On Some Geometric Structures Associated to a k-Symplectic Manifold
aa r X i v : . [ m a t h . DG ] F e b SOME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD ADARA M. BLAGA AND BENIAMINO CAPPELLETTI MONTANO
Abstract.
A canonical connection is attached to any k -symplectic manifold. Westudy the properties of this connection and its geometric applications to k -symplecticmanifolds. In particular, we prove that, under some natural assumptions, any k -symplectic manifold admits an Ehresmann connection, discuss some corollaries ofthis result and find vanishing theorems for characteristic classes on a k -symplecticmanifold. Introduction
The theory of k -symplectic manifolds was initiated by A. Awane ([1]), who defineda k -symplectic structure on an n ( k + 1)-dimensional smooth manifold M as an n -codimensional foliation F and a system of k closed 2-forms vanishing on the subbundleof T M defined by F with transversal characteristic spaces (for a precise definition see § k -symplectic geometry has increased especially in recent yearsby the awareness of its relationship with polysymplectic (or multisymplectic) and n -symplectic geometry, and their applications in field theory (cf. [17], [18], [20]). In factthe k -symplectic formalism is the generalization to field theories of the standard sym-plectic formalism in mechanics, which is the geometric framework for describing mostof autonomous mechanical systems. Especially it can be used for giving a geometricdescription of first order field theories in which the Lagrangian and Hamiltonian dependon the first jet (prolongation) of the field.The definition of a k -symplectic manifold is a generalization of the notion of a symplec-tic manifold foliated by a Lagrangian foliation. Thus it is a natural question whether onecan define an appropriate analogue of the well-known notion of bi-Lagrangian structure tothe context of k -symplectic geometry. We recall that an almost bi-Lagrangian manifoldis a symplectic manifold ( M n , ω ) endowed with two transversal Lagrangian distributions L and L . When L and L are both integrable, we speak of bi-Lagrangian manifold.The peculiarity of these geometric structures is that a canonical symplectic connectioncan be attached to them. This connection was introduced by H. Hess ([13]), who wasworking in geometric quantization, and later on its important geometric properties werepointed out by N. B. Boyom ([9]) and I. Vaisman ([22], [23]).In this work we consider the k -symplectic analogue of bi-Lagrangian structure andattach to a such k -symplectic manifold a canonical connection which plays the same rolein k -symplectic geometry as the Hess connection. Moreover we define on a k -symplectic Mathematics Subject Classification.
Primary 53C12, 53C15, Secondary 53B05, 53D05, 57R30.
Key words and phrases. k -symplectic structures, Ehresmann connections, Lagrangian foliations, char-acteristic classes. manifold a family of tensor fields which can be thought as the proper generalizationin this setting of almost K¨ahler structures, and we prove that under some integrabilityassumptions, the above connection coincides with the Levi-Civita connection of a suitablecompatible metric. Finally, as an application, we prove that under some certain naturalassumption, any k -symplectic manifold admits an Ehresmann connection and we deducesome geometric and topological properties on the k -symplectic manifold in question.2. k -symplectic structures A k -symplectic manifold (cf. [1], [19]) is a smooth manifold M together with k closed2-forms ω , . . . , ω k such that(1) C x ( ω ) ∩ · · · ∩ C x ( ω k ) = { } ,(2) ω α ( X, X ′ ) = 0 for any X, X ′ ∈ Γ ( T F ) and for all α ∈ { , . . . , k } ,where C x ( ω ) = { v ∈ T x M : ω x ( v, w ) = 0 for any w ∈ T x M } and F is an nk -dimensionalfoliation on M . It follows that dim ( M ) = n ( k + 1). We will usually denote by L thetangent bundle of the foliation F . In terms of G -structures, a k -symplectic manifoldcan be defined by an integrable Sp ( k, n ; R )-structure, where Sp ( k, n ; R ) denotes the k -symplectic group, defined by the set of matrices of the following type T S . . . ... T S k t ( T − ) where T ∈ Gl ( n ; R ) and S , . . . , S k are n × n real matrices such that T t S α = S tα T forall α ∈ { , . . . , k } . The canonical model of these structures is the k -cotangent bundle( T k ) ∗ N of an arbitrary manifold N , which can be identified with the vector bundle J ( N, R k ) whose total space is the manifold of 1-jets of maps with target 0 ∈ R k ,and projection τ ∗ ( j x, σ ) = x . In this case, identifying ( T k ) ∗ N with the Whitney sumof k copies of T ∗ N , ( T k ) ∗ N ∼ = T ∗ N ⊕ · · · ⊕ T ∗ N , j x, σ ( j x, σ , . . . , j kx, σ k ), where σ α = π α ◦ σ : N −→ R is the α -th component of σ , the k -symplectic structure on ( T k ) ∗ N is given by ω α = ( τ ∗ α ) ∗ ( ω ) and T F j x, σ = ker( τ ∗ ) ∗ ( j x, σ ), where τ ∗ α : ( T k ) ∗ N −→ T ∗ N is the projection on the α -th copy T ∗ N of ( T k ) ∗ N and ω is the standard symplecticstructure on T ∗ N .Returning to the general case of an arbitrary k -symplectic manifold ( M, ω α , F ), foreach α ∈ { , . . . , k } we set(2.1) L α x := \ β = α C x ( ω β ) . Then we have ([3]):(a) for each α ∈ { , . . . , k } the distribution L α = ( L α x ) x ∈ M is integrable (we denoteby F α the foliation integral to L α );(b) L = L ⊕ · · · ⊕ L k ;(c) for each α ∈ { , . . . , k } the map i α : L α −→ ( N F ) ∗ , X i X ω α , is an isomor-phism, where N F denotes the normal bundle of F .The standard Darboux theorem for Lagrangian foliations holds also for k -symplecticmanifolds: Theorem 2.1 ([1]) . About any point of a k -symplectic manifold ( M, ω α , F ) , α ∈ { , ..., k } ,there exist local coordinates { x , . . . , x n , y , . . . , y kn } such that ω α = P ni =1 dx i ∧ dy ( α − n + i OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 3 and F is described by the equations { x i = const. } . In particular, for each α ∈ { , . . . , k } , L α is generated by ∂∂y ( α − n +1 , . . . , ∂∂y αn . Recall that a vector field X on a symplectic manifold ( M n , ω ) is said to be symplecticif L X ω = 0. For k -symplectic manifolds we prove the following lemma which will be usefulin the sequel. Lemma 2.2.
