Abstract:

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

Key words: para-hyperhermitian structure, tangent bundle, paracomplex space form.

On konstrueeritud peaaegu para-hUperhermiitiliste struktuuride parv peaaegu para-hermiitilise muutkonna puutujakihtkonnal ja uuritud selle integreeruvust. On leitud tarvilikud ja piisavad tingimused, mille korral need struktuurid on para-huper-Kahleri struktuurid.

Key words: para-hyperhermitian structure, tangent bundle, paracomplex space form.

On konstrueeritud peaaegu para-hUperhermiitiliste struktuuride parv peaaegu para-hermiitilise muutkonna puutujakihtkonnal ja uuritud selle integreeruvust. On leitud tarvilikud ja piisavad tingimused, mille korral need struktuurid on para-huper-Kahleri struktuurid.

Article Type:

Report

Subject:

Manifolds (Mathematics)
(Research)

Geometry, Differential (Research)

Geometry, Differential (Research)

Author:

Vilcu, Gabriel Eduard

Pub Date:

09/01/2011

Publication:

Name: Proceedings of the Estonian Academy of Sciences Publisher: Estonian Academy Publishers Audience: Academic Format: Magazine/Journal Subject: Chemistry Copyright: COPYRIGHT 2011 Estonian Academy Publishers ISSN: 1736-6046

Issue:

Date: Sept, 2011 Source Volume: 60 Source Issue: 3

Topic:

Event Code: 310 Science & research

Geographic:

Geographic Scope: Romania Geographic Code: 4EXRO Romania

Accession Number:

268604059

Full Text:

1. 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.sub.1],[J.sub.2],[J.sub.3]}, in which [J.sub.1] is almost complex and [J.sub.2], [J.sub.3] are almost product structures satisfying relations of anti-commutation. An almost para-hyperhermitian structure on a manifold consists of an almost para-hypercomplex 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-Kahler structure, which is also referred to in the literature as neutral hyper-Kahler 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-Kahler 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.sup.4.sub.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.sup.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-Kahler.

2. PRELIMINARIES

An almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P [not equal to] [+ or -]Id, such that

[P.sup.2] = Id,

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 [nabla]P = 0, then (M, P, g) is said to be a para- Kahler manifold.

An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such that

[J.sub.2] = -Id.

An almost para-hypercomplex structure on a smooth manifold M is a triple H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]], where [J.sub.1] is an almost complex structure on M and [J.sub.2], [J.sub.3] are almost product structures on M, satisfying

[J.sub.2][J.sub.1] = - [J.sub.1] [J.sub.2] = [J.sub.3].

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 para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] if it satisfies

g([J.sub.[alpha]]X, [J.sub.[alpha]]Y) = [[epsilon].sub.[alpha]]g(X, Y), [for all][alpha] = [bar.1,3] (1)

for all vector fields X,Y on M, where [[epsilon].sub.1] = 1, [[epsilon].sub.2] = [[epsilon].sub.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 [greater than or equal to] 1, and any adapted metric is necessarily of neutral signature (2m, 2m). If {[J.sub.1], [J.sub.2], [J.sub.3]} are parallel in respect to the Levi-Civita connection of g, then the manifold is called para-hyper-Kahler.

An almost para-hypercomplex manifold (M, H) is called a para-hypercomplex manifold if each [J.sub.[alpha]], [alpha] = 1,2,3, is integrable, that is, if the corresponding Nijenhuis tensors

[N.sub.[alpha]](X,Y) = [[J.sub.[alpha]]X, [J.sub.[alpha]]Y] - [J.sub.[alpha]][X, [J.sub.[alpha]]Y] - [J.sub.[alpha]][[J.sub.[alpha]X],Y] - [[epsilon].sub.[alpha]][X, Y], (2)

[alpha] = 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 TM be the tangent bundle, endowed with the Sasakian metric

G(X, Y) = (g(KX,KY)+ g([[pi].sub.*],X,[[pi].sub.*]Y)) [??] [pi]

for all vector fields X, Y on TM, where [pi] is the natural projection of TM onto M and K is the connection map (see [11]).

