Skew Symmetric Killing Vector Fields on a paraK¨ahelerian Manifold David Carf`ı Dedicated to Professor M.T. Calapso, with admiration David Carf`ı Abstract The aim of this paper is to prove the following theorem: Let M (J, Ω, g) be a 2m-dimensional para K¨ ahlerian manifold with structure 2-form Ω, and let X (resp. Y ) be a skew symmetric Killing vector field on M (resp. the generative of X). Then as in the Riemannian case, the dual forms X [ = α and Y [ = β of X and Y satisfy Rosca’s lemma dα = 2α ∧ β.
Introduction
ParaK¨ ahelerian Manifolds have been for the first time studied by [L] and [P ]. In the last two decades, several authors dealt with such type of manifolds, as for instance [R] , [EG] and some others. On the other hand skew symmetric Killing vector fields (abbr. S.S.K.) have been defined by [R2]. If X is such a vector field, then on a Riemannian manifold it satisfies (0.1)
∇X = X ∧ Y
where ∧ denotes the wedge product of vector fields and Y is defined as the generative of X. If α and β define the dual forms of X and Y respectively, i.e., α = X [,
β =Y[
then the following relation, called Rosca’s lemma, holds good (0.2)
dα = 2α ∧ β.
In the present note if X is a S.S.K. vector field on a paraK¨ahelerian manifold M , it is shown that (0.2) holds also good. Therefore it is proved that Rosca’s lemma has an intuitive character. 1
1.
Preliminaries
Let M (J, Ω, g) be a paraK¨ahelerian manifold endowed with a K¨ahelerian structure defined by the para-complex (1.1)-tensor field J and a structure symplectic form Ω. If ∇ is the Levi-Civita connection one has the following structure equations 2 J = Iλ, g (JZ, Z 0 ) + g (Z, JZ 0 ) = 0 (1.1) , 0 Ω (ZJ, Z ) = g (JZ, Z 0 ) ; (∇Z J) Z 0 = 0 for every Z, Z 0 ∈ κM , where κM is the set of sections of the tangent bundle TM. Let ΓT M and ]
[
[ : T M → T ∗ M,
] : T M → T ∗M
the set of the sections of the tangent bundle and the musical isomorphisms defined by g. Following [P ] we set Aq (M, T M ) = ΓHom (Λq T M, T M ) and we denote by d∇ : Aq (M, T M ) → Aq+1 (M, T M ) the exterior covariant derivative with respect to ∇. It should be noticed that in general 2 d∇ = d∇ ◦ d∇ 6= 0, unlike d2 = d ◦ d. In the case under consideration one assumes that Y is a concurrent vector field having λ as divergence, i.e., ∇Y = λdp, where dp means the soldering form of M . If L means the Lie derivative one derives LY Ω = 2λΩ, and this says that Y defines an infinitesimal conformal transformation of Ω. The case when Y is a null (real) vector field is discussed.
2.
Results
Let M (J, Ω, g) be a 2m-dimensional paraK¨ahelerian manifold. Such manifolds have been defined by P. Libermann [L] and shortly may be considered 2
as neutral pseudo Riemaniann manifolds carrying a para-complex operator J of square +1 and a symplectic form Ω exchangeable with the paraHermitian metric tensor g. Let W = {ha , ha∗ , a = 1, ..., m; a∗ = a + m} be a real witt vector basis that is (2.1)
g (ha , hb∗ ) = δab ,
and let S = {ha } and S ∗ = {ha∗ } be two self orthogonal vector basis. Then W may be split as W = S ⊗ S∗ and the pairing (S, S ∗ ) defines an involutive automorphism of square +1, i.e., (2.2) Jha = ha , Jha∗ = −ha∗ . ∗ ∗ If W ∗ = ω a , ω a is the dual basis of W and ∇ (resp. θba , θba∗ ) the covariant differential operator (resp. the connection form on M ), the matrix Mθ of θ is a Chern-Libermann matrix, i.e., a θb 0 ∗ (2.3) Mθ = . 0 θba∗ One has (2.4)
∗
Ω = ωa ∧ ωa
and g = 2a ω a ⊗ ω a
(2.5)
∗
and the soldering form dp and E. Cartan’s structure equations written in indexless manner are (2.6) dh = θ ⊗ h, (2.7) (2.8)
dω = −θ ∧ ω, dθ = −θ ∧ θ + Θ,
where Θ denotes the curvature forms. If ∗ 2 (2.9) X = X A hA ⇒ kXk = 2X a X a is any vector field on M , then its dual form X [ = α is expressed by ∗ ∗ (2.10) α = X a ωa + X a ωa and one has (2.11)
2
α (X) = kXk = 2X a X a
3
∗
Assume now that X is a skew symmetric Killing vector field (abbr. S.S.K.) in the sense of Rosca [R], i.e. ∇X = X ∧ Y,
(2.12)
where ∧ is the wedge product of vector fields and where the vector fields Y is the generative of X. Setting ∗ ∗ (2.13) β = Y [ = Y a ωa + Y a ωa one may write (2.12) as (2.14)
∇X = β ⊗ X − α ⊗ Y.