In any k -symplectic manifold, L X ω α = 0 , for any X ∈ Γ ( L β ) with α = β .Proof. Using the Cartan formula for the Lie derivative, we have L X ω α = i X dω α + di X ω α = di X ω α , since ω α is closed. But, for any V ∈ Γ (
T M ), i X ω α ( V ) = 2 ω α ( X, V ) =0 from the definition of L α . (cid:3) A canonical connection on k -symplectic manifolds Let (
M, ω α , F ), α ∈ { , . . . , k } , be a k -symplectic manifold. In what follows, Q willdenote an n -dimensional integrable distribution on M transversal to F such that(i) ω α ( Y, Y ′ ) = 0 for any Y, Y ′ ∈ Γ ( Q ) and for all α ∈ { , . . . , k } ,(ii) [ X, Y ] ∈ Γ ( L α ⊕ Q ) for any X ∈ Γ ( L α ) and for any Y ∈ Γ ( Q ).Occasionally, we will denote by G the foliation integral to Q .The geometric interpretation of the condition (i) is that, for each α ∈ { , . . . , k } andfor any x ∈ M , Q x is a Lagrangian subspace of the symplectic vector space ( L α x ⊕ Q x , ω α x ). The condition (ii) is more technical; it will be essential for proving somepreliminary results, like the following Lemma 3.2, and then for the generalization of theHess’s construction to the k -symplectic setting. Its geometric meaning is that for eachfixed α ∈ { , . . . , k } , the subbundle L α ⊕ Q is integrable, hence it defines a foliationwhose leaves are symplectic manifolds with respect to the restriction of the k -symplecticform ω α to the leaves. We also have that ( L α , Q ) is a bi-Lagrangian structure on theleaves of the foliation defined by L α ⊕ Q .A simple example of a k -symplectic manifold endowed with a transversal integrabledistribution verifying (i) and (ii) is given by R n ( k +1) with its standard k -symplectic struc-ture given by Theorem 2.1 and taking as Q the distribution spanned by ∂∂x , . . . , ∂∂x n .We also remark that the splitting T M = L ⊕ Q = L ⊕· · ·⊕ L k ⊕ Q induces a canonicalisomorphism between Q and N F := T M/L , the normal bundle to the foliation F . Inparticular, it follows that Q ∗ = ann( L ) and, arguing in the same way for the foliation L β = α L β ⊕ Q , we get that L ∗ α = ann( L β = α L β ⊕ Q ), for each α ∈ { , . . . , k } . Takinginto account these remarks, we can prove the following preliminary lemmas: Lemma 3.1.
Let
X, X ′ ∈ Γ ( L ) . For each α ∈ { , . . . , k } , the map ϕ XX ′ α : V ( L X i X ′ ω α ) ( V ) = X ( ω α ( X ′ , V )) − ω α ( X ′ , [ X, V ]) , for any V ∈ Γ( T M ) , belongs to Q ∗ .Proof. For any X ′′ ∈ Γ ( L ), ( L X i X ′ ω α ) ( X ′′ ) = X ( ω α ( X ′ , X ′′ )) − ω α ( X ′ , [ X, X ′′ ]) = 0,from which, since Q ∗ = ann ( L ), we get the result. (cid:3) Lemma 3.2.
Let
Y, Y ′ ∈ Γ ( Q ) . For each α ∈ { , . . . , k } , the map ψ Y Y ′ α : V ( L Y i Y ′ ω α ) ( V ) = Y ( ω α ( Y ′ , V )) − ω α ( Y ′ , [ Y, V ]) , for any V ∈ Γ( T M ) , belongs to L ∗ α . A. M. BLAGA AND B. CAPPELLETTI MONTANO
Proof.
Since L ∗ α = ann( L β = α L β ⊕ Q ), we have to prove that ( L Y i Y ′ ω α ) ( X ) = 0 and( L Y i Y ′ ω α ) ( Y ′′ ) = 0 for any X ∈ Γ ( L β ), β = α , and for any Y ′′ ∈ Γ ( Q ). Indeed,( L Y i Y ′ ω α ) ( X ) = Y ( ω α ( Y ′ , X )) − ω α ( Y ′ , [ Y, X ]) = 0 by the definition of L β and by(ii). Next, ( L Y i Y ′ ω α ) ( Y ′′ ) = Y ( ω α ( Y ′ , Y ′′ )) − ω α ( Y ′ , [ Y, Y ′′ ]) = 0 by (i) and by theintegrability of Q . (cid:3) Theorem 3.3.
Let ( M, ω α , F ) , α ∈ { , ..., k } , be a k -symplectic manifold and let Q be an integrable distribution supplementary to T F verifying the above conditions (i),(ii) and such that ( i ∗ ) − ( ψ Y Y ′ ) = · · · = ( i ∗ k ) − ( ψ Y Y ′ k ) for any Y, Y ′ ∈ Γ( Q ) , where ψ Y Y ′ , . . . , ψ Y Y ′ k are the maps defined in Lemma 3.2. Then there exists a unique connec-tion ∇ on M satisfying the following properties: (1) ∇F α ⊂ F α for each α ∈ { , . . . , k } , and ∇ Q ⊂ Q , (2) ∇ ω = · · · = ∇ ω k = 0 , (3) T ( X, Y ) = 0 for any X ∈ Γ ( L ) and for any Y ∈ Γ ( Q ) ,where T denotes the torsion tensor field of ∇ .Proof. According to the decomposition
