© Hindawi Publishing Corp. SKEW-SYMMETRIC VECTOR FIELDS ON A CR-SUBMANIFOLD OF A PARA-KÄHLERIAN MANIFOLD

We deal with a CR-submanifold M of a para-Kahlerian manifold M˜, which carries a J-skew-symmetric vector field X. It is shown that X defines a global Hamiltonian of the symplectic form Ω on M⊤ and JX is a relative infinitesimal automorphism of Ω. Other geometric properties are given.

If M is a para-Kählerian manifold, it has been proved that any coisotropic submanifold M of M is a CR-submanifold (such CR-submanifolds have been denominated CICRsubmanifolds [6]).
In this note, one considers a foliate CICR-submanifold M of a para-Kählerian manifold M(J, Ω, g).It is proved that the necessary and sufficient condition in order that the leaf M of the horizontal distribution D on M carries a J-skew-symmetric vector field X, that is, ∇X = X ∧ JX, is that the vertical distribution D ⊥ on M is autoparallel.
In this case, M may be viewed as the local Riemannian product M = M × M ⊥ , where M is an invariant totally geodesic submanifold of M and M ⊥ is an isotropic totally geodesic submanifold.
Furthermore, if Ω is the symplectic form of M , it is shown that X is a global Hamiltonian of Ω and JX is a relative infinitesimal automorphism of Ω (a similar discussion can be made for proper CR-submanifolds of a Kählerian manifold).

Preliminaries.
Let M(J, Ω, g) be a 2m-dimensional para-Kählerian manifold, where, as is well known [7], the triple (J, Ω, g) of tensor fields is the paracomplex operator, the symplectic form, and the para-Hermitian metric tensor field, respectively.
If ∇ is the Levi-Civita connection on M, these manifolds satisfy Let x : M → M be the immersion of an l-codimensional submanifold M, l < m, in M and let T ⊥ p M and T p M be the normal space and the tangent space at each point p ∈ M.

If J(T ⊥
p M) ⊂ T p M, then M is said to be a coisotropic submanifold of M (see [2]).If W = vect{h a ,h a * ; a = 1,...,m, a * = a + m} is a real Witt vector basis on M, one has (2.2) Next, if W * = {ω a ,ω a * } denotes the associated cobasis of W , then g and Ω are expressed by ) We recall also that W may split as where the pairing ( S, S * ) defines an involutive automorphism of square 1, that is, and the local connection forms It has been proved in [10] that any coisotropic submanifold M of a para-Kählerian manifold M is a CR-submanifold of M and such a submanifold has been called a CICRsubmanifold [6].
Let D : p → D p = T p M \ J(T ⊥ p M) and D ⊥ : p → D ⊥ p = J(T ⊥ p M) ⊂ T p M be the two complementary differentiable distributions on M. One has and D (resp., D ⊥ ) is called the horizontal (resp., vertical) distribution on M.
As in the Kählerian case, the vertical distribution D ⊥ is always involutive.If M is defined by the Pfaffian system then one has (2.10) Further denote by the simple unit form which corresponds to D ⊥ .
Then, in order that the distribution D be also involutive, it is necessary and sufficient that ϕ ⊥ be a conformal integral invariant of D , that is, for a certain scalar function f .By a standard calculation, one derives that the above equation implies and in this case, one may write that is, ϕ ⊥ is exterior recurrent.In this case, as is known [2,10], M is a foliated CR-submanifold of M.
We will investigate now the case when the leaf M of D carries a J-skew-symmetric vector field X, that is, (2.15) One may express ∇X as where (2.17) Recalling Cartan structure equations [4], (2.18) In the above equations, θ, respectively Θ, are the local connection forms in the bundle W , respectively the curvature forms on M.
Then making use of Cartan structure equations, one finds by a standard calculation that (2.16) implies that the vertical distribution D ⊥ is autoparallel, that is, ∇ Z Z ∈ D ⊥ , for all Z ,Z ∈ D ⊥ , which, in terms of connection forms, is expressed by We agree to call θ i r and θ r i the mixed connection forms.Taking account of (2.13) and (2.19), one derives from (2.16) which agrees with the general equation of skew-symmetric killing vector fields [5,8].
Next, by (2.1), one has which shows that JX is a gradient vector field.Hence, we may state the following theorem.
Theorem 2.1.Let x : M → M be an improper immersion of a CR-submanifold in a para-Kählerian manifold M(J, Ω, g) and let D (resp., D ⊥ ) be the horizontal distribution (resp., the vertical distribution) on M. If M is a foliate CR-submanifold, then the necessary and sufficient condition in order that the leaf M of D carries a J-skew-symmetric vector field X is that D ⊥ is an autoparallel foliation.In this case, the CR-submanifold M under consideration may be viewed as the local Riemannian product M = M × M ⊥ , where M is an invariant totally geodesic submanifold of M and M ⊥ is an isotropic totally geodesic submanifold.In addition, in this case, JX is a gradient vector field.

Properties.
In this section, we will pointout some additional properties of X involving the symplectic form Ω of M and the exterior covariant differential d ∇ of ∇X.Operating on (2.16) and (2.21), one derives by a short calculation which gives Therefore, we agree to define X +JX and X − JX as 2-covariant recurrent vector fields.It should also be noticed that by reference to the general formula one finds by (2.15) and (2.21) This shows that the covariant derivative of X ∧ JX with respect to any vector field V is proportional to X ∧ JX.
On the other hand, by the general formula where R denotes the curvature tensor field and V , Z, Z are vector fields, one has (see also [9]) where is the Ricci tensor field of ∇.Since in the case under consideration one must take in (3.6) the para-Hermitian trace, then setting in (3.6) that is, the Ricci curvature of X vanishes.Denote by Ω the symplectic form of M, then Ω = Ω| M is a symplectic form of rank equal to the dimension of M , that is, in our case, 2(m − l).
Then, if Z : Z → −i Z Ω is the symplectic isomorphism, by a short calculation and on behalf of (2.4), one gets and since JX is a gradient vector field, we conclude according to a known definition (see also [1]) that X is a global Hamiltonian of Ω.
In a similar manner, one finds and by (2.20), it follows that d ᏸ JX Ω = 0, (3.10) which shows that JX is a relative infinitesimal automorphism of Ω [1].We state the following theorem.
Theorem 3.1.Let M be a CR-submanifold of a para-Kählerian manifold M and let Ω be the symplectic form on M .If M carries a J-skew-symmetric vector field X, then the following properties hold: (i) X is a global Hamiltonian of Ω and JX is a relative infinitesimal automorphism of Ω; (ii) the Ricci tensor field (X, X) vanishes; (iii) the vector fields X + JX and X − JX are 2-covariant recurrent.