On the other hand, taking the covariant differential of X, one has by (2.3) ∗ ∗ ∗ (2.15) ∇X = dX a + X b θba ⊗ ha + dX a + X b θba∗ ⊗ ha∗ . Hence by equation (2.14) and (2.15) one has dX a + X b θba = X a β − Y a α ∗ ∗ ∗ ∗ ∗ (2.16) dX a + X b θba∗ = X a β − Y a α and taking the exterior differential of (2.10), one gets, in a first step ∗ ∗ ∗ ∗ ∗ (2.17) dα = − X b θba − X a β + Y a α ∧ ω a − X b θba∗ − X a β + Y a α ∧ ω a + ∗ ∗ ∗ +X a ω b θba∗ + X a ω b ∧ θba . Next by a routine matter, carrying out the calculation we define (2.18)
dα = 2α ∧ β
and one refinds Rosca’s lemma [R] regarding S.S.K. vector fields. This says the meaningful fact that this lemma has an intrinsic charater and is independent of the frame one considers. Next, since in the general considerations of S.S.K. vector fields one assumes that the generative vector field Y is closed and since ∗ ∗ β = Y a ωa + Y a ωa one gets by the structure equations (2.7) ∗ ∗ ∗ ∗ (2.19) dY a − Y b θba ∧ ω a + dY a − Y b θba∗ ∧ ω a = 0. In these conditions we assume, in the present note, that Y is a concurrent vector field that is (2.20)
∇Y = λdp, withλ ∈ Λ0 M, 4
and on behalf of (2.19) one may write ∗ dY a = λω a + Y b θba∗ ∗ ∗ ∗ (2.21) . dY a = λω a + Y b θba Since, as is known (see also [R]) one may write Ω = ωa ∧ ωa
(2.22)
∗
one gets at once (2.23)
∗ ∗ iY Ω = Y a ω a − Y a ω a .
But Ω being a closed form one derives by (2.21) and so by (2.23) one has (2.24)
d (iY Ω) = LY Ω = 2λΩ.
Hence the above relation says that the generative Y of X defines an infinitesimal conformal transformation of Ω. This agrees the fact that as is known concurrency is of conformal nature. We also notice that by (2.20) one has (2.25)
LY (∇Y ) = (LY λ) dp.
Recalling that LZ (∇Z) = 0, (for every Z ∈ κM ) defines an affine vector fields, one may say that the necessary and sufficient conditions in order that Y be an affine vector field is that the divergence of Y be constant. On the other hand by (2.20) and (2.19) one has 2 (2.26) [X, Y ] = λ − kY k X + g (X, Y ) Y, which shows that the necessary and sufficient conditions in order that X define an infinitesimal conformal transformation of Y is that 2
λ − kY k = 0 holds good. In particular this happens if Y is a null vector field, because in this case one has 2 λ = 0, kY k = 0 Finally, we can state the following Theorem. Let M (J, Ω, g) be a 2m-dimensional paraK¨ ahlerian manifold with structure 2-form Ω, and let X (resp. Y ) be a skew symmetric Killing vector field on M (resp. the generative of X). Then as in the Riemannian case, the dual forms X [ = α and Y [ = β of X and Y satisfy Rosca’s lemma dα = 2α ∧ β. 5
In addition if Y is a concurrent vector field, then it defines an infinitesimal conformal transformation of Ω, i.e., LY Ω = 2λΩ the necessary and sufficient condition in order that X defines an infinitesimal 2 conformal transformation of Y is that the divergence of Y be equated by kY k .
References
[L]
:
P.Libermann, Sur le probl´eme d’´equivalence de certains structures infinitesimales, Annal.diM at.36(1954)27 − 120.
[P ]
:
E. M. P atterson, Riemannian extension which have K¨ ahlerian metrics, P roc.Roy.Soc.Edinburgh(A)64(1954)113 − 126
[R1]
:
R. Rosca, CR-sous variet´es raisdropes d’une variet´e paraK¨ ahlerienne, C. R. A. S. P aris298(1984), 149 − 151
[EG]
:
[GA]
:
E. Etayo, P. M.Gadea, Paraholomorphically Projective vector fields, Ann.St.U niv.Al.I.T asi38(1992), 201 − 210 P. M.Gadea, M.Ambilihia, Some geometric properties of para-K¨ ahlerian space-forms, Rend.Sem.M at.U niv. Cagliari59(1989).
[R2]
:
R.Rosca, On para Sasakian manifolds, Rend.Sem.M at.U niv. M essina1
[Ch]
:
Geometry of Submanifolds M. Dekker, N ewY ork(1973)
[R3]
:
R. Rosca, On K-left invariant almost contact 3-structure ResultsM ath.27
(1989), 201 − 216.
(1982)117 − 128.
David Carf`ı
Via Canova, 32 98121 Messina, Sicily, Italy
[email protected]
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