TANGENT DIRAC STRUCTURES OF HIGHER ORDER. P. M. Kouotchop Wamba, A. Ntyam, and J. Wouafo Kamga

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1 ARCHIVUM MATHEMATICUM BRNO) Tomus ), TANGENT DIRAC STRUCTURES OF HIGHER ORDER P. M. Kouotchop Wamba, A. Ntyam, and J. Wouafo Kamga Abstract. Let L be an almost Drac structure on a manfold M. In [2] Theodore James Courant defnes the tangent lftng of L on T M and proves that: If L s ntegrable then the tangent lft s also ntegrable. In ths paper, we generalze ths lftng to tangent bundle of hgher order. Introducton Let M be a dfferental manfold dm M = m > 0). Consder the mappng φ M defned by: φ M : T M T M M T M T M R X1, α 1 ), X 2, α 2 ) ) 1 2 X1, α 2 M + X 2, α 1 M ) where M s the canoncal parng defned by: T M M T M R X, α) X, α M An almost Drac structure on M, s a sub vector bundle L of the vector bundle T M T M, whch s sotropc wth respect to the natural ndefnte symmetrc scalar product φ M.e X 1, α 1 ), X 2, α 2 ) ΓL), φ M X 1, α 1 ), X 2, α 2 )) = 0), and such that the rank of L s equal to the dmenson of M. We defne on the set ΓT M T M) of sectons of T M T M a bracket by: X 1, α 1 ), X 2, α 2 ) ΓT M T M) [X 1, α 1 ), X 2, α 2 )] C = [X 1, X 2 ], L X1 α 2 X2 dα 1 ). Ths bracket s called Courant bracket. A Drac structure or generalzed Drac structure) s an almost Drac structure such that: X 1, α 1 ), X 2, α 2 ) ΓL), [X 1, α 1 ), X 2, α 2 )] ΓL). Ths condton s called ntegrablty condton Mathematcs Subject Classfcaton: prmary 53C15; secondary 53C75, 53D05. Key words and phrases: Drac structure, almost Drac structure, tangent functor of hgher order, natural transformatons. Receved August 24, 2009, revsed August Edtor I. Kolář.

2 18 P. M. KOUOTCHOP WAMBA, A. NTYAM AND J. WOUAFO KAMGA For X 3, α 3 ) ΓT M T M), n [2] s defned the 3-tensor T T M T M on the vector bundle T M T M by: T T M T M X 1, α 1 ), X 2, α 2 ), X 3, α 3 )) = φ M [X1, α 1 ), X 2, α 2 )], X 3, α 3 ) ). We put T L = T T M T M ΓL) ΓL) ΓL). The ntegrablty condton of L s determned by the vanshng of the 3-tensor T L on the vector bundle L. For all nteger r, k 1, we have the jet functor Tk r of k-dmensonal velocty of order r and, when k = 1, ths functor s denoted by T r and s called tangent bundle of order r. When r = 1, T 1 s a natural equvalence of tangent functor T. The man results of ths paper are theorems 2 and 3: gvng an almost Drac structure L on M, we construct an almost Drac structure L r on T r M and we prove that: L s ntegrable f and only f L r s ntegrable. All manfolds and maps are assumed to be nfntely dfferentable. r wll be a natural nteger r 1). 1. Other characterzaton of generalzed Drac structure Let V be a real vector space of dmenson m. We consder the map φ V : V V V V R u, u ), v, v ) ) 1 2 u, v + v, u ) where s the dual bracket V V R. Defnton 1. A constant Drac structure on V s a sub vector space L of dmenson m of V V such that: u, u ), v, v ) L, φ V u, u ), v, v ) ) = 0. Theorem 1. A constant Drac structure L on V s determned by a par of lnear maps a: R m V and b: R m V such that: 1) 2) a b + b a = 0 ker a ker b = {0} Proof. Condton 1) s the sotropy of constant Drac structure, and condton 2) s the maxmalty of the sotropy. Remark 1. 1) We say that the constant Drac structure L s determned by the lnear maps a and b. 2) An almost Drac structure on a dfferental manfold M s a sub vector bundle of T M T M such that: x M, the fber L x of L over x s a constant Drac structure on T x M. 3) An almost Drac structure at a pont x M s determned by a par of maps a x : R m T x M, b x : R m T x M such that: { a x b x + b x a x = 0 ker a x ker b x = {0}