T M = L ⊕ · · · ⊕ L k ⊕ Q , we define a connection ∇ L α on each subbundle L α , a connection ∇ Q on Q and then we take the sum of theseconnections for defining a global connection on M . Fix an α ∈ { , . . . , k } . We define ∇ L α Y X := [ Y, X ] L α for any X ∈ Γ ( L α ) and Y ∈ Γ ( Q ). Now we have to define ∇ L α X X ′ for X ∈ Γ ( L ), X ′ ∈ Γ ( L α ). Since i α : L α −→ Q ∗ is an isomorphism for any fixed X ∈ Γ ( L ), X ′ ∈ Γ ( L α ), by Lemma 3.1, there exists a unique section H α ( X, X ′ ) ∈ Γ ( L α ) such that i α ( H α ( X, X ′ )) = ϕ XX ′ α , that is ω α ( H α ( X, X ′ ) , Y ) = X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ X, Y ])for any Y ∈ Γ ( Q ). We set ∇ L α X X ′ := H α ( X, X ′ ) ∈ Γ ( L α ). Now we define the connection ∇ Q . For any X ∈ Γ ( L ) and Y ∈ Γ ( Q ) we put ∇ QX Y := [ X, Y ] Q . It remains to define ∇ QY Y ′ for Y, Y ′ ∈ Γ ( Q ). The isomorphism i α : L α −→ Q ∗ determines an isomorphism i ∗ α between Q and L ∗ α such that i ∗ α ( Y ) ( X ) = ω α ( Y, X ). Then, for any fixed
Y, Y ′ ∈ Γ ( Q ),by Lemma 3.2, there exists a unique section H α ( Y, Y ′ ) ∈ Γ( Q ) such that i ∗ α ( H ( Y, Y ′ )) = ψ Y Y ′ α , that is ω α ( H α ( Y, Y ′ ) , X ) = Y ( ω α ( Y ′ , X )) − ω α ( Y ′ , [ Y, X ]) for any X ∈ Γ ( L α ).Moreover, our assumption ensures that H ( Y, Y ′ ) = · · · = H k ( Y, Y ′ ) =: H ( Y, Y ′ ). Weset ∇ QY Y ′ := H ( Y, Y ′ ) ∈ Γ ( Q ). Now we prove that ∇ Q is a connection on Q and, foreach α ∈ { , . . . , k } , ∇ L α is a connection on L α . For any X ∈ Γ ( L ), Y ∈ Γ ( Q ) and f ∈ C ∞ ( M ) we have ∇ QfX Y = [ f X, Y ] Q = f [ X, Y ] Q − Y ( f ) X Q = f [ X, Y ] Q = f ∇ QX Y, ∇ QX ( f Y ) = [ X, f Y ] Q = f [ X, Y ] Q + X ( f ) Y Q = f ∇ QX Y + X ( f ) Y, and, for any X ∈ Γ ( L α ), Y, Y ′ ∈ Γ ( Q ), ω α ( ∇ QfY Y ′ , X ) = ω α ( H α ( f Y, Y ′ ) , X )= f Y ( ω α ( Y ′ , X )) − ω α ( Y ′ , [ f Y, X ])= f Y ( ω α ( Y ′ , X )) − f ω α ( Y ′ , [ Y, X ]) + X ( f ) ω α ( Y ′ , Y )= f ω α ( H α ( Y, Y ′ ) , X )= ω α ( f ∇ QY Y ′ , X ) , OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 5 from which we get ∇ QfY Y ′ = f ∇ QY Y ′ . Moreover, ω α ( ∇ QY ( f Y ′ ) , X ) = ω α ( H ( Y, f Y ′ ) , X )= Y ( ω α ( f Y ′ , X )) − ω α ( f Y ′ , [ Y, X ])= f Y ( ω α ( Y ′ , X )) + Y ( f ) ω α ( Y ′ , X ) − f ω α ( Y ′ , [ Y, X ])= f ω α ( H ( Y, Y ′ ) , X ) + Y ( f ) ω α ( Y ′ , X )= ω α ( f ∇ QY Y ′ + Y ( f ) Y ′ , X ) , from which we obtain ∇ QY ( f Y ′ ) = f ∇ QY Y ′ + Y ( f ) Y ′ . Now we prove that ∇ L α is aconnection on the subbundle L α , for each α ∈ { , . . . , k } . As before it is easy to showthat ∇ L α fY X = f ∇ L α Y X and ∇ L α Y ( f X ) = f ∇ L α Y X + Y ( f ) X for any X ∈ Γ ( L α ) and Y ∈ Γ ( Q ). Then for any X ∈ Γ ( L ), X ′ ∈ Γ ( L α ) and any Y ∈ Γ ( Q ) ω α ( ∇ L α fX X ′ , Y ) = ω α ( H α ( f X, X ′ ) , Y )= f X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ f X, Y ])= f X ( ω α ( X ′ , Y )) − f ω α ( X ′ , [ X, Y ]) + Y ( f ) ω α ( X ′ , X )= f ω α ( H α ( X, X ′ ) , Y )= ω α ( f ∇ L α X X ′ , Y ) , from which we get ∇ L α fX X ′ = f ∇ L α X X ′ . Moreover, ω α ( ∇ L α X ( f X ′ ) , Y ) = ω α ( H α ( X, f X ′ ) , Y )= X ( ω α ( f X ′ , Y )) − ω α ( f X ′ , [ X, Y ])= f X ( ω α ( X ′ , Y )) + X ( f ) ω α ( X ′ , Y ) − f ω α ( X ′ , [ X, Y ])= f ω α ( H α ( X, X ′ ) , Y ) + X ( f ) ω α ( X ′ , Y )= ω α ( f ∇ L α X X ′ + X ( f ) X ′ , Y )from which we get ∇ L α X ( f X ′ ) = f ∇ L α X X ′ + X ( f ) X ′ . Therefore we can define a globalconnection on M putting, for any V, W ∈ Γ (
T M ),(3.1) ∇ V W = ∇ L V W L + · · · + ∇ L k V W L k + ∇ QV W Q . Now we prove that the connection ∇ satisfies (1)–(3). By construction ∇ preserves thedistributions L α and Q . Then, by (1) we have that, obviously, ( ∇ V ω α ) ( X, X ′ ) = 0 forany X, X ′ ∈ Γ ( L ) and V ∈ Γ (
T M ). For the same reason, ( ∇ V ω α ) ( Y, Y ′ ) = 0 for any Y, Y ′ ∈ Γ ( Q ) and V ∈ Γ( T M ). Now, let X ∈ Γ ( L ), X ′ ∈ Γ ( L α ) and Y ∈ Γ ( Q ). Then( ∇ X ω α ) ( X ′ , Y ) = X ( ω α ( X ′ , Y )) − ω α ( H ( X, X ′ ) , Y ) − ω α ( X ′ , [ X, Y ] Q )= X ( ω α ( X ′ , Y )) − X ( ω α ( X ′ , Y )) + ω α ( X ′ , [ X, Y ]) − ω α ( X ′ , [ X, Y ]) = 0 . Moreover, for any β = α ( ∇ X ω β ) ( X ′ , Y ) = 0 because ∇ X X ′ ∈ Γ ( L α ). Finally, for any X ′ ∈ Γ ( L α ) and Y, Y ′ ∈ Γ ( Q ),( ∇ Y ω α ) ( X ′ , Y ′ ) = Y ( ω α ( X ′ , Y ′ )) − ω α ([ Y, X ′ ] L α , Y ′ ) − ω α ( X ′ , H ( Y, Y ′ ))= Y ( ω α ( X ′ , Y ′ )) − ω α (cid:0) [ Y, X ′ ] L α , Y ′ (cid:1) + Y ( ω α ( Y ′ , X ′ )) − ω α ( Y ′ , [ Y, X ′ ]) = 0 . A. M. BLAGA AND B. CAPPELLETTI MONTANO
