Let Lbe an almost Dirac structure on a manifold M

ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 17–22

TANGENT DIRAC STRUCTURES OF HIGHER ORDER

P. M. Kouotchop Wamba, A. Ntyam, and J. Wouafo Kamga

Abstract. Let Lbe an almost Dirac structure on a manifold M. In [2]

Theodore James Courant defines the tangent lifting ofLonT M and proves that:

IfLis integrable then the tangent lift is also integrable.

In this paper, we generalize this lifting to tangent bundle of higher order.

Introduction

LetM be a differential manifold (dimM =m >0). Consider the mappingφ_M defined by:

φ_M: T M ⊕T^∗M ×M T M⊕T^∗M → R (X₁, α₁),(X₂, α₂)

7→ 1

2 hX₁, α₂i_M +hX₂, α₁i_M whereh·iM is the canonical pairing defined by:

T M×M T^∗M → R (X, α) 7→ hX, αiM

An almost Dirac structure onM, is a sub vector bundleL of the vector bundle T M ⊕T^∗M, which is isotropic with respect to the natural indefinite symmetric scalar product φM (i.e ∀(X1, α1),(X2, α2)∈ Γ(L), φM((X1, α1),(X2, α2)) = 0), and such that the rank of Lis equal to the dimension ofM.

We define on the set Γ(T M⊕T^∗M) of sections ofT M⊕T^∗M a bracket by:

∀(X1, α1),(X2, α2)∈Γ(T M⊕T^∗M)

[(X1, α1),(X2, α2)]C= [X1, X2],LX₁α2−iX₂dα1

.

This bracket is called Courant bracket. A Dirac structure (or generalized Dirac structure) is an almost Dirac structure such that:

∀(X1, α1),(X2, α2)∈Γ(L), [(X1, α1),(X2, α2)]∈Γ(L). This condition is called “integrability condition”.

2010Mathematics Subject Classification: primary 53C15; secondary 53C75, 53D05.

Key words and phrases: Dirac structure, almost Dirac structure, tangent functor of higher order, natural transformations.

Received August 24, 2009, revised August 2010. Editor I. Kolář.

For (X₃, α₃)∈Γ(T M⊕T^∗M), in [2] is defined the 3-tensorT_{T M⊕T}^∗_M on the vector bundleT M⊕T^∗M by:

TT M⊕T^∗M((X1, α1),(X2, α2),(X3, α3)) =φM [(X1, α1),(X2, α2)],(X3, α3) . We putTL=T_{T M⊕T}^∗M|Γ(L)×Γ(L)×Γ(L). The integrability condition ofL is deter- mined by the vanishing of the 3-tensorTL on the vector bundleL.

For all integer r, k ≥1, we have the jet functor T_k^r of k-dimensional velocity of order r and, whenk= 1, this functor is denoted byT^r and is called tangent bundle of orderr. Whenr= 1,T¹is a natural equivalence of tangent functorT.

The main results of this paper are theorems 2 and 3: giving an almost Dirac structure L on M, we construct an almost Dirac structure L^r on T^rM and we prove that:Lis integrable if and only ifL^r is integrable.

All manifolds and maps are assumed to be infinitely differentiable.rwill be a natural integer (r≥1).

1. Other characterization of generalized Dirac structure LetV be a real vector space of dimensionm. We consider the map

φV: V ⊕V^∗×V ⊕V^∗ → R (u, u^∗),(v, v^∗)

7→ ¹₂ hu, v^∗i+hv, u^∗i whereh·iis the dual bracketV ×V^∗→R.

Definition 1. A constant Dirac structure onV is a sub vector spaceLof dimension mofV ⊕V^∗ such that:

∀(u, u^∗),(v, v^∗)∈L , φ_V (u, u^∗),(v, v^∗)

= 0.

Theorem 1. A constant Dirac structureLonV is determined by a pair of linear mapsa:R^m→V andb:R^m→V^∗ such that:

a^∗◦b+b^∗◦a= 0 (1)

kera∩kerb={0}

(2)

Proof. Condition (1) is the isotropy of constant Dirac structure, and condition

(2) is the maximality of the isotropy.

Remark 1.

(1) We say that the constant Dirac structureL is determined by the linear mapsaandb.

(2) An almost Dirac structure on a differential manifold M is a sub vector bundle of T M⊕T^∗M such that: ∀x∈M, the fiberLx of L overxis a constant Dirac structure onTxM.