3 TANGENT DIRAC STRUCTURES OF HIGHER ORDER 19 Corollary. An almost Drac structure s determned n a neghbourhood U of a local trvalzaton L U U R m by a par of vector bundle morphsms a: U R m T U M, b: U R m TU M over U such that: { a x b x + b x a x = 0 x U, ker a x ker b x = {0} We denote by p 1 and p 2 the natural projectons of T M T M onto T M and T M respectvely. Note that a: L T M and b: L T M are really globally defned and are nothng more than the projectons p 1 and p 2. Example 1. Let M be an m-dmensonal manfold. 1) Let ω be a dfferental form on M of degree 2. Γ = {X, X ω), X XM)}. Γ s the set of dfferental sectons of an almost Drac structure on M. It s a Drac structure f and only f ω s pre-symplectc form. 2) Let Π be a bvector feld on M. Γ = { Π α, α), α Ω 1 M)}. Γ s the set of dfferental sectons of an almost Drac structure on M. It s a Drac structure f and only f Π s a Posson bvector. We denote by x, ẋ ) and x, p ) a local coordnates system of T M and T M respectvely. Let L be an almost Drac structure on M defned locally by: We have: a: U R m T M and b: U R m T M. ax, e j ) = a k j x k bx, e j ) = b jk dx k where e j ) denote the canoncal bass of R m. Locally the 3-tensor feld T L s: T L = a p b js x p as k + a p a s ) j x p b ks. cyclc,,j,k 2. Tangent Drac structure of hgher order κ r M : T r T M T T r M and α r M : T T r M T r T M denote the natural transformatons defned n [1] and [7]. We have: κ r M u), v T r M = u, α r M v ) T r M, u, v ) T r T M T r M T T r M where T r M = τ r T r and τ r j r 0ϕ) = dr ϕ dt r t) t=0. We denote by ε r M the nverse map of αr M. Consder the maps a: U R m T M and b: U R m T M. We take ther tangents of order r, to get: T r a: T r U R mr+1) T r T M and T r b: T r U R mr+1) T r T M.

4 20 P. M. KOUOTCHOP WAMBA, A. NTYAM AND J. WOUAFO KAMGA We apply natural transformatons κ r M and εr M respectvely, to get the vector bundle maps over d T r U defned by: a r : T r U R mr+1) T T r M and b r : T r U R mr+1) T T r M. Theorem 2. The par of maps a r and b r determnes a generalzed almost Drac structure L r on T r M, whch we call the tangent lft of order r of the generalzed almost Drac structure on M determned by a and b. Proof. Frstly, we prove that: a r ) b r +b r ) a r = 0. Let j r 0ψ, j r 0ϕ T r U R m ), where ϕ, ψ : R U R m dfferentals. We have: By the same way, we have: a r ) b r j r 0ϕ), j r 0ψ = b r j r 0ϕ), a r j r 0ψ) = ε r M T r b, κ r M T r aj r 0ψ) = T r bj r 0ϕ), T r aj r 0ψ) T r M = τ r j r 0 b ϕ, a ψ M ) = τ r j r 0 a b ϕ, ψ M ). b r ) aj r 0ϕ), j r 0ψ = τ r j r 0 b a ϕ, ψ M ) we deduce that: a r ) b r + b r ) a)j0ϕ), r j0ψ r = τ r j0 r a b + b a) ϕ, ψ ) M = 0. Secondly we prove that: ker a r ker b r = {0}. We prove ths case for r = 2. The proof for r 3 s smlar. In the local coordnates system, we have: a : U R m U R m x, e) x, ae) and b : U R m U R m ) x, e) x, be) a 2 x, ẋ, ẍ, e, ė, ë) = x, ẋ, ẍ, ae, ȧe + aė, äe + ȧė + aä) b 2 x, ẋ, ẍ, e, ė, ë) = x, ẋ, ẍ, be + ḃė + bë, ḃe + bė, be) a 0 0 e b ḃ b e a 2 e, ė, ë) = ȧ a 0 ė and b 2 e, ė, ë) = ḃ b 0 ė. ä ȧ a ë b 0 0 ë If a 2 e, ė, ë) = b 2 e, ė, ë) = 0, we have: ae = 0 be = 0 e ker a ker b = {0}. and t follows that e = 0. { bė + ḃe = 0 aė + ȧe = 0 { bė = 0 aė = 0

