Natural affinors and torsion of connections on Weil like functors on double vector bundles

doi: 10.17951/a.2022.76.2.1-13 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA L U B L I N – P O L O N I A VOL. LXXVI, NO. 2, 2022 SECTIO A 1–13 MIROSLAV DOUPOVEC, JAN KUREK and WŁODZIMIERZ M. MIKULSKI Na u al a ino s and o sion o connec ions on Weil like unc o s on double ec o bundles Abs ac . We desc ibe comple ely all na u al a ino s on p oduc p ese ing gauge bundle unc o s on double ec o bundles. Nex , we s udy o sion o double-linea connec ions. 1. In oduc ion. We assume ha any mani old conside ed in he pape is Hausdo , second coun able, ini e dimensional, wi hou bounda y and smoo h (i.e. o class C∞). All maps be ween mani olds a e assumed o be smoo h (o class C∞). The concep o double ec o bundles was in oduced in [12] and u he s udied in [1, 7, 9], e c. The amewo k o double ec o bundles is con- enien o many cons uc ions like linea o ms, linea Poisson s uc u es, linea connec ions, e c. The equi alen concep o double ec o bundles can be ound in [10]. We ci e i in Sec ion 2 o he p esen no e. The gene al concep o gauge bundle unc o s can be ound in [5]. The concep o p oduc p ese ing gauge bundle unc o s (ppgb- unc o s) on he ca ego y o double ec o bundles can be ound in [10], oo. We ci e i in Sec ion 3. 2010 Ma hema ics Subjec Classi ica ion. 58A20, 53A55. Key wo ds and ph ases. Double ec o bundle, p oduc p ese ing gauge bundle unc- o , na u al a ino , o sion o double-linea connec ion. The i s au ho was suppo ed by he p ojec no. FSI-S-23-8161. 2 M. Doupo ec, J. Ku ek and W. M. Mikulski In [10], i is p o ed ha he ppgb- unc o s Fon he ca ego y o double ec o bundles a e in bijec ion wi h he AF-bilinea maps ⋄F:UF×VF→WF, whe e AFa e Weil algeb as and UFand VFand WFa e ini e dimensional (o e R)AF-modules. Mo eo e , gi en a ppgb- unc o Fon double ec o bundles and a poin c∈AF, in [10] an a ino (i.e. enso ield o ype (1,1)) a (c):TFK →TFK on FK is cons uc ed o any double ec o bundle K. The main esul o he p esen no e is he ollowing one ex ending [6]. Theo em 1.1. Le Fbe a ppgb- unc o on double ec o bundles. The canonical a ino s a (c) o all c∈AFa e all na u al a ino s on F K. Canonical (la e called na u al) a ino s on some o he bundle unc o s a e desc ibed in [2, 3, 6, 8], e c. We poin ou ha na u al a ino s play an impo an ole in di e en ial geome y. Fo example, na u al a ino s a e use ul in he p olonga ion o ec o ields o p oduc p ese ing bundles, see e.g. [5]. Na u al a ino s can be also used o de ine he gene al concep o a o sion o a connec ion, [6]. 