(3) An almost Dirac structure at a pointx∈M is determined by a pair of mapsax: R^m→TxM,bx:R^m→T_x^∗M such that:

(a^∗_x◦b_x+b^∗_x◦a_x= 0 kerax∩kerbx={0}

Corollary. An almost Dirac structure is determined in a neighbourhoodU of a local trivializationL|U ≈U ×R^m by a pair of vector bundle morphisms a:U ×R^m→ T_UM,b:U×R^m→T_U^∗M overU such that:

∀x∈U ,

(a^∗_x◦bx+b^∗_x◦ax= 0 kera_x∩kerb_x={0}

We denote byp1 andp2the natural projections ofT M⊕T^∗M ontoT M and T^∗M respectively. Note thata: L→ T M andb: L →T^∗M are really globally defined and are nothing more than the projectionsp₁andp₂.

Example 1. LetM be anm-dimensional manifold.

(1) Letω be a differential form onM of degree 2.

Γ ={(X, iXω), X ∈X(M)}.

Γ is the set of differential sections of an almost Dirac structure onM. It is a Dirac structure if and only if ωis pre-symplectic form.

(2) Let Π be a bivector field onM.

Γ⁰ ={(iΠα, α), α∈Ω¹(M)}.

Γ⁰ is the set of differential sections of an almost Dirac structure onM. It is a Dirac structure if and only if Π is a Poisson bivector.

We denote by (xⁱ,x˙ⁱ) and (xⁱ, p_i) a local coordinates system ofT M andT^∗M respectively. LetL be an almost Dirac structure onM defined locally by:

a: U×R^m→T M and b:U×R^m→T^∗M . We have:







a(xⁱ, ej) =a^k_j ∂

∂x^k b(xⁱ, ej) =bjkdx^k

where (e_j) denote the canonical basis ofR^m. Locally the 3-tensor fieldT_L is:

TL= X

cyclic,i,j,k

a^p_i∂bjs

∂x^pa^s_k+a^p_i∂a^s_j

∂x^pbks

.

2. Tangent Dirac structure of higher order

κ^r_M:T^rT M →T T^rM andα^r_M:T^∗T^rM →T^rT^∗M denote the natural trans- formations defined in [1] and [7]. We have:

hκ^r_M(u), v^∗iT^rM =hu, α^r_M(v^∗)i⁰_TrM, (u, v^∗)∈T^rT M ×T^rMT^∗T^rM whereh · i⁰_TrM =τ_r◦T^rh·i and τ_r(j₀^rϕ) = d^rϕ

dt^r(t)|_t=0. We denote byε^r_M the inverse map ofα^r_M.

Consider the mapsa:U×R^m→T M andb:U×R^m→T^∗M. We take their tangents of order r, to get:

T^ra:T^rU×R^m(r+1)→T^rT M and T^rb: T^rU×R^m(r+1)→T^rT^∗M .

We apply natural transformationsκ^r_M andε^r_M respectively, to get the vector bundle maps overid_TrU defined by:

a^r:T^rU×R^m(r+1)→T T^rM and b^r:T^rU×R^m(r+1)→T^∗T^rM . Theorem 2. The pair of mapsa^r andb^r determines a generalized almost Dirac structure L^r on T^rM, which we call the tangent lift of order rof the generalized almost Dirac structure on M determined by aandb.

Proof. Firstly, we prove that: (a^r)^∗◦b^r+(b^r)^∗◦a^r= 0. Letj₀^rψ,j₀^rϕ∈T^r(U×R^m), whereϕ,ψ:R→U×R^mdifferentials. We have:

h(a^r)^∗◦b^r(j₀^rϕ), j^r₀ψi=hb^r(j₀^rϕ), a^r(j₀^rψ)i

=hε^r_M◦T^rb, κ^r_M ◦T^ra(j₀^rψ)i

=hT^rb(j₀^rϕ), T^ra(j₀^rψ)i⁰_TrM

=τ^r◦j₀^r(hb◦ϕ, a◦ψiM)

=τ^r◦j₀^r(ha^∗◦b◦ϕ, ψiM). By the same way, we have:

h(b^r)^∗◦a(j₀^rϕ), j₀^rψi=τ^r◦j₀^r(hb^∗◦a◦ϕ, ψiM) we deduce that:

((a^r)^∗◦b^r+ (b^r)^∗◦a)(j^r₀ϕ), j₀^rψ

=τ^r◦j₀^r

(a^∗◦b+b^∗◦a)◦ϕ, ψ

M

= 0. Secondly we prove that: kera^r∩kerb^r={0}. We prove this case for r= 2. The proof forr≥3 is similar.

In the local coordinates system, we have:

a:U×R^m → U×R^m

(x, e) 7→ (x, ae) and b:U×R^m → U×(R^m)^∗ (x, e) 7→ (x, be)

a²(x,x,˙ x, e,¨ e,˙ ¨e) = (x,x,˙ x, ae,¨ ae˙ +ae,˙ ¨ae+ ˙ae˙+a¨a) b²(x,x,˙ x, e,¨ e,˙ ¨e) = (x,x,˙ x,¨ ¨be+ ˙be˙+b¨e,be˙ +be, be)˙

a²(e,e,˙ e) =¨





a 0 0

˙ a a 0

¨ a a˙ a







 e

˙ e

¨ e



 and b²(e,e,˙ ¨e) =





¨b b˙ b b˙ b 0 b 0 0







 e

˙ e

¨ e



.