Thus we conclude that ( ∇ V ω α ) ( X, Y ) = 0 for any X ∈ Γ ( L α ), Y ∈ Γ ( Q ) and V ∈ Γ (
T M ). Analogously, one can compute for all the other cases, concluding that ∇ ω α = 0for all α ∈ { , . . . , k } . Finally, for any X ∈ Γ ( L α ) and Y ∈ Γ ( Q ) we have T ( X, Y ) =[
X, Y ] Q − [ Y, X ] L α − [ X, Y ] = [
X, Y ] L α ⊕ Q − [ X, Y ] = 0, since by (ii) [
X, Y ] ∈ Γ( L α ⊕ Q ).It remains to prove the uniqueness of this connection up to the properties (1)–(3). Let X ∈ Γ ( L ) and Y ∈ Γ ( Q ). For any X ′ ∈ Γ ( L ) we have, by (1) and (3), ω α ( ∇ X Y, X ′ ) = ω α ( ∇ Y X + [ X, Y ] , X ′ ) = ω α ([ X, Y ] , X ′ ), for all α ∈ { , . . . , k } , from which we get ∇ X Y = [ X, Y ] Q . Then, using (3) again, we obtain ∇ Y X = [ Y, X ] L . Moreover, for any X ∈ Γ ( L ), X ′ ∈ Γ ( L α ) and Y ∈ Γ ( Q ) by (2) we have ω α ( ∇ X X ′ , Y ) = X ( ω α ( X ′ , Y )) − ω α ( X ′ , ∇ X Y ) = X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ X, Y ] Q ) = X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ X, Y ]) = ω α ( H α ( X, X ′ ) , Y ), from which, since ∇ X X ′ , H α ( X, X ′ ) ∈ Γ ( L α ), we get ∇ X X ′ = H α ( X, X ′ ). Similarly, one can find that ∇ Y Y ′ = H ( Y, Y ′ ) for any Y, Y ′ ∈ Γ ( Q ). (cid:3) Proposition 3.4.
The connection ∇ defined in Theorem 3.3 is torsion free along theleaves of the foliations F and G .Proof. Let X ∈ Γ ( L β ) and X ′ ∈ Γ ( L α ) and assume that α = β . We have T ( X, X ′ ) = H α ( X, X ′ ) − H β ( X ′ , X ) − [ X, X ′ ] ∈ Γ ( L ). Then for any Y ∈ Γ ( Q ) ω α ( T ( X, X ′ ) , Y ) = ω α ( H α ( X, X ′ ) − [ X, X ′ ] , Y )= X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ X, Y ]) − ω α ([ X, X ′ ] , Y )= 3 dω α ( X, X ′ , Y ) = 0since each ω α is closed. Analogously, ω α ( T ( X, X ′ ) , Y ) = 0. Moreover, for each γ = α, βω γ ( T ( X, X ′ ) , Y ) = − ω γ ([ X, X ′ ] , Y ) = 3 dω γ ( X, X ′ , Y ) = 0 . Then T ( X, X ′ ) ∈ C ( ω ) ∩ · · · ∩ C ( ω k ) = { } . If X, X ′ ∈ Γ ( L α ), we have T ( X, X ′ ) = H α ( X, X ′ ) − H α ( X ′ , X ) − [ X, X ′ ] ∈ Γ ( L α ) and ω α ( T ( X, X ′ ) , Y ) = X ( ω α ( X ′ , Y )) − ω α ( X ′ , [ X, Y ]) − X ′ ( ω α ( X, Y ))+ ω α ( X, [ X ′ , Y ]) − ω α ([ X, X ′ ] , Y )= 3 dω α ( X, X ′ , Y ) = 0 , hence T ( X, X ′ ) = 0. Analogously, one can prove that T ( Y, Y ′ ) = 0 for any Y, Y ′ ∈ Γ ( Q ). (cid:3) Proposition 3.5.
The curvature tensor field of the connection ∇ defined in Theorem3.3 vanishes along the leaves of the foliations F and G .Proof. For any
X, X ′ ∈ Γ ( L ) and Y ∈ Γ ( Q ), using the integrability of L , we have R X,X ′ Y = ∇ X [ X ′ , Y ] Q − ∇ X ′ [ X, Y ] Q − ∇ [ X,X ′ ] Y = ∇ X [ X ′ , Y ] Q − ∇ X ′ [ X, Y ] Q − [[ X, X ′ ] , Y ] Q = 0by the Jacobi identity. Then, for any X, X ′ ∈ Γ ( L ) and X ′′ ∈ Γ ( L α ) we have(3.2) R X,X ′ X ′′ = H α ( X, H α ( X ′ , X ′′ )) − H α ( X ′ , H α ( X, X ′′ )) − H α ([ X, X ′ ] , X ′′ ) OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 7 Now, for any Y ∈ Γ ( Q ) ω α ( H α ( X, H α ( X ′ , X ′′ )) , Y ) = X ( ω α ( H α ( X ′ , X ′′ ) , Y )) − ω α ( H α ( X ′ , X ′′ ) , [ X, Y ])= X ( ω α ( H ( X ′ , X ′′ ) , Y )) − ω α ( H ( X ′ , X ′′ ) , [ X, Y ])= X ( X ′ ( ω α ( X ′′ , Y ))) − X ( ω α ( X ′′ , [ X ′ , Y ])) − X ′ ( ω α ( X ′′ , [ X ′ , [ X, Y ]]) + ω α ( X ′′ , [ X ′ , [ X, Y ]]) ,ω α ( H α ( X ′ , H α ( X, X ′′ )) , Y ) = X ′ ( ω α ( H α ( X, X ′′ ) , Y )) − ω α ( H α ( X, X ′′ ) , [ X ′ , Y ])= X ′ ( ω α ( H ( X, X ′′ ) , Y )) − ω α ( H ( X, X ′′ ) , [ X ′ , Y ])= X ′ ( X ( ω α ( X ′′ , Y ))) − X ′ ( ω α ( X ′′ , [ X, Y ])) − X ( ω α ( X ′′ , [ X, [ X ′ , Y ]])) + ω α ( X ′′ , [ X, [ X ′ , Y ]])and ω α ( H α ([ X, X ′ ] , X ′′ ) , Y ) = [ X, X ′ ] ( ω α ( X ′′ , Y )) − ω α ( X ′′ , [[ X, X ′ ] , Y ]) . Therefore ω α ( R X,X ′ X ′′ , Y ) = [ X, X ′ ] ( ω α ( X ′′ , Y )) + ω α ( X ′′ , [ X ′ , [ X, Y ]]) − ω α ( X ′′ , [ X, [ X ′ , Y ]]) − [ X, X ′ ] ( ω α ( X ′′ , Y )) + ω α ( X ′′ , [[ X, X ′ ] , Y ])= ω α ( X ′′ , [[ X, X ′ ] , Y ] + [[ X ′ , Y ] , X ] + [[ Y, X ] , X ′ ]) = 0by the Jacobi identity. This shows that R X,X ′ = 0 for any X, X ′ ∈ Γ ( L ). In the sameway, one can prove the flatness along the leaves of the foliation defined by Q . (cid:3) Corollary 3.6.