2. On double ec o bundles. De ini ion 2.1 ([10]). An almos double ec o bundle is a sys em K= (K , Kl, E , El)o ec o bundles K = (K, τ , E ),Kl= (K, τl, El),E = (E , τl, M)and El= (El, τ , M)such ha τl◦τ =τ ◦τl. I K′= (K′ , K′ l, E′ , E′ l)is ano he almos double ec o bundle, an almos double ec o bundle map K→K′is a map :K→K′such ha he e a e maps :E →E′ , l:El→E′ land :M→M′such ha ( , ): K →K′ ,( , l):Kl→K′ l,( , ):E →E′ and ( l, ):El→E′ la e ec o bundle maps. We call M he basis o Kand :M→M′ he base map o . We ha e he i ial almos double ec o bundle Rm1,m2,n1,n2= (K , Kl, E , El), whe e Kl= (Rm1×Rm2×Rn1×Rn2, τl,Rm1×Rn1),K = (Rm1× Rm2×Rn1×Rn2, τ ,Rm1×Rm2),E = (Rm1×Rm2, τl,Rm1)and El= (Rm1×Rn1, τ ,Rm1), and whe e τ ,τl,τ ,τla e he ob ious p ojec ions. De ini ion 2.2 ([10]). A double ec o bundle is a locally i ial almos dou- ble ec o bundle K, ha is, he e a e non-nega i e in ege s m1, m2, n1, n2 such ha o any x∈M he e is an open neighbo hood Ω⊂Mo xsuch ha K|Ω=Rm1,m2,n1,n2modulo an almos double ec o bundle isomo - phism. Na u al a ino s and o sion o connec ions... 3 A e y impo an example o a double ec o bundle is he angen bun- dle TE = (TE, T E, E, T M)o a ec o bundle E= (E, π, M), whe e τ :=pT E :TE →E,τl:=Tπ :TE →TM,τ :=pT M :T M →M, τl:=π:E→M. Ano he such example is he co angen bundle T∗E= (T∗E, T ∗E, E, E∗)o a ec o bundle E, see [9]. Double ec o bundle s uc- u es on TTM and TT∗Mmake possible he Lag angian o mula ion o he dynamics in classical mechanics, see [13]. All double ec o bundles and almos double ec o bundle maps be ween hem o m a ca ego y which we deno e by DVB. (In [10], he no ion o 2-VB ins ead o DVB is used.) Any DVB-map :Rm1,m2,n1,n2→Rm′ 1,m′ 2,n′ 1,n′ 2is o he o m (1) (x, u, , w) =a(x),X j aj(x)uj,X k bk(x) k,X j,k cjk(x)uj k+X l dl(x)wl o some maps a:Rm1→Rm′ 1, aj:Rm1→Rm′ 2,bk:Rm1→Rn′ 1,cjk : Rm1→Rn′ 2,dl:Rm1→Rn′ 2,j= 1, . . . , m2,k= 1, . . . , n1,l= 1, . . . , n2, whe e x∈Rm1,u= (u1, . . . , um2)∈Rm2, = ( 1, . . . , n1)∈Rn1,w= (w1, . . . , wn2)∈Rn2. By he local desc ip ion (p esen ed in [7]) o double ec o bundles in he sense o [9], he double ec o bundles in ou sense a e equi alen o he one o [9]. 3. On ppgb- unc o s on double ec o bundles. Le FM deno e he ca ego y o ib ed mani olds and ib ed maps. The gene al concep o (gauge) bundle unc o s can be ound in he book [5]. We need he ollowing pa icula case o i . De ini ion 3.1 ([10]). A gauge bundle unc o on DVB is a co a ian unc o F:DVB → FM sending any double ec o bundle Kwi h he basis M in o ib ed mani old pK:FK →Mo e Mand any double ec o bundle map :K→K′wi h he base map :M→M′in o ib ed map F : FK →FK′o e :M→M′and sa is ying he ollowing condi ions: (i) (Localiza ion condi ion) Fo e e y double ec o bundle Kwi h he basis Mand any open subse U⊂M, he inclusion map iK|U:K|U→K induces di eomo phism FiK|U:F(K|U)→p−1 K(U), and (ii) (Regula i y condi ion)F ans o ms smoo hly pa ame ized amilies o DVB-maps in o smoo hly pa ame ized amilies o FM-maps. A