Para-hyperhermitian structures on tangent bundles

In this paper we construct a family of almost para-hyperhermitian structures on the tangent bundle of an almost parahermitian manifold and study its integrability. Also, the necessary and sufficient conditions are provided for these structures to become para-hyper-Kähler.


INTRODUCTION
The almost para-hypercomplex structures, also named almost quaternionic structures of second kind, were introduced by Libermann in 1954 under the latter name [20] as a triple of endomorphisms of the tangent bundle {J 1 , J 2 , J 3 }, in which J 1 is almost complex and J 2 , J 3 are almost product structures satisfying relations of anti-commutation.An almost para-hyperhermitian structure on a manifold consists of an almost parahypercomplex structure and a compatible semi-Riemannian metric necessarily of neutral signature.If all three structures involved in the definition of an almost para-hyperhermitian structure are parallel with respect to the Levi-Civita connection of the compatible metric, one arrives at the concept of para-hyper-Kähler structure, which is also referred to in the literature as neutral hyper-Kähler or hypersymplectic structure [4,7,10,13].
The quaternionic structures of second kind are of great interest in theoretical physics, because they arise in a natural way both in string theory and integrable systems [2,6,9,14,23] and, consequently, to find new classes of manifolds endowed with structures of this kind is an interesting topic.Kamada [19] proves that any primary Kodaira surface admits para-hyper-Kähler structures, whose compatible metrics can be chosen to be flat or nonflat.On the other hand, an integrable para-hyperhermitian structure has been constructed in [17] on Kodaira-Thurston properly elliptic surfaces and also on the Inoe surfaces modelled on Sol 4  1 .In higher dimensions, para-hyperhermitian structures on a class of compact quotients of 2-step nilpotent Lie groups can be found in [12].A procedure to construct para-hyperhermitian structures on R 4n with complete and not necessarily flat associated metrics is given in [1].Also, some examples of integrable almost para-hyperhermitian structures which admit compatible linear connections with totally skew symmetric torsion are given in [18].Recently, in [15], a natural para-hyperhermitian structure was constructed on the tangent bundle of an almost para-hermitian manifold and on the circle bundle over a manifold with a mixed 3-structure.The main purpose of this paper is to generalize this construction to obtain an entire class of such structures; we also investigate its integrability and obtain the necessary and sufficient conditions for these structures to become para-hyper-Kähler.

