X(M).

Since the i(X) jointly separate points, the equation follows.6. K horizontal implies that Θ(K) ◦h∗−h∗◦Θ(K) ◦h∗is algebraic.

So again itsuffices to check that both sides of equation 6 agree when applied to an arbitrary 1-formω ∈Ω1(M). This will again be proved by induction on k. We start with the case k = 0:Let K = X and Y ∈X(M).Θ(X)h∗ω(Y ) −h∗Θ(X) h∗ω(Y ) ==Θ(X)(ω ◦h)(Y −hY ) = X.ω(hφY ) −(ω ◦h)([X, φY ]) = 0 −ω(h[X, φY ])Note that [X, φ](Y ) = [X, φY ]−φ[X, Y ], so h[X, φ](Y ) = h[X, φY ], and thereforeih(0−h[X, φ])ω(Y ) = −ω(h[X, φY ]).Now we treat the case k > 0.

For X ∈X(M) we have in turn:i(X)Θ(K)h∗ω =(−1)kΘ(K)i(X) + Θ(i(X)K) + (−1)ki([X, K])(h∗ω)by [6,1.6]= (−1)kΘ(K)(ω(hX)) + Θ(i(X)K)h∗ω + (−1)kω(h[X, K]).

8PETER W. MICHORWe noticed above that Θ(K)f = h∗Θ(K)f for f ∈C∞(M) since K is horizontal.i(X)h∗Θ(K)h∗ω = ih(hX)Θ(K)h∗ω = h∗i(hX)Θ(K)h∗ω,by 2.3.4.= (−1)kh∗Θ(K)(ω(hX)) + h∗Θ(i(hX)K)h∗ω + (−1)kh∗(ω(h[X, K])),see above.i(X)Θ(K)h∗ω −h∗Θ(K)h∗ω== Θ(i(X)K)h∗ω −h∗Θ(i(hX)K)h∗ω + (−1)kω(h[X, K]) −h∗(ω ◦h ◦[hX, K])== −ih(h[i(X)K, φ])ω + (−1)k−1ih(h ◦i( ¯R)i(X)K)ω++ (−1)kω(h ◦[X, K]) −(−1)kω(h ◦(h∗[hX, K])),by induction, where we also used i(X)K = i(hX)K, which holds for horizontal K.−i(X)ih(h ◦[K, φ])ω = −i(X)(ω ◦h ◦[K, φ]) = ω ◦h ◦(i(X)[K, φ]) == ω ◦h ◦[i(X)K, φ] + (−1)k[K, i(X)φ] −(−1)ki([K, X])φ + (−1)k−1i([φ, X])K== −ih(h ◦[i(X)K, φ])ω + (−1)kω(h ◦[φX, K]) + 0 + (−1)kω(h ◦(i([φ, X])K),where we used [6, 1.10.2] and h ◦φ = 0.i(X)ih(h ◦i( ¯R)K)ω = i(X)(ω ◦h ◦i( ¯R)K) = ω ◦h ◦(i(X)i( ¯R)K).i(X)(left hand side −right hand side)ω == ω(−1)k−1h ◦i( ¯R) ◦i(X) ◦K + (−1)kh ◦[X, K] −(−1)kh ◦(h∗[hX, K])−−(−1)kh ◦[φX, K] −(−1)kh ◦i([φ, X])K −(−1)kh ◦(i(X)i( ¯R)K)= (−1)k(ω ◦h)−i([ ¯R, X]∧)K + [X −φX, K] −h∗[hX, K] −i([φ, X])K== (−1)k(ω ◦h)i(0 −i(X) ¯R)K + [hX, K] + h∗[hX, K] −i([φ, X])K−hi(i(X) ¯R)K(Y1, . .

. , Yk) = −kXj=1hK(Y1, .

. .

, ¯R(X, Yj), . .

. , Yk) == −kXj=1hK(Y1, .

. .

, [φX, φYj], . .