We remark that for each u [member of] [T.sub.x]M, x [member of] M, we have a direct sum decomposition

[T.sub.u]TM = [T.sup.h.sub.u]TM [direct sum] [T.sup.v.sub.u]TM,

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the horizontal subspace of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the vertical subspace of [T.sub.u]TM. Moreover, the elements of [T.sup.h.sub.u]TM are called horizontal vectors at u and the elements of [T.sup.v.sub.u]TM are said to be vertical vectors at .

We can see that if u,X [member of] [T.sub.x]M and [X.sup.h.sub.u] (resp. [X.sup.v.sub.u]) denotes the horizontal lift (resp. vertical lift) of X to [T.sub.u]TM then

[[pi].sub.*][X.sup.h.sub.u] = X, [[pi].sub.*][X.sup.v.sub.u] = 0, K[X.sup.h.sub.u] = 0, K[X.sup.v.sub.u] = X.

Remark 2.2. If A is a vector field along [pi] (i.e. a map A : TM [right arrow] TM such that [pi] [??] A = [pi]) and [A.sup.h] (resp. [A.sup.v]) denotes the horizontal lift (resp. vertical lift) of A (that is: [A.sup.h] : u [??] [A.sup.h.sub.u] = A[(u).sup.h.sub.u] and [A.sup.v] : u [??] [A.sup.v.sub.u] = A[(u).sup.v.sub.u]), then any horizontal (vertical) vector field X on TM can be written as X = [A.sup.h] (X = [A.sup.v]) for a unique vector A along [pi]. If A, B are vector fields along [pi], 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where the covariant derivative of a vector field C along [pi] in the direction of [xi] [member of] [T.sub.u]TM, u [member of] TM, is defined as the tangent vector to M at x = [pi](u) given by

[[nabla].sub.[xi]]C = (K [??]dC)([xi]).

We can also remark that every tensor field T of type on M is a vector field along [pi]. Moreover, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

and, if T is parallel,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

We note that the identity map id : u [??] u on TM is a parallel tensor field of type on M. Moreover, if [[parallel] x [parallel].sup.2] is the function u [??] [[parallel]u[parallel].sup.2] = g(u,u) on TM, then we have

[A.sup.h][[parallel] x [parallel].sup.2] = 0, [A.sup.v] [[parallel] x [parallel].sup.2] = 2g(A,id). (8)

Remark 2.3. If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields [J.sub.1], [J.sub.2], [J.sub.3] on TM by the equalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to see that [J.sub.1] is an almost complex structure and [J.sub.2], [J.sub.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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] is an almost para-hypercomplex structure on TM 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-Kahler manifold.

In the next section, following the same techniques as in [3,21,22,24-26], we deform the almost para-hyperhermitian 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.

3. 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 Ji be a tensor field of type (i, i) on TM, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

for all vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t = [[parallel]u[parallel].sup.2] and a,b,c,m,n,p are differentiable real functions. Then [J.sub.1] defines an almost complex structure if and only if

am + 1 = 0, an(a + tb) - b = 0, ap(a - tc) - c = 0. (10)

Proof. The conditions follow from the property [J.sub.2.sub.1] = -Id. []

Lemma 3.2. Let (M, P, g) be an almost para-hermitian manifold and let [J.sub.2] be a tensor field of type (1, 1) on TM, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

for all vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t = [[parallel]u[parallel].sup.2] and a, b, c, q, r, s are differentiable real functions. Then [J.sub.2] defines an almost product structure if and only if

aq - i = 0, ar(a - tc) - c = 0, as(a + tb) - b = 0. (i2)

Proof. The conditions follow from the property [J.sub.2.sub.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 TM.