gauge bundle unc o Fon DVB is called a Weil like unc o (o ppgb- unc o ) i F(K1×K2) = F(K1)×F(K2) o any DVB-objec s K1and K2. An example o a ppgb- unc o on DVB is he angen unc o Tsending any DVB-objec Kwi h basis Min o he angen bundle TK ( ea ed as 4 M. Doupo ec, J. Ku ek and W. M. Mikulski he ib ed mani old o e M) and any DVB-map :K→K′in o T : TK →TK′. In [10], i is p o ed ha he ppgb- unc o s Fon he ca ego y o double ec o bundles a e in bijec ion wi h he AF-bilinea maps ⋄F:UF×VF→WF, whe e AFa e Weil algeb as and UFand VFand WFa e ini e dimensional (o e R)AF-modules. We ha e FRm1,m2,n1,n2= (AF)m1×(UF)m2×(VF)n1×(WF)n2, and i :Rm1,m2,n1,n2→Rm′ 1,m′ 2,n′ 1,n′ 2is o he o m (1), hen F :(AF)m1×(UF)m2×(VF)n1×(WF)n2→ →(AF)m′ 1×(UF)m′ 2×(VF)n′ 1×(WF)n′ 2 is o he simila o m (2) F (x, u, , w) =aAF(x),X j aAF j(x)uj,X k bAF k(x) k ,X j,k cAF jk(x)uj⋄F k+X l dAF l(x)wl , x∈(AF)m1,u= (u1, . . . , um2)∈(UF)m2, = ( 1, . . . , n1)∈(VF)n1, w= (w1, . . . , wn2)∈(WF)n2, whe e aAF=TAFa:TAFRm1= (AF)m1→ TAFRm′ 1= (AF)m′ 1,aAF j=TAFaj:(AF)m1→(AF)m′ 2,bAF k=TAFbk: (AF)m1→(AF)n′ 1,cAF ij =TAFcjk :(AF)m1→(AF)n′ 2,dAF l=TAFdl: (AF)m1→(AF)n′ 2a e he alues o a, aj, bk, cjk, dlby he (usual) Weil unc o TAFo Weil algeb a AF. So, Fhas alues in DVB, i.e. F:DVB → DVB. 4. Tangen bundle o a ppgb- unc o on double ec o bundles. I is obse ed ha any ppgb- unc o Fon DVB has alues in DVB. So, we can compose ppgb- unc o s F1and Fand ob ain ppgb- unc o F1Fon DVB. In pa icula , he composi ion T F o he angen unc o Tand a ppgb- unc o Fon DVB is again a ppgb- unc o on DVB. We ha e AT F =AF×AF, UT F =UF×UF, V T F =VF×VF, WT F =WF×WF and he algeb a mul iplica ion (o AT F ) and he module mul iplica ions (o UT F and VT F and WT F ) and he AT F -bilinea map ⋄T F sa is y (3) (a1, a2)(b1, b2)=(a1b1, a2b1+a1b2), (a1, a2)(u1, u2)=(a1u1, a2u1+a1u2), (a1, a2)( 1, 2)=(a1 1, a2 1+a1 2), (a1, a2)(w1, w2)=(a1w1, a2w1+a1w2), (u1, u2)⋄T F ( 1, 2)=(u1⋄F 1, u2⋄F 1+u1⋄F 2) Na u al a ino s and o sion o connec ions... 5 o any a1, a2, b1, b2∈AF,u1, u2∈UF, 1, 2∈VF,w1, w2∈WF. In [10], o any c∈AF, i is cons uc ed a DVB-in a ian a ino a (c):TFK →TFK on FK o any DVB-objec K. I K=Rm1,m2,n1,n2, hen a (c)((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1), c(a2, u2, 2, w2)) o any a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈ (WF)n2, whe e he s anda d iden i ica ion TX =X×X o ec o spaces Xis applied. The in a iance means ha i :K→K1is a DVB-map, hen TF ◦a (c) = a (c)◦TF . 