. , Yk), since K is horizontal.−h[K, hX]((Y1, .

. .

, Yk) = −h[K(Y1, . .

. , Yk), hX] +kXj=1hK(Y1, .

. .

, [Yj, hX], . .

. , Yk).h(h∗[K, hX])(Y1, .

. .

, Yk) = h[K, hX](hY1, . .

. , hYk) == h[K(hY1, .

. .

, hYk), hX] −XhK(hY1, . .

. , [hYj, hX], .

. .

, hYk) == h[K(Y1, . .

. , Yk), hX] −XhK(Y1, .

. .

, [hYj, hX], . .

. , Yk),since K is horizontal.

We also have h[φ, X](Y ) = h([φY, X] −φ[Y, X]) = h[φY, X], andthus we finally get−h(i([φ, X]K)(Y1, . .

. , Yk) = −h(i(h[φ, X])K)(Y1, .

. .

, Yk) == −XhK(Y1, . .

. , h[φ, X](Yj), .

. .

, Yk) = −XhK(Y1, . .

. , [φYj, X], .

. .

, Yk).

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS9All these sum to zero and equation 6 follows.7.This is equation 5 minus equation 6.8. We compute as follows:[Θh(K1), Θh(K2)] == h∗Θ(K1)h∗Θ(K2)h∗−(−1)k1k2h∗Θ(K2)h∗Θ(K1)h∗==h∗Θ(K1) −h∗i(φ ◦[K1, φ])θ(K2)h∗−−(−1)k1k2h∗Θ(K2) −h∗i(φ ◦[K2, φ])Θ(K1)h∗,by 5,= h∗Θ(K1)Θ(K2) −(−1)k1k2Θ(K2)Θ(K1)h∗−−h∗i(φ ◦[K1, φ])Θ(K2) −(−1)k1k2i(φ ◦[K2, φ])Θ(K1)h∗== h∗Θ([K1, K2])h∗+ Remainder = Θh([K1, K2]) + Remainder.For the following note that i(φ ◦[K1, φ])h∗= ih(h ◦φ ◦[K1, φ]) = 0, by 2.3.4.Remainder = −h∗i(φ ◦[K1, φ])Θ(K2)h∗+ (−1)k1k2i(φ[K2, φ])Θ(K1)h∗++ (−1)k1k2h∗Θ(K2)i(φ ◦[K1, φ])h∗−h∗Θ(K1)i(φ ◦[K2, φ])h∗,which are 0,= −h∗[i(φ ◦[K1, φ]), Θ(K2)]h∗+ (−1)k1k2h∗[i(φ ◦[K2, φ]), Θ(K1)]h∗== h∗Θ(i(φ ◦[K1, φ])K2 = 0) + (−1)k2i([φ ◦[K1, φ], K2])h∗++ (−1)k1k2h∗Θ(i(φ ◦[K2, φ])K1 = 0) + (−1)k1i([φ ◦[K2, φ], K1)h∗== −ihh ◦[K1, φ ◦[K2, φ]] ◦Λh −(−1)k1k2h ◦[K2, φ ◦[K1, φ]] ◦Λh,by 2.3.4.2.7.

Corollary. (1) h∗◦Θ(φ) = ih(R).

(2) Θh(φ) = 0. (3) [h∗, Θ(h)] = −2ih(R) −ih( ¯R).

(4) h∗◦Θ(h) = Θh(h) −2ih(R). (5) Θh(h) + ih( ¯R) = Θ(h) ◦h∗.

(6) [Θh(h), Θh(h)] = 2Θh(R) = 2h∗◦i(R) ◦d ◦h∗. (7) Θh( ¯R) = 0.Proof.

1 follows from 2.6.2. 2 follows from 2.6.3.

3 follows from 2.6.7. 4 follows from2.6.5.5 is 4 minus 3.6 follows from 2.6.8 and some further computation and 7 issimilar.□2.8.