Proof. We define a tensor field [J.sub.3] of type on TM by [J.sub.3] = [J.sub.2][J.sub.1], where [J.sub.1], [J.sub.2] are given by (9) and (11), such that (10) and (12) are satisfied. We can easily see now that H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] is an almost para-hypercomplex structure on TM. []

Proposition 3.4. Let (M, P, g) be an almost para-hermitian manifold, let H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] be the almost para-hypercomplex structure on TM given above and let G be a semi-Riemannian metric on TM defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all vectors X, Y [member of] [T.sub.[pi](u)]M, u [member of] [T.sub.x]M, x [member of] M, where t = [[parallel]u[parallel].sup.2] and [alpha], [beta], [gamma], [delta], [epsilon], [theta] are smooth real functions such that [alpha], [alpha] + t[beta], [alpha] - t[gamma] or [delta], [delta] + t[epsilon], [delta] - t[theta] are nowhere null. Then [??] is adapted to the almost para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Proof. The conditions (i3) are obtained by direct computations using the property (1). []

Corollary 3.5. There exists an infinite class of almost para-hyperhermitian structures on the tangent bundle of an almost para-hermitian manifold.

4. THE STUDY OF INTEGRABILITY

Let (M, P, g) be a para-Kahler manifold. A plane [PI] [subset] [T.sub.p]M, p [member of] M, is called para-holomorphic if it is left invariant by the action of P, that is P[PI] [subset] [PI]. The para-holomorphic sectional curvature is defined as the restriction of the sectional curvature to para-holomorphic non- degenerate planes. A para-Kahler 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-Kahler manifold (M, P, g) is a paracomplex space form, denoted M(k), if and only if its curvature tensor is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where a, b, c are differentiable real functions such that 1/a, b/(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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable if and only if

4b(a - 2ta') - 8aa' + k = 0, 4ac + k = 0. (18)

Proof. First of all we remark that if two of the structures [J.sub.1], [J.sub.2], [J.sub.3] are integrable, then the third structure is also integrable because the corresponding Nijenhuis tensors are related by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any even permutation ([alpha], [beta], [gamma]) of (1,2,3), where [[epsilon].sub.1] = 1, [[epsilon].sub.2] = [[epsilon].sub.3] = -1.

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 (i,0), denoted by [K.sup.(1,0)], is involutive, i.e. it satisfies [[K.sup.(1,0)], [K.sup.(1,0)]] [subset] [K.sup.(1,0)]. Now, using (2), (3)-(8), and (i4), we obtain for any vector field A, B along [pi], satisfying g(A, id) = g(B, id) = g(A, P [??] id) = g(B, P [??] id) = 0 on TM

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, [J.sub.1] is integrable if and only if the next three relations are satisfied:

k/4 t + a(a + tb) - 2a't(a + tb)/a = a, (19)

-k/4 t + a(a - tc)/a = a, (20)

kt + (a + tb)(a - tc) - 2t(a + tb)(a - tc)'/a - tc = a - tc. (21)

Thirdly, an almost product structure on a manifold is integrable if and only if the eigendistributions [K.sup.+] and [K.sup.-] corresponding to the eigenvalues i and -1, respectively, are integrable. We similarly obtain that [J.sub.2] is integrable if and only if the same three relations hold.

Finally, we obtain the conclusion because the relation (2i) is involved by the relations (i9) and (20). []

Example 4.2. If M is a flat paracomplex space form, then we set

a = A, b = 0, c = 0,

where A is an arbitrary non-zero real constant, and we can easily see that the conditions (i8) are satisfied and a, b, c, 1/a, b/a(a+tb), c/(a-tc) are clearly differentiable, being constants. Consequently, the almost para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable.

Example 4.3. If M(k) is a non-flat paracomplex space form, then we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable.

Remark 4.4. From Proposition 3.4 we have a class of compatible semi-Riemannian metric on the tangent bundle of a paracomplex space form M(k), given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where [alpha], [beta] are differentiable real functions such that [alpha] and [alpha] + t[beta] are nowhere null. From Theorem 4.i 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.