5. The na u al a ino s on ppgb- unc o s on double ec o bun- dles. Le DVBm1,m2,n1,n2be he ca ego y o all DVB-objec s Kbeing lo- cally isomo phic wi h Rm1,m2,n1,n2wi h local DVB-isomo phisms be ween hem as mo phisms. De ini ion 5.1. ADVBm1,m2,n1,n2-na u al a ino on Fis a DVBm1,m2,n1,n2- in a ian amily B:TF →TF o a ino s B:TFK →TFK on FK o any DVBm1,m2,n1,n2-objec K. I means ha TF ◦B=B◦TF o any DVBm1,m2,n1,n2-map :K→K′. Theo em 5.2. I m1≥2, hen he na u al a ino s a (c):TF →TF o c∈AFa e all DVBm1,m2,n1,n2-na u al a ino s on a ppgb- unc o F. P oo . Le Bbe a DVBm1,m2,n1,n2-na u al a ino on a ppgb- unc o Fon DVB. Clea ly, Bis de e mined by he a ino B:TFRm1,m2,n1,n2→TFRm1,m2,n1,n2 on FRm1,m2,n1,n2= (AF)m1×(UF)m2×(VF)n1×(WF)n2. Then B:FRm1,m2,n1,n2×FRm1,m2,n1,n2→FRm1,m2,n1,n2×FRm1,m2,n1,n2 modulo he s anda d iden i ica ion. So, we can w i e B(x, y)=(x, ˜ B(x, y)) o all x, y ∈FRm1,m2,n1,n2, whe e ˜ B(x, y)∈FRm1,m2,n1,n2is linea in y. Because o he in a iance o Bwi h espec o he homo he ies ·idRm1,m2,n1,n2 o > 0,˜ B( x, y) = ˜ B(x, y), i.e. ˜ B( x, y) = ˜ B(x, y). Consequen ly, ˜ B(x, y)is independen o x. So, we can w i e B((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1),(α(a2, u2, 2, w2), β(a2, u2, 2, w2), γ(a2, u2, 2, w2), δ(a2, u2, 2, w2))) 6 M. Doupo ec, J. Ku ek and W. M. Mikulski o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1and w1, w2∈ (WF)n2, whe e α(a2, u2, 2, w2)∈(AF)m1is linea in (a2, u2, 2, w2)and β(a2, u2, 2, w2)∈(UF)m2is linea in (a2, u2, 2, w2)and γ(a2, u2, 2, w2)∈ (VF)n1is linea in (a2, u2, 2, w2)and δ(a2, u2, 2, w2)∈(WF)n2is linea in (a2, u2, 2, w2). Le φ , 1, 2, 3:Rm1,m2,n1,n2→Rm1,m2,n1,n2be gi en by φ , 1, 2, 3(x, y1, y2, y3)=( x, 1y1, 2y2, 3y3) o all x∈Rm1and y1∈Rm2and y2∈Rn1and y3∈Rn2, whe e , 1, 2, 3 a e posi i e eal numbe s. I is a DVBm1,m2,n1,n2-map. Then, by he in a i- ance o Bwi h espec o φ , 1, 2, 3, we de i e α( a2, 1u2, 2 2, 3w2) = α(a2, u2, 2, w2). Consequen ly, α(a2, u2, 2, w2)is linea in a2and independen o u2, 2, w2. Simila ly, β(a2, u2, 2, w2)is linea in u2and independen o a2, 2, w2, and γ(a2, u2, 2, w2)is linea in 2and independen od a2, u2, w2, and δ(a2, u2, 2, w2)is linea in w2and independen o b, u2, 2. Hence we can w i e B((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1),(α(a2), β(u2), γ( 2), δ(w2))) o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈(WF)n2, whe e α(a2)∈(AF)m1is linea in a2and β(u2)∈(UF)m2is linea in u2 and γ( 2)∈(VF)n1is linea in 2and δ(w2)∈(WF)n2is linea in w2. Le :Rm1,m2,n1,n2→Rm1,m2,n1,n2be gi en by (x, y1, y2, y3)=(x+x1x, y1+x1y1, y2+x1y2, y3+x1y3) o all x= (x1, . . . , xm1)∈Rm1and y1∈Rm2and y2∈Rn1and y3∈Rn2. I is a DVBm1,m2,n1,n2-map on he open and dense subse in Rm1,m2,n1,n2, x1=−1. Then, by he in a iance o Bwi h espec o and (in pa icula ) o mula (2) o TF ins ead o Fand o mulas (3), we ge ((a1+a1 1a1, u1+a1 1u1, . . .),(α(a2+a1 1a2+a1 2a1), β(u2+a1 1u2+a1 2u1), . . .)) = ((a1+a1 1a1, u1+a1 1u1, . . .),(α(a2) + a1 1α(a2) + α1(a2)a1, β(u2) + a1 1β(u2) + α1(a2)u1, . . .)) o all a1, a2∈(AF)m1and u1, u2∈(UF)m2and ..., whe e (α1(b), . . . , αm1(b)) =α(b)∈(AF)m1and (b1, . . . , bm1) = b∈(AF)m1. Then α(a1 1a2) + α(a1 2a1) = a1 1α(a2) + α1(a2)a1, β(a1 1u2) + β(a1 2u1) = a1 1β(u2) + α1(a2)u1, γ(a1 1 2) + γ(a1 2 1) = a1 1γ( 2) + α1(a2) 1, δ(a1 1w2) + δ(a1 2w1) = a1 1δ(w2) + α1(a2)w1. Na u al a ino s and o sion o connec ions... 7 I a1 1= 1, hen α(a1 2a1) = α1(a2)a1, β(a1 2u1) = α1(a2)u1, γ(a1 2 1) = α1(a2) 1, δ(a1 2w1) = α1(a2)w1. I a2= (1,0,...,0) ∈(AF)m1, we ge α(a) = c1a , β(u) = c1u , γ( ) = c1 , δ(w) = c1w o any a= (a1, . . . , am1)∈(AF)m1wi h a1= 1 and u∈(UF)m2and ∈(VF)n1and w∈(WF)n2, whe e c1:=α1(1,0,...,0) ∈AF. Simila ly, eplacing 1by i∈ {1, . . . , m1}, we de i e α(a) = cia , β(u) = ciu , γ( ) = ci , δ(w) = ciw o any a= (a1, . . . , am1)∈(AF)m1wi h ai= 1 and u∈(UF)m2and ∈(VF)n1and w∈(WF)n2, whe e ci:=αi(0,...,1,...,0) ∈AF(1in i- h posi ion). F om he linea i y o αand m1≥2we ob ain α(a) = ca , β(u) = cu , γ( ) = c , δ(w) = cw o any a= (a1, . . . , am)∈(AF)m1and u∈(UF)m2and ∈(VF)n1and w∈(WF)n2, whe e c:=c1=. . . =cm∈AF. Tha c1=. . . =cmis a simple consequence o he in a iance o Bwi h espec o he pe mu a ions o he base coo dina es. Then B((a1, u1, 1, w1),(a2, u2, 2, w2)) = ((a1, u1, 1, w1), c(a2, u2, 2, w2)) o all a1, a2∈(AF)m1,u1, u2∈(UF)m2, 1, 2∈(VF)n1,w1, w2∈(WF)n2, whe e c∈AFis as abo e. Then B= a (c), as well and he p oo is comple e. Q.E.D. □ 6. On double-linea ec o ields. Le Kbe a double ec o bundle wi h basis M. A ec o ield Zon Kis called double-linea i he low o Z is o med by local DVB-isomo phisms. Le x1, . . . , xm1,u1, . . . , um2, 1, . . . , n1,w1, . . . , wn2be (local) DVB- coo dina es on K. A map :K→Kis a DVB-map i and only i i is o he o m (1). Consequen ly, a ec o ield Zon Kis double linea i and only i i is o he o m (4) Z= m1 X i=1 ai(x)∂ ∂xi+ m2 X j,j1=1 bj1 j(x)uj∂ ∂uj1+ n1 X k,k1=1 ck1 k(x) k∂ ∂ k1 + n2 X l,l1=1 el1 l(x)wl∂ ∂wl1+ m2 X j2=1 n1 X k2=1 n2 X l2=1 l2 j2k2(x)uj2 k2∂ ∂wl2. 8 M. Doupo ec, J. Ku ek and W. M. Mikulski So, we ha e: Lemma 6.1 ([11]). The space o all double-linea ec o ields on Kis he Lie subalgeb a in he Lie algeb a o ec o ields on K. Le Fbe a ppgb- unc o on DVB. Then F K is again a DVB-objec (see Sec ion 3). Lemma 6.2. Le Zbe a double-linea ec o ield on FK and c∈AFbe a poin . Then he ec o ield a (c)(Z)on F K is also double-linea . P oo . We may assume ha K=Rm1,m2,n1,n2. Then F K =Am1×Um2× Vn1×Wn2. Fixing he bases (o e R) o Am1and Um2and Vn1and Wn2, we can w i e FK =RM1,M2,N1,N2. Le x1, . . . , xM1, u1, . . . , uM2, 1, . . . , N1, w1, . . . , wN2be he usual coo dina es on RM1,M2,N1,N2. Then Zis o he o m (5) Z= M1 X i=1 ai(x)∂ ∂xi+ M2 X j,j1=1 bj1 j(x)uj∂ ∂uj1+ N1 X k,k1=1 ck1 k(x) k∂ ∂ k1 + N2 X l,l1=1 el1 l(x)wl∂ ∂wl1+ M2 X j2=1 N1 X k2=1 N2 X l2=1 l2 j2k2(x)uj2 k2∂ ∂wl2. To p o e ha a (c)(Z)is double-linea , i is su icien o show ha