Theorem: 1. For K ∈Ωk(M; TM) and L ∈Ωl+1(M; TM)h we have[ih(L), Θh(K)] = Θh(i(L)K) + (−1)kih(h ◦[L, K] ◦Λh).2.For L ∈Ωl+1(M; TM)h and Ki ∈Ωki(M; TM) we haveih(L)(h∗[K1, K2]) = h∗[i(L)K1, K2] + (−1)k1lh∗[K1, i(L)K2]−−(−1)k1lih(h∗[K1, L])K2 + (−1)(k1+l)k2ih(h∗[K2, L])K1.

10PETER W. MICHOR3.For L ∈Ωl+1(M; TM)h and Ki ∈Ωkihor(M; TM) we haveih(L)(h∗[K1, K2]) = h∗[ih(L)K1, K2] + (−1)k1lh∗[K1, ih(L)K2]−−(−1)k1lih(h∗[K1, L])K2 + (−1)(k1+l)k2ih(h∗[K2, L])K1.4.For K ∈Ωkhor(M; TM) and Li ∈Ωli+1(M; TM)h we haveh ◦[K, [L1, L2]∧,h] ◦Λh == [h ◦[K, L1] ◦Λh, L2]∧,h + (−1)kl1[L1, h ◦[K, L2] ◦Λh]∧,h−−(−1)kl1h ◦[ih(L1)K, L2] ◦Λh −(−1)(l1+k)l2h ◦[ih(L2)K, L1] ◦Λh.5.Finally for Ki ∈Ωki(M; TM) and Li ∈Ωki+1(M; TM)h we have[Θh(K1) + ih(L1), Θh(K2) + ih(L2)] == Θh[K1, K2] + i(L1)K2 −(−1)k1k2i(L2)K1++ ih[L1, L2]∧,h + (−1)k2h ◦[L1, K2] ◦Λh −(−1)k1(k2+1)h ◦[L,K1] ◦Λh−−h ◦[K1, φ ◦[K2, φ]] ◦Λh + (−1)k1k2h ◦[K2, φ ◦[K1, φ]] ◦Λh.Proof: 1. [ih(L), Θh(K)] = ih(L)h∗Θ(K)h∗−(−1)klh∗Θ(K)h∗ih(L) == h∗i(L)Θ(K) −(−1)klΘ(K)i(L)h∗,by 2.3.4.= h∗Θ(i(L)K) + (−1)ki([L, K])h∗,by [6,1.6]= Θh(i(L)K) + (−1)kih(h ◦[L, K] ◦Λh),by 2.3.4 and 2.4.2.ih(L)(h∗[K1, K2]) = h∗i(L)[K1, K2] by 2.3.4 and the rest follows from [6, 1.10].3.Note that for horizontal K we have i(L)K = ih(L)K and use this in formula 2.4.This follows from 1 by writing out the graded Jacobi identity for the gradedcommutators, uses horizontality of K and 2.3 several times.5.Collect all terms from 2.6.8 and 1.□2.9.

The space of derivations over h∗of Ω(M) is a graded module over the graded algebraΩ(M) with the action (ω ∧D)ψ = ω ∧Dψ. The subspace DerhΩ(M) is stable underω ∧· if and only if ω ∈Ωhor(M).Theorem.

1. For derivations D1, D2 over h∗of degree k1, k2, respectively, and ω ∈Ωqhor(M) we have [ω ∧D1, D2] = ω ∧[D1, D2] −(−1)(q+k1)k2D2ω ∧h∗D1.2.

For L ∈Ω(M; TM) and ω ∈Ω(M) we have ω ∧ih(L) = ih(ω ∧L).3. For K ∈Ωk(M) and ω ∈Ωq(M) we have(ω ∧Θ(K)) ◦h∗= Θh(ω ∧K) ◦h∗+ (−1)q+k−1ih(dω ∧(h ◦K)).4.

