C. Morales

M. Vilches

Morales C.

Vilches M.

Morales, C.

Vilches, M.

Tokyo University of Science Repository for Academic Resources / 東京理科大学学術リポジトリ

SUT Journal of MathematicsVol. 43, No. 1 (2007), 107–114A curvature form for pseudoconnectionsC. Morales and M. Vilches(Received January 8, 2007)Abstract. We obtain the curvature form F∇ = P ◦d∇◦∇−d∇◦P ◦∇+d∇◦∇◦Pfor a vector bundle pseudoconnection ∇, where d∇ is the exterior derivativeassociated to ∇. We use F∇ to obtain the curvature of ∇. We also provethat F∇ = 0 is a necessary (but not sufficient) condition for d∇ to be a chaincomplex. Instead we prove that F∇ = 0 and d∇ ◦ d∇ ◦∇ = 0 are necessary andsufficient conditions for d∇ to be a chain 2-complex, i.e., d∇ ◦ d∇ ◦ d∇ = 0.AMS 2000 Mathematics Subject Classification. Primary 53C05, Secondary55R25.Key words and phrases. Curvature Form, Pseudoconnection, Vector Bundle.§1. IntroductionLet M be a differentiable manifold. Denote by Ω0(M) the ring of C∞ realvalued maps in M . Denote by χ∞(M) and Ωk(M) respectively the Ω0(M)-modules of C∞ vector fields and k-forms defined on M , k ≥ 0. Denote byd : Ωk(M) → Ωk+1(M) the standard exterior derivation of k-forms of M .We denote by Hom(A,B) the set of homomorphisms from the modulus A tothe modulus B. If ξ is a real smooth vector bundle over M we denote byΩk(ξ) the set of ξ-valued k-forms of M , namely, Ω0(ξ) is the Ω0(M)-moduleof smooth sections of ξ and Ωk(ξ) = Ωk(M) ⊗ Ω0(ξ) for all k ≥ 1. If η isanother vector bundle over M we denote by HOM(ξ, η) the set of bundlehomomorphims from ξ to η over the identity. Every P ∈ HOM(ξ, η) inducesa homomorphism P ∈ Hom(Ω0(ξ),Ω0(η)) of Ω0(M) modules in the usualway. It also defines a homomorphism P ∈ Hom(Ωk(ξ),Ωk(η)) for all k ≥ 1 bysetting P (ω ⊗ s) = ω ⊗ P (s) at every generator ω ⊗ s ∈ Ωk(ξ).A pseudoconnection of a vector bundle ξ over M is an IR-linear map ∇ :Ω0(ξ) → Ω0(ξ) for which there is a bundle homomorphism P ∈ HOM(ξ, ξ)107108 C. MORALES AND M. VILCHEScalled the principal homomorphism of ∇ such that the Leibnitz rule belowholds:∇(f · s) = df ⊗ P (s) + f · ∇(s), ∀(f, s) ∈ Ω0(M)× Ω0(ξ).(See [2].) An ordinary connection is a pseudoconnection whose principal ho-momorphism is the identity. By classical arguments we shall associate to anypseudoconnection ∇ an exterior derivative, that is, a sequence of linear mapsd∇ : Ωk(ξ) → Ωk+1(ξ) which reduces to ∇ when k = 0 and satisfies a Leibnitzrule. We shall use it to define the curvature form F∇ : Ω0(ξ) → Ω2(ξ) as thealternating sumF∇ = P ◦ d∇ ◦ ∇ − d∇ ◦ P ◦ ∇+ d∇ ◦ ∇ ◦ P.Notice that F∇ above reduces to the classical curvature form if ∇ were anordinary connection. We shall prove that F∇ is a tensor, that is, F∇ ∈Hom(Ω0(ξ),Ω2(ξ)), and explain how to obtain the Abe’s curvature [2] fromF∇. We also prove that F∇ = 0 is a necessary (but not sufficient) condition ford∇ to be a chain complex. Instead we prove that F∇ = 0 and d∇ ◦ d∇ ◦∇ = 0are necessary and sufficient conditions for d∇ to be a chain 2-complex, i.e.,d∇ ◦ d∇ ◦ d∇ = 0.