ON A CLASS OF EVEN-DIMENSIONAL MANIFOLDS STRUCTURED BY AN AFFINE CONNECTION

We deal with a 2m-dimensional Riemannian manifold (M,g) structured by an affine connection and a vector field , defining a -parallel connection. It is proved that is both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature 2-forms are established. It is shown that M is endowed with a conformal symplectic structure Ω and defines a relative conformal transformation of Ω.


Introduction.
In [5], a class of odd-dimensional manifolds endowed with a -parallel connection was investigated.
In the present paper, we consider a 2m-dimensional Riemannian manifold (M, g), structured by an affine connection defined by the torsion 2-forms S A , A ∈ {1, 2,..., 2m}.If {e A } and {ω A } are a vector and a covector basis, respectively, and -(T A ) a vector field (called the structure vector field of M), we assume thatdefines a -parallel connection, in the sense of [9] (see also [2,4]), that is, the connection forms associated with {e A } and {ω A } satisfy where ∧ means the wedge product of vector fields, which implies ∇ -e A = 0. Next, we assume that the torsion forms S A are exterior recurrent (abbreviated ER) [1] with α =as recurrence form, that is, Assuming that T A are also ER with a certain Pfaffian u as recurrence form, that is, dT A = T A u, and denoting 2t = -2 , we have where dp is the soldering form of M [3], which says thatis a torse forming vector field [8,11,12].We derive that is,is an exterior concurrent vector field [10] (see also [4]).
Setting S = S 1 ∧ S 2 ∧•••∧S 2m , we find that the 4m-form S associated with M is ER with 4mα as recurrence form.
It is shown that the curvature 2-forms Θ A B are ER having the closed 1-form 2(u + α) as recurrence form.We agree to define such a manifold as an exterior recurrent curvature 2-form manifold.
Finally, assuming that M carries an almost symplectic form Ω, that is, a nondegenerate differential 2-form, we prove that Ω is a conformal symplectic form.
It is shown thatdefines a relative conformal transformation of the conformal symplectic form Ω (see [5]).
The above results are stated in Theorem 3.1.

Preliminaries.
Let (M, g) be a 2m-dimensional oriented Riemannian manifold structured by an affine differential operator ∇.
Let Γ (T M) be the set of sections of the tangent bundle and : T M → T * M and : T * M → T M the classical musical isomorphisms defined by g (i.e., is the index lowering operator and is the index raising operator).
Following [7], we denote by the set of vector-valued q-forms (q ≤ dim M) and we write for the affine operator ∇ is the canonical vector-valued 1-form of M, then as an extension of the Levi-Civita operator and by [3], we agree to call dp the soldering form of M.
Let the unit vector fields {e A } be an orthonormal vector basis and {ω A } its corresponding cobasis on M, A = 1,...,2m.Then, if θ A B , S A , and Θ A B denote the connection forms, the torsion 2-forms and the curvature 2-forms, respectively, Cartan's structure equations are expressed by ) ) (2.5) We recall the following definitions (cf.[4]).
A vector fieldis said to be a torse forming vector field [12] if it satisfies Also, the vector fieldis called exterior concurrent [10] if If Z, Z ∈ Γ (T M), we also have the following formula: where ᏸ is the Lie derivative.
Since dp = ω A ∧ e A , then it follows that (2.9)

Manifolds with affine connection.
In the present paper, we assume first that the 2m-dimensional Riemannian manifold (M, g) carries a structure vector field -(T A ) which defines a --parallel connection, in the sense of [9] (see also [2,4]).Such a connection is expressed by Since we quickly find from (3.1) that this agrees with the definition of --parallel connection.Setting 2t = T 2 , we derive where α =is the dual 1-form of -.Also, we find by (3.1) and (2.4) that Second, we assume that the torsion forms S A are exterior recurrent [1] having α as recurrence form, that is, and T A are ER with the Pfaffian u as recurrence form, that is, We obtain dα = 0, that is, α =is a closed vector field.Under these conditions, it follows from (3.3) and (3.6) that this proves thatis a torse forming vector field [4,8,11,12].Since the operator ∇ acts inductively and clearly by (3.6), then we infer This means that the vector fieldis an exterior concurrent vector field [6,10].
By [6], (3.9) implies that where denotes the Ricci tensor field on M. By (3.9) and by standard calculation, we derive and therefore we may say that the vector fieldis an element of On the other hand, recall that the Bianchi forms in the sense of Tachibana are defined by where Ω α q+1 αq are 2-forms.Thus, setting we find that dS = 4mα ∧ S.
(3.15) Therefore, we may say that the 4m-form S associated with M is ER with 4mα as recurrence form.
By (3.4) we may set and by (3.1) and the structure equations (2.5) we get after some calculations Next, performing the exterior differentiation of Θ A B , we derive, taking account of (3.8) This shows that all curvature forms Θ A B are ER and have the closed 1-form 2(u+α) as recurrence form.
We agree to define such an even-dimensional manifold M as an exterior recurrent curvature 2-form manifold.
Finally, assume that M carries an almost symplectic form Ω. Then, we may express Ω as Taking the exterior differentiation of Ω, we find by (3.4) and (3.16) that dΩ = 2(α + u) ∧ Ω. (3.20)This shows that the manifold under consideration is endowed with a conformal symplectic structure having α + u as covector of Lee.
Moreover, taking the Lie differentiation of Ω with respect to the structure vector field -, we infer Hence, by [4], the above equation says thatdefines a relative conformal transformation of the conformal symplectic form Ω.
Summing up, we state the following theorem.
Theorem 3.1.Let (M, g) be a 2m-dimensional Riemannian manifold structured by an affine connection defined by the torsion 2-forms S A , A = 1,...,2m.Let -(T A ) be a structure vector field, which defines a --parallel connection and assume that S A are exterior recurrent, havingas recurrence form (-= α is a closed Pfaffian).
Then the following properties hold: (i)is both a torse forming and an exterior concurrent vector field; (ii) the structure curvature 2-forms Θ A B are exterior recurrent with the closed Pfaffian 2(u + α) as recurrence form; (iii) the manifold M is endowed with a conformal symplectic structure Ω having u + α as covector of Lee; (iv) the vector fielddefines a relative conformal transformation of Ω, that is, dᏸ -Ω = 8tu ∧ Ω, where 2t = -2 .