5 TANGENT DIRAC STRUCTURES OF HIGHER ORDER 21 e and ė are constant, t follows that ė = 0. { bë = 0 ë = 0. aë = 0 Thus ker a 2 ker b 2 = {0}. Theorem 3. The almost Drac structure L on M s ntegrable f and only f the almost Drac structure L r on T r M s ntegrable. Proof. Consder the local coordnates system {x 1,..., x m } of M, we have: We have: ax, e j ) = a k x k and bx, e j ) = b k dx k. a j... 0 a r =..... r) a j... a j and br = r) b j... b j b j... 0 We get a r = A j ) 1,j mr+1) and b r = B j ) 1,j mr+1). For q, d = 0, 1,... r, we have:, j) {qm + 1,..., mq + 1)} {dm + 1,..., md + 1)}, A j = a mq j md )q d) and B j = b mq,j md ) r q d) We adopt the followng notaton: x p = ) x p mα = α x p mα )α) αm + 1 p αm + 1). The Courant tensor T jk of the almost Drac structure s gven by: T jk = cyclc,,j,k A p B js x p As k + A p A s j x p B ks, we wsh to verfy that T jk = 0. We take hm + 1 mh + 1), lm + 1 j ml + 1) and tm + 1 k mt + 1) for h, l, t = 0, 1,..., r. We have: T jk = r r qm+1) dm+1) q=0 d=0 p=qm+1 s=dm+1 = a p mq A p B js x p As k + A p A s ) j x p B ks b j ml,s md) r l d) mh )q h) x p mq a s md k mt )d t) q + a p mq mh as md j ml )d l) )q h) x p mq q b k mt,s md ) r d t)

6 22 P. M. KOUOTCHOP WAMBA, A. NTYAM AND J. WOUAFO KAMGA = a p mq mh )q h) b ) r l d q) j ml,s md a s md x p mq k mt )d t) = mh )q h) a s md j ml + a p mq a p md mh = a p mq mh b j ml,s md x p mq x p mq ) d l q) bk mt,s md ) r d t) ) r l h t) a s md k mt + b j ml,s md x p mq a s md k mt + ap mq mh a s md j ml a p mq mh a s md j ml x p mq b k mt,s md ) r l h t) x p mq b k mt,s md) r l h t) the calculaton above shows that T L = 0 f and only f T L r = 0. Remark 2. Ths constructon generalzes the tangent lfts of hgher order of Posson and pre-symplectc structure to tangent bundle of hgher order. References [1] Cantrjn, F., Crampn, M., Sarlet, W., Saunders, D., The canoncal somorphsm between T k T and T T k, C. R. Acad. Sc., Pars, Sér. II ), [2] Courant, T., Tangent Drac Structures, J. Phys. A: Math. Gen ) 1990), [3] Courant, T., Tangent Le Algebrods, J. Phys. A: Math. Gen ) 1994), [4] Gancarzewcz, J., Mkulsk, W., Pogoda, Z., Lfts of some tensor felds and connectons to product preservng functors, Nagoya Math. J ), [5] Grabowsk, J., Urbansk, P., Tangent lfts of Posson and related structures, J. Phys. A: Math. Gen ) 1995), [6] Kolář, I., Mchor, P., Slovák, J., Natural Operatons n Dfferental Geometry, Sprnger-Verlag, [7] Mormoto, A., Lftng of some type of tensors felds and connectons to tangent bundles of p r -veloctes, Nagoya Math. J ), [8] Ntyam, A., Wouafo Kamga, J., New versons of curvatures and torson formulas of complete lftng of a lnear connecton to Wel bundles, Ann. Pol. Math. 82 3) 2003), Department of Mathematcs, The Unversty of Yaoundé 1, P.O BOX, 812, Yaoundé, Cameroon E-mal: wambapm@yahoo.fr Department of Mathematcs, ENS Yaoundé, P.O BOX, 47 Yaoundé, Cameroon E-mal: antyam@uy1-unnet.cm Department of Mathematcs, The Unversty of Yaoundé 1, P.O BOX, 812, Yaoundé Cameroon E-mal: wouafoka@yahoo.fr