For K ∈Ωkhor(M; TM) and ω ∈Ωqhor(M) we haveω ∧Θh(K) = Θh(ω ∧K) + (−1)q+k−1ih(dhω ∧(h ◦K)).5. Ω(M; TM)h is stable under multiplication with ω ∧.

if and only if ω ∈Ωhor(M).For Lj ∈Ωlj+1(M; TM)h and ω ∈Ωqhor(M) we have[ω ∧L1, L2]∧,h = ω ∧[L1, L2]∧,h −(−1)(q+l1)l2 ih(L2)ω ∧(h ◦L1).The proof consists of straightforward computations.

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS113. Derivations on Principal FiberBundles and Associated Bundles.Liftings.3.1.

Let (E, p, M, S) be a fiber bundle and let φ ∈Ω1(M; TM) be a connection for itas described in 1.1. The cocurvature ¯R is then zero.

We consider the horizontal liftingχ : TM ×M E →TE and use it to define the mapping χ∗: Ωk(M; TM) →Ωk(E; TE)by(χ∗K)u(X1, . .

. , Xk) := χK(Tup.X1, .

. .

, Tup.Xk), u.Then χ∗K is horizontal with horizontal values: h ◦χ∗K = χ∗K = χ∗K ◦Λh.Forω ∈Ωq(M) we have χ∗(ω∧K) = p∗ω∧χ∗K, so χ∗: Ω(M; TM) →Ω(E; TE)h is a modulehomomorphism of degree 0 over the algebra homomorphism p∗: Ω(M) →Ωhor(E).Theorem. In this setting we have for K, Ki ∈Ω(M; TM):(1) p∗◦i(K) = i(χ∗K) ◦p∗= ih(χ∗K) ◦p∗: Ω(M) →Ωhor(E).

(2) p∗◦Θ(K) = Θ(χ∗K) ◦p∗= Θh(χ∗K) ◦p∗: Ω(M) →Ωhor(E). (3) i(χ∗K1)χ∗K2 = ih(χ∗K1)χ∗K2 = χ∗(i(K1)K2).

(4) The following is a homomorphism of graded Lie algebras:χ∗: (Ω(M; TM), [,]∧) −→(Ω(E; TE)h, [,]∧,h) ⊂(Ω(E; TE), [,]∧)(5) χ∗[K1, K2] = h ◦[χ∗K1, χ∗K2] = h ◦[χ∗K1, χ∗K2] ◦Λh.Proof. (1).

p∗◦i(K) and i(χ∗K) ◦p∗are graded module derivations : Ω(M) →Ω(E)over the algebra homomorphism p∗in a sense similar to 2.1: p∗i(K)(ω ∧ψ) = p∗i(K)ω ∧p∗ψ + (−1)(k−1)|ω|p∗ω ∧p∗i(K)ψ and similarly for the other expression. Both are zeroon C∞(M) and it suffices to show that they agree on ω ∈Ω1(M).This is an easycomputation.

Furthermore ih(χ∗K)|Ωhor(E) = i(χ∗K)|Ωhor(E), so the rest of 1 follows. (2) follows from (1) by expanding the graded commutators.

(3). Plug in the definitionsand use 2.3.1.

(4) follows from 2.3.3 and [6,1.2.2]. (5) follows from 2 and some furtherconsiderations.□3.2.

Let (P, p, M, G) be a principal fiber bundle with structure group G, and writer : P ×G →P for the principal right action. A connection φP : TP →V P in the sense of1.1 is a principal connection if and only if it is G-equivariant i.e., T(rg)◦φP = φP ◦T(rg)for all g ∈G.Then φPu = Te(ru).ϕu, where ϕ , a 1-form on P with values in theLie algebra of G, is the usual description of a principal connection.

Here we used theconvention r(u, g) = rg(u) = ru(g) for g ∈G and u ∈P.The curvature R of 1.3corresponds to the negative of the usual curvature dϕ + 12[ϕ, ϕ] and the Bianchi identityof 1.4 corresponds to the usual Bianchi identity.The G-equivariant graded derivations of Ω(P) are exactly the Θ(K) + i(L) with Kand L G-equivariant i.e., T(rg) ◦K = K ◦ΛT(rg) for all g ∈G. This follows from [6,2.1].