PRELIMINARIES
An almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P = ±Id, such that where Id is the identity tensor field of type (1,1) on M.
An almost para-hermitian structure on a differentiable manifold M is a pair (P, g), where P is an almost product structure on M and g is a semi-Riemannian metric on M satisfying g(PX, PY ) = −g(X,Y ) for all vector fields X,Y on M.
In this case, (M, P, g) is said to be an almost para-hermitian manifold.It is easy to see that the dimension of M is even.Moreover, if ∇P = 0, then (M, P, g) is said to be a para-Kähler manifold.
An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such that An almost para-hypercomplex structure on a smooth manifold M is a triple H = (J α ) α=1,3 , where J 1 is an almost complex structure on M and J 2 , J 3 are almost product structures on M, satisfying In this case (M, H) is said to be an almost para-hypercomplex manifold.
A semi-Riemannian metric g on (M, H) is said to be compatible or adapted to the almost parahypercomplex structure H = (J α ) α=1,3 if it satisfies for all vector fields X,Y on M, where ε 1 = 1, ε 2 = ε 3 = −1.Moreover, the pair (g, H) is called an almost para-hyperhermitian structure on M and the triple (M, g, H) is said to be an almost para-hyperhermitian manifold.It is clear that any almost para-hyperhermitian manifold is of dimension 4m, m ≥ 1, and any adapted metric is necessarily of neutral signature (2m, 2m).If {J 1 , J 2 , J 3 } are parallel in respect to the Levi-Civita connection of g, then the manifold is called para-hyper-Kähler.
An almost para-hypercomplex manifold (M, H) is called a para-hypercomplex manifold if each J α , α = 1, 2, 3, is integrable, that is, if the corresponding Nijenhuis tensors α = 1, 2, 3, vanish for all vector fields X,Y on M. In this case H is said to be a para-hypercomplex structure on M.Moreover, if g is a semi-Riemannian metric adapted to the para-hypercomplex structure H, then the pair (g, H) is said to be a para-hyperhermitian structure on M and (M, g, H) is called a para-hyperhermitian manifold.We note that the existence of para-hyperhermitian structures on compact complex surfaces was recently investigated in [8].
Remark 2.1.Let (M, P, g) be an almost para-hermitian manifold and T M be the tangent bundle, endowed with the Sasakian metric for all vector fields X,Y on T M, where π is the natural projection of T M onto M and K is the connection map (see [11]).We remark that for each u ∈ T x M, x ∈ M, we have a direct sum decomposition Moreover, the elements of T h u T M are called horizontal vectors at u and the elements of T v u T M are said to be vertical vectors at u.We can see that if u, X ∈ T x M and X h u (resp.X v u ) denotes the horizontal lift (resp.vertical lift) of X to , then any horizontal (vertical) vector field X on T M can be written as X = A h (X = A v ) for a unique vector A along π.If A, B are vector fields along π, then, by generalizing the well-known Dombrowski's lemma [11], Ii and Morikawa [16] showed that the brackets of the horizontal and vertical lifts are given by where the covariant derivative of a vector field C along π in the direction of ξ ∈ T u T M, u ∈ T M, is defined as the tangent vector to M at x = π(u) given by We can also remark that every tensor field T of type (1,1) on M is a vector field along π.Moreover, we have and, if T is parallel, We note that the identity map id : Remark 2.3.If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields J 1 , J 2 , J 3 on T M by the equalities: It is easy to see that J 1 is an almost complex structure and J 2 , J 3 are almost product structures.We also have the following result (see [15]).
Theorem 2.4.Let (M, P, g) be an almost para-hermitian manifold.Then the triple H = (J α ) α=1,3 is an almost para-hypercomplex structure on T M which is para-hyperhermitian with respect to the Sasakian metric G.Moreover, H is integrable if and only if (M, P) is a flat para-Kähler manifold.
In the next section, following the same techniques as in [3,21,22,[24][25][26], we deform the almost parahyperhermitian structure given above in order to obtain an entire family of structures of this kind on the tangent bundle of an almost para-hermitian manifold.

A FAMILY OF ALMOST PARA-HYPERHERMITIAN STRUCTURE ON THE TANGENT
BUNDLE OF A PARA-HERMITIAN MANIFOLD Lemma 3.1.Let (M, P, g) be an almost para-hermitian manifold and let J 1 be a tensor field of type (1, 1) on T M, defined by for all vectors X ∈ T π(u) M, u ∈ T x M, x ∈ M, where t = u 2 and a, b, c, m, n, p are differentiable real functions.Then J 1 defines an almost complex structure if and only if Proof.The conditions follow from the property J 2 1 = −Id.
Lemma 3.2.Let (M, P, g) be an almost para-hermitian manifold and let J 2 be a tensor field of type (1, 1) on T M, defined by for all vectors X ∈ T π(u) M, u ∈ T x M, x ∈ M, where t = u 2 and a, b, c, q, r, s are differentiable real functions.Then J 2 defines an almost product structure if and only if Proof.The conditions follow from the property J 2 2 = Id.
Proposition 3.3.Let (M, P, g) be an almost para-hermitian manifold.Then there exists an infinite class of almost para-hypercomplex structures on T M.
Proof.We define a tensor field J 3 of type (1,1) on T M by J 3 = J 2 J 1 , where J 1 , J 2 are given by ( 9) and (11), such that (10) and ( 12) are satisfied.We can easily see now that H = (J α ) α=1,3 is an almost parahypercomplex structure on T M.
Proposition 3.4.Let (M, P, g) be an almost para-hermitian manifold, let H = (J α ) α=1,3 be the almost parahypercomplex structure on T M given above and let G be a semi-Riemannian metric on T M defined by Proof.The conditions ( 13) are obtained by direct computations using the property (1).