The leaves of the foliations F and G admit a canonical flat affine struc-ture. Now we give an interpretation of the connection stated in Theorem 3.3 in termsof some geometric structures which can be attached to a k -symplectic manifold. Solet ( M, ω α , F ), α ∈ { , ..., k } , be a k -symplectic manifold and let Q be a distribu-tion transversal to F such that ω α ( Y, Y ′ ) = 0 for any Y, Y ′ ∈ Γ( Q ). Assume that M admits a Riemannian metric g such that the distributions L , . . . , L k , Q are mutu-ally orthogonal. For each α ∈ { , ..., k } , since ω α is non-degenerate on L α ⊕ Q , onecan find a linear map A α : L α ⊕ Q −→ L α ⊕ Q such that ω α ( X, Y ) = g ( X, A α Y ),for any X, Y ∈ Γ( L α ⊕ Q ). The operator A α , α ∈ { , . . . , k } , is skew-symmetric and A α A ∗ α , α ∈ { , . . . , k } , is symmetric and positive definite, thus it diagonalizes with pos-itive eigenvalues ( λ α ) i , i ∈ { , . . . , n } , A α A ∗ α = B α diag { ( λ α ) , . . . , ( λ α ) n } B − α . Set p A α A ∗ α := B α diag { p ( λ α ) , . . . , p ( λ α ) n } B − α which is also symmetric and positivedefinite. Set J α := (cid:26) ( p A α A ∗ α ) − A α , on L α ⊕ Q ;0 , on L β , β = α .Then ( J , . . . , J k ) is a family of endomorphisms of the tangent space satisfying(i) L α = T β = α ker( J β ),(ii) J α = − I on L α ⊕ Q and J α L α = Q , J α Q = L α ,(iii) ω α ( X, Y ) = g ( X, J α Y ) for any X, Y ∈ Γ( T M ).Note also that the Riemannian metric g satisfies g ( J α X, J α Y ) = g ( X, Y ) for each α ∈{ , . . . , k } and for any X, Y ∈ Γ( T M ). We call ( J , . . . , J k , g ) a compatible almost k -K¨ahler structure . Now assume to be under the assumptions of Theorem 3.3. Note thatfor each α ∈ { , . . . , k } , the leaves of the foliation defined by L α ⊕ Q , endowed with the A. M. BLAGA AND B. CAPPELLETTI MONTANO tensor fields induced by J α , are almost K¨ahler manifolds. Then we have that [ J α , J α ] = 0if and only if each leaf of the foliation L α ⊕ Q is K¨ahlerian. When [ J α , J α ] = 0, for each α ∈ { , . . . , k } , that is the leaves of all the foliations L α ⊕ Q are K¨ahler manifolds, wesay that ( M, ω α , F , J α , g ) is a k -K¨ahler manifold . Then we have the following result. Theorem 3.7.
Let ( M, ω α , F , J α , g ) , α ∈ { , . . . , k } , be a k -K¨ahler manifold. If theLevi-Civita connection ∇ g preserves the distributions L α , then it preserves also Q and itcoincides with the canonical connection ∇ .Proof. We show that the Levi-Civita connection ∇ g satisfies the properties (1), (2), (3)which, according to Theorem 3.3, define uniquely the canonical connection ∇ . First of allwe prove that ∇ g preserves Q . Let Y ∈ Γ ( Q ). Then, since ∇ g g = 0, for any V ∈ Γ (
T M )and X ∈ Γ ( T F ), we have0 = ( ∇ gV g ) ( X, Y ) = V ( g ( X, Y )) − g ( ∇ gV X, Y ) − g ( X, ∇ gV Y ) = − g ( X, ∇ gV Y ) , since ∇ g F ⊂ F . Thus ∇ g Q ⊂ Q . Finally we have to prove that ∇ g ω α = 0, for each α ∈ { , . . . , k } . We observe, firstly, that ∇ g J α = 0, for each α ∈ { , . . . , k } . This is aconsequence of the definition of J α , of the fact that the leaves of the foliation definedby L α ⊕ Q are K¨ahlerian manifolds, and of the above properties that ∇ g L α ⊂ L α and ∇ g Q ⊂ Q . Now we can prove that ( ∇ gV ω α )( W, W ′ ) = 0, for any V, W, W ′ ∈ Γ (
T M ).This equality holds immediately for
W, W ′ ∈ Γ ( L ) and for W, W ′ ∈ Γ ( Q ) because L and Q are preserved by ∇ g . So it remains to show that ( ∇ gV ω α ) ( X, Y ) = 0, for any X ∈ Γ ( L ) and Y ∈ Γ ( Q ). In fact, since ∇ g J α = 0 and ∇ g g = 0,( ∇ gV ω α ) ( X, Y ) = V ( g ( X, J α Y )) − g ( ∇ gV X, J α Y ) − g ( X, J α ∇ gV Y )= V ( g ( X, J α Y )) − g ( ∇ gV X, J α Y ) − g ( X, ∇ gV J α Y )= ( ∇ gV g ) ( X, J α Y ) = 0 . This concludes the proof. (cid:3) Applications
In this section we will examine some consequences of Theorem 3.3. It can be use-ful to find the connection defined in Theorem 3.3 in Darboux coordinates { x , . . . , x n ,y , . . . , y kn } according to Theorem 2.1. There exist functions t αji such that Q =span { X , . . . , X n } , where X i := ∂∂x i − P kα =1 P nj =1 t αji ∂∂y ( α − n + j . We put Y αi := ∂∂y ( α − n + i . Then by a straightforward computation we have that ∇ Y αi Y βj = 0 , ∇ Y αi X j = 0 , ∇ X i Y αj = k X β =1 n X h =1 ∂t βhi ∂y ( α − n + j Y βh , ∇ X i X j = − n X h =1 ∂t αji ∂y ( α − n + h X h , OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 9 where the functions t αji satisfy the conditions ∂t αji ∂y ( β − n + h = 0 for α = β , and ∂t ji ∂y h = · · · = ∂t kji ∂y ( k − n + h , for all i, j, h ∈ { , . . . , n } . Moreover, the curvature is given by R Y αi ,Y βj = 0 , R X i ,X j = 0 , (4.1) R Y αi ,X j Y βh = k X γ =1 n X l =1 ∂ t γ l j ∂y ( α − n + i ∂y ( β − n + h Y γl , (4.2) R Y αi ,X j X k = − n X l =1 ∂ t α k i ∂y ( α − n + i ∂y ( α − n + l X l . (4.3)Then we have that the curvature 2-form of ∇ has the following very simple expression :Ω = X Ω α i j dx i ∧ dy ( α − n + j . from which it follows that Ω h vanishes for h > n . Thus if f ∈ I h ( G ) is an ad ( G )-invariant polynomial of degree h , where G = Sp ( k, n ; R ), we have that f (Ω) = 0 for h = deg ( f ) > n . This proves the following result. Proposition 4.1.