The G-equivariant derivations in DerhΩ(P) are exactly the Θh(K) + ih(L) withK ∈Ωhor(P; TP), L ∈Ω(P; TP)h and K,L G-equivariant. Theorem 3.1 can be appliedand can be complemented by G-equivariance.3.3.

In the situation of 2.2 let us suppose furthermore, that we have a smooth left actionof the structure group G on a manifold S, ℓ: G × S →S. We consider the associatedbundle (P[S] = P ×G S, p, M, S) with its G-structure and induced connection φP [S].

Let

12PETER W. MICHORq : P × S →P[S] be the quotient mapping, which is also the projection of a principalG-bundle. Then φP [S]Tq(Xu, Ys) = Tq(φP Xu, Ys) by [7, 2.4].Now we want to analyze q∗◦(Θh(K) + ih(L)) : Ω(P[S]) →Ω(P × S).

For that weconsider the associated bundle (P × S = P ×G (G × S), p ◦pr1, M, G × S, G), wherethe structure group G acts on G × S by left translation on G alone. Then the inducedconnection is φP ×S = φP × IdS : T(P × S) = TP × TS →V P × TS and we have thefollowingLemma.

Tq◦hP ×S = hP [S] ◦Tq and DhP ×S ◦q∗= q∗◦DhP [S] for the classical covariantderivatives.The proof is obvious from the description of induced connections given in [7, 2.4].3.4. Theorem.

In the situation of 3.3 we have:1. q∗◦(hP [S])∗= (hP ×S)∗◦q∗: Ω(P[S]) →Ω(P × S).2. For K ∈Ωk(P[S]; T(P[S])) we haveq∗◦ihP [S](K) = ihP ×S(χP ×S∗K) ◦q∗: Ω(P[S]) →Ωhor(P × S)q∗◦ΘhP [S](K) = ΘhP ×S(χP ×S∗K) ◦q∗: Ω(P[S]) →Ωhor(P × S).Proof.

1.Use lemma 3.3 and the definition of h∗in 2.2.For 2 use also 2.3.1 andTq ◦χ∗K = K ◦ΛTq. The last equation is easy.□3.5.

For a principal bundle (P, p, M, G) let ρ : G →GL(V ) be a linear representation ina finite dimensional vector space V . We consider the associated vector bundle (E = P ×GV = P[V ], p, M, V ), a principal connection φP on P and the induced G-connection on E,which gives rise to the covariant exterior derivative ∇E ∈Der1Ω(M; E) as investigatedin [6, section 3].

We also have the mapping q♯: Ωk(M; E) →Ωk(P, V ), given by(q♯Ψ)u(X1, . .

. , Xk) = qu−1(Ψp(u)(Tup.X1, .

. .

, Tup.Xk).Recall that qu : {u} × V →Ep(u) is a linear isomorphism. Then q♯is an isomorphism ofΩ(M; E) onto the subspace Ωhor(P, V )G of all horizontal and G-equivariant forms.Theorem.

1. q♯◦∇E = DhP ◦q♯= dhP ◦q♯.2. For K ∈Ωk(M; TM) we have:q♯◦i(K) = i(χP∗K) ◦q♯= ihP (χP∗K) ◦q♯q♯◦Θ∇E(K) = ΘhP (χP∗K) ◦q♯: Ω(M; E) →Ωhor(P, V )G.Proof.

1. A proof of this is buried in [3, p 76, p 115].

A global proof is possible, butit needs a more detailed description of the passage from φP to ∇E, e.g. the connectorK : TE →E.

We will not go into that here.2. q♯◦i(K) and i(χP ∗P) ◦q♯are both derivations from the ω(M)-module Ω(M; E)into the Ω(P)-module Ω(P, V ) over the algebra homomorphism q∗: Ω(M) →Ω(P).Both vanish on 0-forms.