THE STUDY OF INTEGRABILITY
Let (M, P, g) be a para-Kähler manifold.A plane Π ⊂ T p M, p ∈ M, is called para-holomorphic if it is left invariant by the action of P, that is PΠ ⊂ Π.The para-holomorphic sectional curvature is defined as the restriction of the sectional curvature to para-holomorphic non-degenerate planes.A para-Kähler manifold is said to be a paracomplex space form if its para-holomorphic sectional curvatures are equal to a constant, say k.It is well known that a para-Kähler manifold (M, P, g) is a paracomplex space form, denoted M(k), if and only if its curvature tensor is for all vector fields X,Y, Z on M.
From the above section we deduce that the tangent bundle of a paracomplex space form M(k) can be endowed with a class of almost para-hypercomplex structure H = (J α ) α=1,3 given by where a, b, c are differentiable real functions such that 1 a , b a(a+tb) , c a(a−tc) are well defined and also differentiable real functions.Theorem 4.1.Let M(k) be a paracomplex space form.Then the almost para-hypercomplex structure H = (J α ) α=1,3 given above is integrable if and only if Proof.First of all we remark that if two of the structures J 1 , J 2 , J 3 are integrable, then the third structure is also integrable because the corresponding Nijenhuis tensors are related by for any even permutation (α, β , γ) of (1,2,3), where Secondly, it is well known that an almost complex structure on a manifold is integrable if and only if the distribution of the complex tangent vector fields of type (1,0), denoted by X (1,0) , is involutive, i.e. it satisfies [X (1,0) , X (1,0) ] ⊂ X (1,0) .Now, using (2), ( 3)-( 8), and ( 14), we obtain for any vector field A, B along π, satisfying g(A, id Consequently, J 1 is integrable if and only if the next three relations are satisfied: Thirdly, an almost product structure on a manifold is integrable if and only if the eigendistributions X + and X − corresponding to the eigenvalues 1 and −1, respectively, are integrable.We similarly obtain that J 2 is integrable if and only if the same three relations hold. Finally, we obtain the conclusion because the relation ( 21) is involved by the relations (19) and (20).
where A is an arbitrary real constant, and we can easily verify that the conditions (18) are satisfied and the functions a, b, c, 1 a , b a(a+tb) , c a(a−tc) are differentiable.Consequently, the almost para-hypercomplex structure H = (J α ) α=1,3 given above is integrable.
where α, β are differentiable real functions such that α and α + tβ are nowhere null.From Theorem 4.1 we may state now the following result.
Corollary 4.5.There exists an infinite class of para-hyperhermitian structures on the tangent bundle of a paracomplex space form.
For any vector field A and B along π, satisfying g(A, id) = g(B, id) = g(A, P • id) = g(B, P • id) = 0 on T M, using (3)-( 8), we obtain and But, because ω 1 is closed, from ( 24) and ( 25) we obtain where C is a real constant.On the another hand, J 1 , J 2 , J 3 being integrable, from Theorem 4.1 we deduce that the functions a, b, c also satisfy the conditions (18).The conclusion follows now easily since a, b, c, α, β , 1  a , b a(a+tb) , c a(a−tc) must be differentiable functions satisfying (18) and (26).

CONCLUSIONS
We constructed an infinite class of almost para-hyperhermitian structures on the tangent bundle of an almost para-hermitian manifold (M, P, g).Moreover, if (M, P, g) is a paracomplex space form, we also obtained necessary and sufficient conditions for the above structures to become para-hyper-Kähler.These results can have important applications both in differential geometry and theoretical physics, since the existence of para-hyper-Kähler structures is of great importance in many geometric and physics problems (see e.g.[5]).A possible extension of this paper is to construct a class of paraquaternionic Kähler structures on the tangent bundle of a paracomplex space form.

Example 4 . 2 .
If M is a flat paracomplex space form, then we seta = A, b = 0, c = 0,where A is an arbitrary non-zero real constant, and we can easily see that the conditions (18) are satisfied and a, b, c,1  a , b a(a+tb) , c a(a−tc) are clearly differentiable, being constants.Consequently, the almost parahypercomplex structure H = (J α ) α=1,3 given above is integrable.Example 4.3.If M(k) is a non-flat paracomplex space form, then we set