Under the assumptions of Theorem 3.3, we have that
Pont j ( T M ) = 0 for all j > n , where Pont(
T M ) denotes the Pontryagin algebra of the bundle T M . Another strong consequence of Theorem 3.3 is the existence of an Ehresmann con-nection. We recall the concept of
Ehresmann connection for foliations. Let ( M, F ) bea foliated manifold and D a distribution on M which is supplementary to the tangentbundle L of the foliation F at every point. A horizontal curve is a piecewise smoothcurve β : [0 , b ] −→ M , b ∈ R , such that β ′ ( t ) ∈ D β ( t ) for all t ∈ [0 , b ]. A vertical curve is a piecewise smooth curve α : [0 , a ] −→ M , a ∈ R , such that α ′ ( t ) ∈ L α ( t ) for all t ∈ [0 , a ], i.e. which lies entirely in one leaf of F . A rectangle is a piecewise smooth map σ : [0 , a ] × [0 , b ] −→ M such that for every fixed s ∈ [0 , b ] the curve σ s := σ | [0 ,a ] ×{ s } is vertical and for every fixed t ∈ [0 , a ] the curve σ t := σ | { t }× [0 ,b ] is horizontal. Thecurves σ = σ ( · , σ b = σ ( · , b ), σ = σ (0 , · ) and σ a = σ ( a, · ) are called, respectively,the initial vertical edge , the final vertical edge , the initial horizontal edge and the finalhorizontal edge of σ . We say that the distribution D is an Ehresmann connection for F if for every vertical curve α and horizontal curve β with the same initial point, thereexists a rectangle whose initial edges are α and β (cf. [6]). This rectangle is unique andis called the rectangle associated to α and β . It is known ([5]) that every totally geodesicfoliation of a complete Riemannian manifold admits an Ehresmann connection, namelythe distribution orthogonal to the leaves of the foliation. Furthermore, by the dualityRiemannian – totally geodesic, the orthogonal bundle to a Riemannian foliation is alsoan Ehresmann connection for this foliation.Recall that given a foliated manifold ( M, F ) and a supplementary subbundle D to T F (not necessarily an Ehresmann connection), any horizontal curve τ : [0 , −→ M definesa family of diffeomorphisms { ϕ t : V −→ V t } t ∈ [0 , such that(1) each V t is a neighborhood of τ ( t ) in the leaf of F through τ ( t ), for all t ∈ [0 , ϕ t ( τ (0)) = τ ( t ) for all t ∈ [0 , p ∈ V the curve t ϕ t ( x ) is horizontal, Throughout all this work, if no confusion is feared, we identify forms on M with their lifts to principalbundle of linear frames LM . (4) ϕ : V −→ V is the identity map.This family of diffeomorphisms is called an element of holonomy along τ ([6]). It isshown in ([14]) and in ([5]) that an element of holonomy along τ exists and is unique, inthe sense that any two elements of holonomy must agree on some neighborhood of τ (0)in the leaf through τ (0). When the leaves of F have a geometric structure – such as aRiemannian metric or a linear connection – we say that D preserves the geometry of theleaves if the element of holonomy along any horizontal curve is a local isomorphism ofthe particular geometric structure.Using the canonical connection which we have defined in § Theorem 4.2.
Let ( M, ω α , F ) , α ∈ { , . . . , k } , be a compact connected k -symplecticmanifold and let Q be an integrable distribution transversal to F and satisfying the as-sumptions of Theorem 3.3. If the leaves of F are complete affine manifolds, then thedistribution Q is an Ehresmann connection for F . Furthermore, if the canonical con-nection ∇ on M induced by Q is everywhere flat, then the Ehresmann connection Q preserves ∇ .Proof. Let α : [0 , a ] −→ M and β : [0 , b ] −→ M be, respectively, a vertical and ahorizontal curve such that α (0) = x = β (0). We need to show that there exists a fullrectangle σ : [0 , a ] × [0 , b ] −→ M whose initial vertical and horizontal edges are just α and β , respectively. First we will show it under the further assumption that α is a geodesic(with respect to the connection ∇ ). Fix an s ∈ [0 , b ]. We transport by parallelism thevector α ′ (0) along the curve β , obtaining a vector v s ∈ T β ( s ) M which is in turn tangentto F since the ∇ -parallel transport preserves the foliation F (note also that the vector v s does not depend on the curve because the curvature vanishes identically). Let τ s bethe geodesic determined by the initial conditions τ s (0) = β ( s ) and τ ′ s (0) = v s . Sincethe foliation F is totally geodesic (with respect to ∇ ), τ s is a curve lying on the leaf L s of F passing for β ( s ), and the assumption on the completeness of L s implies thatwe can extend τ s for all the values of the parameter t . In this way we obtain a map σ : [0 , a ] × [0 , b ] −→ M , defined by σ ( t, s ) := τ s ( t ), and it is easy to show that it isjust the rectangle we are looking for. Now we have to prove the theorem dropping theassumption that the curve α is a geodesic. Because M is compact and the leaves of F are complete affine manifolds with respect to ∇ , we find ǫ > x ∈ M ,the ǫ -ball B ( x, ǫ ) is convex. As the leaves are totally geodesic, the ǫ -balls B L ( x, ǫ ) in anyleaf L coincide with the corresponding connected components of B ( x, ǫ ) ∩ L . Therefore,for any x ∈ M , there exists ǫ > ǫ -balls B L ( x, ǫ ) are convex. Suppose nowthat α : [0 , a ] −→ M is a vertical curve contained in B L ( x, ǫ ), with x = α (0). Let α t denote the geodesic on L joining x with α ( t ), for any fixed t ∈ [0 , a ]. Then we define σ ( t, s ) := σ α t ,β | [0 ,s ] ( t, s ) , for any ( t, s ) ∈ [0 , a ] × [0 , b ], where σ α t ,β | [0 ,s ] denotes the rectangle associated to thecurves α t and β | [0 ,s ] . By the first part of the proof, σ is just the rectangle whose initialedges are α and β . Finally, if α is any leaf curve on M , not necessarily contained in B L ( x, ǫ ), then we can always find a partition of [0 , a ], say 0 = t < t < · · · < t m = a ,with the property that, for any i ∈ { , . . . , m − } , α ( t i ) , α ( t i +1 ) ∈ B ( α ( t i ) , ǫ ). Let σ (0) be the rectangle corresponding to α | [0 ,t ] and β . The curve β := σ (0) | { t }× [0 ,b ] ishorizontal and β (0) = α ( t ), so we can find a rectangle σ (1) whose edges are α | [ t ,t and β . After m steps we have m rectangles σ (0) , σ (1) , . . . , σ ( m − and we can define OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 11 σ := σ (0) ∪ σ (1) ∪ · · · ∪ σ ( m − obtaining the rectangle whose initial edges are α and β .The last part of the statement follows directly from ([6, Proposition 5.3]). (cid:3) The existence of an Ehresmann connection implies strong consequences for the fo-liation. Many of them have been studied in ([6]), from which we have the followingresults.