§2. ResultsLet ξ, η be real smooth vector bundles over a differentible manifold M . An O-derivative operator from ξ to η with principal homomorphism P ∈ HOM(ξ, η)is an IR-linear map ∇ : Ω0(ξ) → Ω0(η) satisfying the Leibnitz rule∇(f · s) = df ⊗ P (s) + f · ∇(s), ∀(f, s) ∈ Ω0(M)× Ω0(ξ).(See [1].) Notice that a pseudoconnection of ξ is nothing but an O-derivativeoperator from ξ to itself.As in [1] we denote by O(ξ, η;P ) the set whose elements are the O-derivativeoperators with principal homomorphism P from ξ to η. We even write O(ξ;P )instead of O(ξ, ξ;P ) and defineO(ξ, η) =⋃PO(ξ, η;P ) and O(ξ) =⋃PO(ξ;P ).Every α ∈ HOM(ξ, η) induces an alternating product∧α : Ωk(M)× Ωl(ξ) → Ωk+l(η)defined at the generators byβ ∧α (ω ⊗ s) = (β ∧ ω)⊗ α(s).A CURVATURE FORM FOR PSEUDOCONNECTIONS 109If η = ξ and α is the identity, then ∧α reduced to the ordinary alternatingproduct ∧ ([3]).Lemma 2.1. For every ∇ ∈ O(ξ, η;P ) there is a unique sequence of linearmaps d∇ : Ωk(ξ) → Ωk+1(η), k ≥ 0, satisfying the following properties:1. If k = 0, thend∇ = ∇.2. If k, l ≥ 0, ω ∈ Ωk(M) and S ∈ Ωl(ξ), thend∇(ω ∧ S) = dω ∧P S + (−1)kω ∧ d∇S.Proof. First define the map D∇ : Ωk(M)× Ω0(ξ) → Ωk+1(η) byD∇(ω, s) = dω ⊗ P (s) + (−1)kω ∧∇s, ∀(ω, s) ∈ Ωk(M)× Ω0(ξ).Clearly D∇ is linear and satisfiesD∇(f · ω, s) = D∇(ω, f · s)for all f ∈ Ω0(M) and (ω, s) ∈ Ωk(M)×Ω0(ξ). Therefore D∇ induces a linearmap d∇ : Ωk(ξ) → Ωk+1(η) whose value at the generator ω ⊗ s of Ωk(ξ) isgiven byd∇(ω ⊗ s) = dω ⊗ P (s) + (−1)kω ∧∇s.It follows that d∇ and ∇ coincide at the generators (for k = 0) by the Leibnitzrule of ∇. Therefore (1) holds. The proof of (2) follows as in [3]. This endsthe proof.The sequence d∇ in the lemma above will be refered to as the exteriorderivative of ∇ ∈ O(ξ, η). Next we state the following definition.Definition 2.2. Let ∇ be a pseudoconnection with principal homomorphismP on a vector bundle ξ. We define the following mapsE∇, F∇: Ω0(ξ) → Ω2(ξ), L∇: Ω0(ξ) → Ω1(ξ) and G∇: Ω0(ξ) → Ω3(ξ)by• E∇ = d∇ ◦ ∇;• F∇ = P ◦ d∇ ◦ ∇ − d∇ ◦ P ◦ ∇+ d∇ ◦ ∇ ◦ P ;• L∇ = P ◦ ∇ −∇ ◦ P ;• G∇ = d∇ ◦ d∇ ◦ ∇.110 C. MORALES AND M. VILCHESThe map F∇ will be refered to as the curvature form of ∇.The maps in the definition above are related by the expressionsF∇ = P ◦E∇ − d∇ ◦ L∇, G∇ = d∇ ◦ E∇.(2.1)As already observed the curvature form F∇ of a pseudoconnection ∇ reducesto the classical curvature form when ∇ is an ordinary connection [3].Theorem 2.3. If ∇ is a pseudoconnection of ξ, then F∇ ∈ Hom(Ω0(ξ),Ω2(ξ))and L∇ ∈ Hom(Ω0(ξ),Ω1(ξ)).Proof. It is not difficult to see that L∇ ∈ Hom(Ω0(ξ),Ω1(ξ)). On the otherhand,P (ω ∧ S) = ω ∧P S, ∀ω ∈ Ω1(M),∀S ∈ Ω1(ξ)soE∇(f · s) = df ∧ L∇(s) + f · E∇(s), ∀f ∈ Ω0(M),∀s ∈ Ω0(ξ).Then, (2.1) impliesF∇(f ·s) = P (E∇(f ·s)))−d∇(L∇(f ·s)) = P (df∧L∇(s)+f ·E∇(s))−d∇(f ·L∇(s))= df ∧P L∇(s)− df ∧P L∇(s) + f · F∇(s) = f · F∇(s)∀f ∈ Ω0(M), ∀s ∈ Ω0(ξ). Therefore F∇ ∈ Hom(Ω0(ξ),Ω2(ξ)) and we aredone. This ends the proof.Lemma 2.4. If ∇ is a pseudoconnection of a vector bundle ξ and i ≥ 0, thend∇ ◦ d∇ ◦ d∇(ω ⊗ s) = dω ∧ F∇(s) + (−1)iω ∧G∇(s)(2.2)for every generator ω ⊗ s ∈ Ωi(ξ).Proof. First notice thatd∇ ◦ d∇(ω ⊗ s) = d∇(d∇(ω ⊗ s)) = d∇(dω ⊗ P (s) + (−1)iω ∧∇s) == d2ω ⊗ P 2(s) + (−1)i+1dω ∧∇P (s) + (−1)i(dw ∧P ∇s++(−1)iω ∧ d∇(∇s)) = (−1)i[dω ∧ (P∇s−∇Ps) + (−1)iω ∧ d∇(∇s)].Therefored∇ ◦ d∇(ω ⊗ s) = ω ∧E∇(s) + (−1)idω ∧ L∇(s).Applying d∇ to this expression we get (2.2). The proof follows.A CURVATURE FORM FOR PSEUDOCONNECTIONS 111As is well known the curvature form F∇ of an ordinary connection ∇ mea-sures how the exterior derivative d∇ of ∇ deviates from being a chain complex,i.e., d∇◦d∇ = 0. Indeed, d∇ is a chain complex if and only if F∇ = 0. However,the analogous result for pseudoconnections is false in general by Proposition2.8 below. Despite we shall obtain a pseudoconnection version of this resultbased on the following definition.Definition 2.5. A pseudoconnection ∇ is called:1. strongly flat if E∇ = 0 and L∇ = 0,2. weakly flat if F∇ = 0 and G∇ = 0.For ordinary connections one has F∇ = E∇, L∇ = 0 and then the notionsof flatness above coincide with the classical flatness [3]. The exterior derivatived∇ of a pseudoconnection ∇ is said to be a chain 2-complex if d∇◦d∇ ◦d∇ = 0.With these definitions we have the following result.Theorem 2.6. A pseudoconnection ∇ is weakly flat (resp. strongly flat) ifand only if d∇ is a chain 2-complex (resp. chain complex).Proof. We only prove the result for weakly flat since the proof for strongly flatis analogous.Fix a pseudoconnection ∇ with principal homomorphism P on a vectorbundle ξ. If ∇ is weakly flat then d∇ is a chain 2-complex by (2.2) in Lemma2.4. Conversely, if d∇ is a chain 2-complex, then both d∇ ◦ d∇ ◦ d∇ andd∇ ◦ d∇ ◦ ∇ vanish hence ω ∧ F∇(s) = 0 for all exact form ω of M andall s ∈ Ω0(ξ) by (2.2) in Lemma 2.4. Since every form in M is locally aΩ0(M)-linear combination of alternating product of exact forms we obtainthat ω ∧ F∇(s) = 0 for all k-form ω of M (k ≥ 1) and all s ∈ Ω0(ξ). Fromthis we obtain that F∇ = 0 so ∇ is weakly flat. The proof follows.The following is a direct corollary of the above theorem.Corollary 2.7. If the exterior derivative d∇ of a pseudoconnection ∇ is achain complex, then F∇ = 0.The converse of the above corollary is false by the following proposition.Proposition 2.8. There is a pseudoconnection ∇ with F∇ = 0 such that d∇is not a chain complex.Proof. Choose a suitable vector bundle ξ over M = IR3, Φ2,Φ3 ∈ HOM(ξ, ξ)such that Φ3 ◦ Φ2 6= Φ2 ◦ Φ3 and three 1-forms ω1, ω2, ω3 ∈ Ω1(M) such thatω1 ∧ ω2 ∧ ω3 never vanishes. Define the map ∇ : Ω0(ξ) → Ω1(ξ) by∇s = ω1 ⊗ s + ω2 ⊗ Φ2(s) + ω3 ⊗ Φ3(s).112 C. MORALES AND M. VILCHESWe have that ∇ ∈ Hom(Ω0(ξ),Ω1(ξ)) therefore ∇ is a pseudoconnection withzero principal homomorphism so F∇ = 0. On the other hand, an straightfor-ward computation yieldsG∇(s) = (ω1 ∧ ω2 ∧ ω3)⊗ (Φ3 ◦ Φ2 − Φ2 ◦ Φ3)(s), ∀s ∈ Ω0(ξ)therefore G∇ 6= 0 and so∇ is not weakly flat. Then, d∇ cannot be a chain com-plex by Theorem 2.6 since a chain complex is necessarily a chain 2-complex.This