Thus it suffices to check that they coincide on Ψ ∈Ω1(M; E),which may be done by plugging in the definitions. i(χP ∗K) and ihP (χP ∗K) coincideon horizontal forms, so the first assertion follows.

The second assertion follows from thefirst one, lemma 3.3, and [6, 3.7]. Note finally that q♯◦Θ∇E(K) ̸= Θ(χP ∗K) ◦q♯ingeneral.□

GRADED DERIVATIONS ASSOCIATED WITH CONNECTIONS133.6. In the setting of 3.6, for Ξ ∈Ωk(M; L(E, E)), let q♯Ξ ∈Ω(P; L(V, V )) be given by(q♯Ξ)u(X1, .

. .

, Xk) = qu−1 ◦Ξp(u)(Tup.X1, . .

. , Tup.Xk) ◦qu : V →V,where L(E, E) = P[L(V, V ), Ad ◦ρ] and q♯Ξ defined here coincides with that of 3.5.Furthermore recall from [6, section 3] that any graded Ω(M)-module homomorphismΩ(M; E) →Ω(M; E) of degree k is of the form µ(Ξ) for unique Ξ ∈Ωk(M; L(E, E)),where(µ(Ξ)Ψ)(X1, .

. .

, Xk+q) =1k! q!Xσε(σ).Ξ(Xσ1, .

. .

, Xσk).Ψ(Xσ(k+1), . .

. , Xσ(k+q)).Then any graded derivation D of the Ω(M)-module Ω(M; E) can be uniquely written inthe form D = Θ∇E(K) + i(L) + µ(Ξ).Theorem.

For Ξ ∈Ω(M; L(E, E)) we haveq♯◦µ(Ξ) = µ(q♯Ξ) ◦q♯= µhP (q♯Ξ) ◦q♯: Ω(M; E) →Ωhor(P, V )G,where for ξ ∈Ωk(P, L(V, V )) the module homomorphism µ(ξ) of Ω(P, V ) is as above forthe trivial vector bundle P × V →P and µh(ξ) is given by(µh(ξ)ω)(X1, . .

. , Xk+q) ==1k!

q!Xσ∈Sk+qε(σ)ξ(Xσ1, . .

. , Xσk)ω(hP Xσ(k+1), .

. .

, hP Xσ(k+q)).The proof is straightforward.References[1] A. Fr¨olicher, A. Nijenhuis, Theory of vector valued differential forms. Part I, Indagationes Math.18 (1956), 338-359.

[2] W. Greub, S. Halperin, R. Vanstone, Connections, curvature, and cohomology. Vol II, AcademicPress, 1973.

[3] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry. Vol.

I, J. Wiley, 1963. [4] I.Kolaˇr, P. W. Michor, Determination of all natural bilinear operators of the type of the Fr¨olicher-Nijenhuis bracket., Suppl.

Rendiconti Circolo Mat. Palermo, Series II No 16 (1987), 101-108.

[5] L. Mangiarotti, M. Modugno, Fibered spaces, Jet spaces and Connections for Field Theories, Pro-ceedings of the International Meeting on Geometry and Physics, Florence 1982, Pitagora, Bologna,1983. [6] P. W. Michor, Remarks on the Fr¨olicher-Nijenhuis bracket, Proceedings of the Conference on Dif-ferential Geometry and its Applications, Brno 1986, D. Reidel, 1987, pp.

198-220. [7] P. W. Michor, Gauge theory for diffeomorphism groups., Proceedings of the Conference on Differen-tial Geometric Methods in Theoretical Physics, Como 1987, K. Bleuler and M. Werner (eds), Kluwer,Dordrecht, 1988, pp.

345–371. [8] J. Monterde, A. Montesinos, Integral curves of derivations., to appear, J.

Global Analysis andGeometry.

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