a (c)(Z)is o he o m (5), oo. O cou se, i is su icien o show ha a (c)( ∂ ∂xi)is he linea combina ion o ∂ ∂x1,..., ∂ ∂xM1wi h eal coe icien s and ha a (c)( ∂ ∂uj)is he linea combina ion o ∂ ∂u1,..., ∂ ∂uM2wi h eal co- e icien s and ha a (c)( ∂ ∂ k)is he linea combina ion o ∂ ∂ 1,..., ∂ ∂ N1wi h eal coe icien s and ha a (c)( ∂ ∂wl)is he linea combina ion o ∂ ∂w1,..., ∂ ∂wN2wi h eal coe icien s. Fo example, we p o e ha a (c)( ∂ ∂u1)is he linea combina ion o ∂ ∂u1,..., ∂ ∂uM2wi h eal coe icien s. Le (x, u, , w)∈Am1×Um2×Vn1× Wn2. Le e1, . . . , eM2be he usual basis in RM2˜=Um2. We can w i e ∂ ∂uj|(x,u, ,w)= ((x, u, , w),(0, ej,0,0)). Then a (c)∂ ∂u1|(x,u, ,w) = ((x, u, , w),(0, c ·e1,0,0)) . On he o he hand, c·e1∈Um2(as e1∈Um2), and hen c·e1is he linea combina ion o e1, . . . , eM2wi h eal coe icien s. The p oo o he p oposi ion is comple e. Q.E.D. □ 7. The F-N-b acke and double-linea (semi-basic) angen al- ued p- o ms. I K→Mis a ib ed mani old, a p ojec able semi-basic angen alued p- o m on Kis a sec ion φ:K→ ∧pT∗M⊗T K such Na u al a ino s and o sion o connec ions... 9 ha φ(X1, . . . , Xp)is a p ojec able ec o ield on K o any ec o ields X1, . . . , Xpon M. Gi en a p ojec able semi-basic angen alued p- o m φ:K→ ∧pT∗M⊗ TK we ha e he unde lying angen alued p- o m φ:M→ ∧pT∗M⊗T M on Msuch ha φ(X1, . . . , Xp)is he unde lying ec o ield o φ(X1, . . . , Xp) o any ec o ields X1, . . . , Xpon M. Lemma 7.1. Le K→Mbe a ib ed mani old. Gi en a p ojec able semi- basic angen alued p- o m φ:K→ ∧pT∗M⊗T K on Kand a p o- jec able semi-basic angen alued q- o m ψ:K→ ∧qT∗M⊗T K on K, he F oliche –Nijenhuis b acke (F-N-b acke ) [[φ, ψ]] is (again) a p ojec able semi-basic angen alued (p+q)- o m [[φ, ψ]] :K→ ∧p+qT∗M⊗T K on Ksa is ying (6) [[φ, ψ]](X1, . . . , Xp+q) =1 p!q!X σ sign σ[φ(Xσ1, . . . , Xσp), ψ(Xσ(p+1), . . . , Xσ(p+q))] +−1 p!(q−1)! X σ sign σψ([φ(Xσ1, . . . , Xσp), Xσ(p+1)], Xσ(p+2), . . .) +(−1)pq (p−1)q!X σ sign σφ([ψ(Xσ1, . . . , Xσq), Xσ(q+1)], Xσ(q+2), . . .) +(−1)p−1 (p−1)!(q−1)!2! X σ sign σψ(φ([Xσ1, Xσ2], Xσ3, . . .), Xσ(p+2), . . .) +(−1)(p−1)q (p−1)!(q−1)!2! X σ sign σφ(ψ([Xσ1, Xσ2], Xσ3, . . .), Xσ(q+2), . . .) o any ec o ields X1, . . . , Xp+qon M, whe e sums a e o e all pe mu a- ions σ:{1, . . . , p +q}→{1, . . . , p +q}. P oo . I is well-known ac , see e.g. [4]. Q.E.D. □ Le Fbe a ppgb- unc o on DVB and Kbe a DVB-objec wi h basis M. Then we ha e he ib ed mani old FK →M. We ha e also he DVB-objec FK wi h basis FM. De ini ion 7.2. A double-linea semi-basic angen alued p- o m on F K → Mis a p ojec able semi-basic angen alued p- o m φ:F K → ∧pT∗M⊗ TFK on ( ibe ed mani old) F K (wi h basis M) such ha (addi ionally) φ(X1, . . . , Xp)is a double-linea ec o ield on DVB-objec F K (wi h basis FM) o any ec o ields X1, . . . , Xpon M.