ends the proof.To finish we explain how the Abe’s curvature [2] can be obtained from thecurvature form F∇. For this we need some short definitions (see [3]).Given a vector bundle ξ over M and k vector fields X1, · · · , Xk ∈ χ∞(M)we define the evaluation map EvX1,···,Xk : Ωk(ξ) → Ω0(ξ) by definingEvX1,···,Xk(ω ⊗ s) = w(X1, · · · , Xk) · sat each generator ω ⊗ s ∈ Ωk(ξ). If ∇ ∈ O(ξ) and X,Y ∈ χ∞(M) we define∇X : Ω0(ξ) → Ω0(ξ) by∇Xs = EvX(∇s)and F∇X,Y : Ω0(ξ) → Ω0(ξ) byF∇X,Y = EvX,Y (F∇(s)), ∀s ∈ Ω0(ξ).Theorem 2.9. If ∇ ∈ O(ξ;P ) thenF∇X,Y (s) = ∇X∇Y (Ps)−∇Y∇X(Ps)−∇XP (∇Y s) + P∇X∇Y s++∇Y P (∇Xs)− P∇Y∇Xs− P(∇[X,Y ]P (s)),for all X,Y ∈ χ∞(M) and all s ∈ Ω0(ξ).Proof. By definition we haveF∇X,Y (s) =(2.3)= EvX,Y (P (d∇(∇s)))−EvX,Y (d∇(P (∇s))) + EvX,Y (d∇(∇(Ps))).Let us compute the three sumands separated way. First of all if ω ⊗ s ∈Ω1(ξ) is a generator thenEvX,Y (d∇(ω ⊗ s)) = dw(X,Y ) · P (s)− ω(X) · ∇Y s + ω(Y ) · ∇Xs.(2.4)Now, as ∇s ∈ Ω1(ξ) and {ω⊗ s′ : (ω, s′) ∈ Ω1(M)×Ω0(ξ)} is a generatingset of Ω1(ξ) we obtain∇s =k∑r=1ωr ⊗ sr,(2.5)A CURVATURE FORM FOR PSEUDOCONNECTIONS 113for some (ωr, sr) ∈ Ω1(M)× Ω0(ξ), r = 1, · · · , k. Then (2.4) impliesEvX,Y (d∇(∇s)) =(2.6)=k∑r=1{dwr(X,Y ) · P (sr)− ωr(X) · ∇Y sr + ωr(Y ) · ∇Xsr}.On the other hand, (2.5) yields∇Xs =k∑r=1ωr(X) · srtherefore∇Y∇Xs =k∑r=1{d[ωr(X)](Y ) · P (sr) + ωr(Y ) · ωr(X) · ∇Y ss} .But ∇[X,Y ]s =k∑r=1ωr([X,Y ]) · sr, soP(∇[X,Y ]s)=k∑r=1ωr([X,Y ]) · P (sr)and thenEvX,Y (d∇(∇s)) = ∇X∇Y s−∇Y∇Xs− P(∇[X,Y ]s)(2.7)because of (2.6). Replacing s by P (s) in (2.7) we obtainEvX,Y (d∇(∇P (s))) = ∇X∇Y P (s)−∇Y∇XP (s)− P(∇[X,Y ]P (s)).(2.8)Besides (2.5) impliesP (∇s) =k∑r=1ωr ⊗ P (sr)thusEvX,Y (d∇(P∇s)) =k∑r=1EvX,Y (d∇(ωr ⊗ P (sr))) ==k∑r=1[dwr(X,Y ) · P2(sr)− ωr(X)∇XP (sr) + +ωr(Y ) · ∇XP (sr)]and then∇Y P (∇Xs) =k∑r=1{d[wr(X)](Y ) · P2(sr) + wr(X) · ∇Y P (sr)}.114 C. MORALES AND M. VILCHESAs P 2(∇[X,Y ]s)=k∑r=1ωr([X,Y ]) · P2(sr) we obtainEvX,Y (d∇(P (∇s))) = ∇XP (∇Y s)−∇Y P (∇Xs)− P2(∇[X,Y ]s).(2.9)As the maps P and EvX,Y commute we can apply P to (2.7) and use (2.3),(2.8), (2.9) to obtain the result.Remark 2.10. F∇X,Y (s) in Theorem 2.9 is the curvature K(∇)X,Y (s) definedin [2] p. 328.References[1] Abe, N., Geometry of certain first order differential operators and its applicationsto general connections, Kodai Math. J. 11 (1988), no. 2, 205–223.[2] Abe, N., General connections on vector bundles, Kodai Math. J. 8 (1985), 322–329.[3] Madsen, I., Tornehave, J., From calculus to cohomology. de Rham cohomologyand characteristic classes, Cambridge University Press, Cambridge, 1997.C. MoralesInstituto de Matematica, Universidade Federal do Rio de JaneiroP. O. Box 68530, 21945-970 Rio de Janeiro, BrazilM. VilchesDepartamento de Ana´lise Matema´tica, IME, Universidade do Estado do Rio de Janeiro20550-013, Rio de Janeiro, Brazil

A curvature form for pseudoconnections

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A curvature form for pseudoconnections

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