Corollary 4.3.
Let ( M, ω α , F ) , α ∈ { , . . . , k } , be a k -symplectic manifold satisfyingthe assumptions of Theorem 4.2. Then the following statements hold: (a) Any two leaves of F can be joined by a horizontal curve. (b) The universal covers of any two leaves of F are isomorphic. (c) The universal cover ˜ M of M is topologically a product ˜ L × ˜ Q , where ˜ L is theuniversal cover of the leaves of F and ˜ Q the universal cover of the leaves of thefoliation integral to Q . In general, to each leaf L of a foliation admitting an Ehresmann connection D , it isattached a group H D ( L , x ), x ∈ L , defined as follows ([6]). Let Ω x be the set of all hori-zontal curves β : [0 , −→ M with starting point x . Then there is an action of the funda-mental group π ( L , x ) of L on Ω x given in the following way: for any δ = [ τ ] ∈ π ( L , x )and for any β ∈ Ω x , τ · β is the final horizontal edge of the rectangle correspondingto τ and β . It can be proved that this definition does not depend on the vertical loop τ in x representing δ . Let K D ( L , x ) = { δ ∈ π ( L , x ) : τ · β = β for all β ∈ Ω x } . Then K D ( L , x ) is a normal subgroup of π ( L , x ) and we define H D ( L , x ) := π ( L , x ) /K D ( L , x ) . It is known that H D ( L , x ) does not depend on the Ehresmann connection D , thus it isan invariant of the foliation. Then we have the following result. Corollary 4.4.
Let ( M, ω α , F ) , α ∈ { , . . . , k } , be a k -symplectic manifold satisfyingTheorem 4.2. If F has a compact leaf L with finite H D ( L , x ) , then every leaf L of F is compact with finite H D ( L , x ) .Proof. It is a direct consequence of ([7, Theorem 1]). (cid:3)
Another consequence of Theorem 4.2 is the following result.
Corollary 4.5.
Let ( M, ω α , F ) , α ∈ { , . . . , k } , be a k -symplectic manifold satisfying theassumptions of Theorem 4.2. Then F has no vanishing cycles. Moreover, the homotopygroupoid of F is a Hausdorff manifold.Proof. The assertions follow from ([25, Theorem 2]) and ([25, Corollary 2]). (cid:3)
Now we study more deeply k -symplectic manifolds whose canonical connections areflat. From (4.1)–(4.3) it follows that the geometric interpretation of the flatness of ∇ isthat the functions t α j i are leaf-wise affine. Usually this condition is expressed saying that Q is an affine transversal distribution for F (see, for instance, [22], [23]). In the followingtheorem we give a normal form for flat k -symplectic manifolds: Theorem 4.6.
Let ( M, ω α , F ) , α ∈ { , . . . , k } , be a k -symplectic manifold and Q adistribution satisfying the assumptions of Theorem 3.3. If the corresponding canonicalconnection ∇ is flat, then there exist local coordinates { x , . . . , x n , y , . . . , y kn } with re-spect to which each -form ω α is given by (4.4) ω α = n X i =1 dx i ∧ dy ( α − n + i , F is described by the equations { x = const., . . . , x n = const. } and Q is spanned by ∂∂x , . . . , ∂∂x n .Proof. Let x ∈ M be a point and U ⊂ M a chart containing x . One can consider anadapted basis { e , . . . , e n ( k +1) } of T x M such that, for each α ∈ { , . . . , k } , { e ( α − n +1 , . . . ,e αn } is a basis of L α x , { e kn +1 , . . . , e n ( k +1) } is a basis of Q x , and ω α (cid:0) e ( β − n + i , e ( γ − n + j (cid:1) = ω α ( e kn + i , e kn + j ) = 0 , (4.5) ω α (cid:0) e ( β − n + i , e kn + j (cid:1) = − δ αβ δ ij , (4.6)for all α, β, γ ∈ { , . . . , k } , i, j ∈ { , . . . , n } . For each l ∈ { , . . . , n ( k + 1) } we define avector field E k on U by the ∇ -parallel transport along curves. More precisely, for any y ∈ U we consider a curve γ : [0 , −→ U such that γ (0) = x , γ (1) = y and define E l ( y ) := τ γ ( e l ), τ γ : T x M −→ T y M being the parallel transport along γ . Note that E l ( y ) does not depend on the curve joining x and y , since R ≡
0. Thus we obtain n ( k +1)vector fields on U , E , . . . , E n ( k +1) such that, for each α ∈ { , . . . , k } , i ∈ { , . . . , n } , E ( α − n + i ∈ Γ( L α ) and E kn + i ∈ Γ( Q ), since the connection ∇ preserves the subbundles L α and Q . Moreover, by (4.5)–(4.6) we have for any y ∈ U and α, β, γ ∈ { , . . . , k } , i, j ∈ { , . . . , n } ω α (cid:0) E ( β − n + i , E ( γ − n + j (cid:1) = ω α ( E kn + i , E kn + j ) = 0 , (4.7) ω α (cid:0) E ( β − n + i , E kn + j (cid:1) = − δ αβ δ ij . (4.8)Indeed, for all l, m ∈ { , . . . , n ( k + 1) } , ddt ω α ( E l ( γ ( t )) , E m ( γ ( t ))) = ω α ( ∇ γ ′ E l , E m ) + ω α ( E l , ∇ γ ′ E m ) = 0because ω α is parallel with respect to ∇ . Thus ω α x ( e l , e m ) = ω α y ( E l ( y ) , E m ( y )),for any y ∈ U . Note that, by construction, we have ∇ E l E m = 0 for all l, m ∈{ , . . . , n ( k + 1) } . ¿From this, Theorem 3.3 and Proposition 3.4 it follows that the vec-tor fields E , . . . , E n ( k +1) commute each other. Therefore there exist local coordinates { x , . . . , x n , y , . . . , y kn } , α ∈ { , . . . , k } , such that E ( α − n + i = ∂∂y α i and E kn + j = ∂∂x j ,for any i, j ∈ { , . . . , n } . Note that by (4.7)–(4.8) we get that ω α = P ni =1 dx i ∧ dy ( α − n + i .Thus, with respect this coordinate system,(i) each L α is spanned by ∂∂y ( α − n +1 , . . . , ∂∂y αn ,(ii) Q is spanned by ∂∂x , . . . , ∂∂x n ,(iii) the k -symplectic forms ω α are given by ω α = P ni =1 dx i ∧ dy ( α − n + i .This proves the assertion. (cid:3) Remark . Theorem 4.6 should be compared with Theorem 2.1. It should be remarkedthat according to Theorem 2.1 there always exist local coordinates { x , . . . , x n , y , . . . , y kn } verifying (4.4) and such that the foliation F is locally given by the equations { x =const. , . . . , x n = const. } . On the other hand, by the general theory of foliations there al-ways exists local coordinates { x ′ , . . . , x ′ n , y ′ , . . . , y ′ kn } with respect to which the foliationdefined by Q is described by the equations { y ′ = const. , . . . , y ′ kn = const. } . In generalthese two types of coordinate systems do not coincide. Theorem 4.6 just states that asufficient condition for this is expressed by the flatness of the canonical connection. Notethat this condition is also necessary, as easily it follows from (4.1)–(4.3). OME GEOMETRIC STRUCTURES ASSOCIATED TO A k -SYMPLECTIC MANIFOLD 13 Acknowledgments
The authors thank the referees for their useful remarks.
References [1] A. Awane, k -symplectic structures , J. Math. Phys. (1992), 4046–4052.[2] A. Awane, G -spaces k -symplectic homog`enes , J. Geom. Phys. (1994), 139-157.[3] A. Awane, Some affine properties of the k -symplectic manifolds , Beitr. Algebra Geom. (1998),75–83.[4] A. Awane, M. Goze, Pfaffian systems, k -symplectic systems , Kluwer Academic Publishers, 2000.[5] R. Blumenthal, J. Hebda, De Rham decomposition theorems for foliated manifolds , Ann. Inst.Fourier (1983), 183–198.[6] R. Blumenthal, J. Hebda, Ehresmann connections for foliations , Indiana Univ. Math. J. n. 4(1984), 597–611.[7] R. Blumenthal, J. Hebda, Complementary distributions which preserve the leaf geometry and ap-plications to totally geodesic foliations , Quart. J. Math. Oxford (1984), 383–392.[8] R. Bott, Lectures on characteristic classes and foliations , Lect. Notes Math. (1972), 1–94.[9] N. B. Boyom,
Metriques Kahleriennes affinement plates de certaines varietes symplectiques I , Proc.London Math. Soc. (1993), 35880.[10] G. B. Byrnes, H. W. Capel, F. A. Haggar, G. R. W. Quispel, k -integrals and k -Lie symmetries indiscrete dynamical systems , Physica A (1996), 379–394.[11] F. Etayo, R. Santamaria, The canonical connection of a bi-Lagrangian manifold , J. Phys. A.: Math.Gen. (2001), 981–987.[12] C. G¨unther, The polysymplectic Hamiltonian formalism in field theory and calculus of variationsI: The local case , J. Differential Geom. (1987), 23-53.[13] H. Hess, Connections on symplectic manifolds and geometric quantization , Lect. Notes Math. (1980), 153–166.[14] D. Johnson, L. Whitt -
Totally geodesic foliations , J. Differential Geom. (1980), 225–235[15] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol. II , Interscience Publishers,1969.[16] M. de Le´on, E. Merino, J. A. Oubi˜na, P. R. Rodrigues, M. A. Salgado,
Hamiltonian systems on k -cosymplectic manifolds , J. Math. Phys. (1998), 876–893.[17] M. de Le´on, M. McLean, L. K. Norris, A. Rey Roca, M. Salgado, Geometric structures in fieldtheory , arXiv.math-ph/0208036 v1 (2002).[18] E. Merino, k -symplectic and k -cosymplectic geometries. Applications to classical field theory . Pub-blicaciones del Departemento de Geometria y Topologia, Universitad de Santiago de Compostela , 1997.[19] M. Puta, Some remarks on the k -symplectic manifolds , Tensor (1988), 109–115.[20] A. M. Rey, N. Rom´an-Roy, M. Salgado, G¨unter’s formalism ( k -symplectic formalism) in classicalfield theory: Skinner-Rusk approach and the evolution operator , J. Math. Phys. (2005).[21] V. Rovenskij, Foliations on Riemannian manifolds and submanifolds , Birkh¨auser, 1998.[22] I. Vaisman, d f -cohomology of Lagrangian foliations , Monash. Math. (1998), 221–244.[23] I. Vaisman, Basic of Lagrangian foliations , Publ. Mat. (1989), 559–575.[24] R. Wolak, Ehresmann connections for Lagrangian foliations , J. Geom. Phys. (1995), 310–320.[25] R. Wolak, Graphs, Ehresmann connections and vanishing cycles , Proc. Conf. Differential geometryand applications (1996), 345–352.[26] R. Wolak,
On leaves of Lagrangian foliations , Russian Mathematics (Izvestiya VUZ. Matematika) n. 6 (1998), 24–31. Facultatea de Matematic˘a s¸i Informatic˘a, Universitatea de Vest din Timis¸oara, Bd. VasilePˆarvan 4, 300223 Timis¸oara, Romania
E-mail address : [email protected] Dipartimento di Matematica, Universit`a degli Studi di Bari, Via Edoardo Orabona 4, 70125Bari, Italy
E-mail address ::