Ifa²(e,e,˙ ¨e) =b²(e,e,˙ ¨e) = 0, we have:

ae= 0 be= 0 ⇒ e∈kera∩kerb={0}. and it follows that e= 0.

be˙+ ˙be= 0 ae˙+ ˙ae= 0 ⇒

be˙= 0 ae˙= 0

eand ˙eare constant, it follows that ˙e= 0.

b¨e= 0

a¨e= 0 ⇒ e¨= 0.

Thus kera²∩kerb²={0}.

Theorem 3. The almost Dirac structureL onM is integrable if and only if the almost Dirac structureL^r on T^rM is integrable.

Proof. Consider the local coordinates system {x¹, . . . , x^m} ofM, we have:

a(xⁱ, ej) =aⁱ_k ∂

∂x^k and b(xⁱ, ej) =bikdx^k. We have:

a^r=









aⁱ_j . . . 0 ... . . . ...

(r)

aⁱ_j . . . aⁱ_j









and b^r=









(r)

bij . . . bij

... . . . ... bij . . . 0







 .

We geta^r= (Aⁱ_j)1≤i,j≤m(r+1)andb^r= (Bij)1≤i,j≤m(r+1). Forq, d= 0,1, . . . r, we have:

∀(i, j)∈{qm+ 1, . . . , m(q+ 1)} × {dm+ 1, . . . , m(d+ 1)},











Aⁱ_j= (a^i−mq_j−md)^(q−d)

and B_ij= (b_{i−mq,j−md})^(r−q−d) We adopt the following notation:

∂

∂x^p = ∂

∂x^p−mαα

= ( ∂

∂x^p−mα)^(α) αm+ 1≤p≤α(m+ 1) .

The Courant tensorTijk of the almost Dirac structure is given by:

Tijk= X

cyclic, i,j,k

A^p_i∂Bjs

∂x^p A^s_k+A^p_i∂A^s_j

∂x^pBks, we wish to verify that Tijk = 0.

We takehm+ 1≤i≤m(h+ 1),`m+ 1≤j≤m(`+ 1) andtm+ 1≤k≤m(t+ 1) forh, `, t= 0,1, . . . , r. We have:

Tijk=

r

X

q=0 r

X

d=0 q(m+1)

X

p=qm+1 d(m+1)

X

s=dm+1

A^p_i∂Bjs

∂x^p A^s_k+A^p_i∂A^s_j

∂x^pBks

= (a^p−mq_i−mh)^(q−h)∂(bj−m`,s−md)<sup>(r−`−d)</sup>

∂x^p−mqq

(a^s−md_k−mt)^(d−t)

+ (a^p−mq_i−mh)^(q−h)∂(a^s−md<sub>j−m`</sub>)<sup>(d−`)</sup>

∂x^p−mqq

(bk−mt,s−md)^(r−d−t)

= (a^p−mq_i−mh)^(q−h)∂b_{j−m`,s−md}

∂x^p−mq

(r−`−d−q)

(a^s−md_k−mt)^(d−t)

+ (a^p−mq_i−mh)^(q−h)∂a^s−md_j−m`

∂x^p−mq

(d−`−q)

(b_{k−mt,s−md})^(r−d−t)

=

a^p−md_i−mh∂b_{j−m`,s−md}

∂x^p−mq a^s−md_k−mt(r−`−h−t)

+

a^p−mq_i−mh∂a^s−md_j−m`

∂x^p−mqbk−mt,s−md

(r−`−h−t)

= (a^p−mq_i−mh∂b_{j−m`,s−md}

∂x^p−mq a^s−md_k−mt+a^p−mq_i−mh∂a^s−md_j−m`

∂x^p−mqb_{k−mt,s−md})^{(r−`−h−t)}

the calculation above shows thatTL= 0 if and only if TL^r = 0.

Remark 2. This construction generalizes the tangent lifts of higher order of Poisson and pre-symplectic structure to tangent bundle of higher order.

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Department of Mathematics, The University of Yaoundé 1, P.O BOX, 812, Yaoundé, Cameroon

E-mail:[email protected]

Department of Mathematics, ENS Yaoundé, P.O BOX, 47 Yaoundé, Cameroon

E-mail:[email protected]

Department of Mathematics, The University of Yaoundé 1, P.O BOX, 812, Yaoundé Cameroon

E-mail:[email protected]