Theorem 4.6. Let M(k) be a paracomplex space form. Then the almost para- hyperhermitian structure ([??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) on TM defined by (15), (16), (17), and (22) is para-hyper-Kahler if and only if M is flat and

a = [C.sub.1], [alpha] = [C.sub.2], b = c = [beta] = 0, (23)

where [C.sub.1], [C.sub.2] are non-null real constants.

Proof. If M is flat and the relations (23) hold, then it is clear that (TM, [??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) is a para-hyper-Kahler manifold.

Conversely, if (TM, [??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) is a para-hyper-Kahler manifold, then each Ja is integrable and the fundamental 2- forms [[omega].sub.[alpha]], given by

[[omega].sub.[alpha]](X, Y)= [??]([J.sub.[alpha]]X, Y),

for all vector fields X, Y on TM, are closed for all [alpha] [member of] {1, 2, 3}.

For any vector field A and B along [pi], satisfying g(A, id) = g(B, id) = g(A, P [??] id) = g(B, P [??] id) = 0 on TM, using (3)-(8), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

But, because [[omega].sub.1] is closed, from (24) and (25) we obtain

[alpha]/a = [alpha] + t[beta]/a + tb = [alpha] + t[beta]/a - tc = C, (26)

where C is a real constant.

On the another hand, [J.sub.1], [J.sub.2], [J.sub.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, [alpha], [beta], 1/a, b/a(a+tb), c/a(a-tc) must be differentiable functions satisfying (18) and (26). []

5. 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-Kahler. These results can have important applications both in differential geometry and theoretical physics, since the existence of para-hyper-Kahler 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 Kahler structures on the tangent bundle of a paracomplex space form.

doi: 10.3176/proc.2011.3.04

ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to Professor Liviu Ornea for carefully reading this paper and offering me helpful suggestions. This work was partially supported by CNCSIS--UEFISCSU, project PNII--IDEI code 8/2008, contract No. 525/2009.

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Gabriel Eduard Vilcu

University of Bucharest, Faculty of Mathematics and Computer Science, Research Center in Geometry, Topology and Algebra, Academiei Str. 14, Sector 1, Bucharest 70109, Romania; and Petroleum-Gas University of Ploiesti, Department of Mathematics and Computer Science, Bulevardul Bucuresti 39, Ploiesti 100680, Romania; gvilcu@upg-ploiesti.ro

Received 22 June 2010, revised 22 February 2011, accepted 23 February 2011

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.sub.1],[J.sub.2],[J.sub.3]}, in which [J.sub.1] is almost complex and [J.sub.2], [J.sub.3] are almost product structures satisfying relations of anti-commutation. An almost para-hyperhermitian structure on a manifold consists of an almost para-hypercomplex 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-Kahler structure, which is also referred to in the literature as neutral hyper-Kahler 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-Kahler 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.sup.4.sub.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.sup.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-Kahler.

2. PRELIMINARIES

An almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P [not equal to] [+ or -]Id, such that

[P.sup.2] = Id,

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 [nabla]P = 0, then (M, P, g) is said to be a para- Kahler manifold.

An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such that

[J.sub.2] = -Id.

An almost para-hypercomplex structure on a smooth manifold M is a triple H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]], where [J.sub.1] is an almost complex structure on M and [J.sub.2], [J.sub.3] are almost product structures on M, satisfying

[J.sub.2][J.sub.1] = - [J.sub.1] [J.sub.2] = [J.sub.3].

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 para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] if it satisfies

g([J.sub.[alpha]]X, [J.sub.[alpha]]Y) = [[epsilon].sub.[alpha]]g(X, Y), [for all][alpha] = [bar.1,3] (1)

for all vector fields X,Y on M, where [[epsilon].sub.1] = 1, [[epsilon].sub.2] = [[epsilon].sub.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 [greater than or equal to] 1, and any adapted metric is necessarily of neutral signature (2m, 2m). If {[J.sub.1], [J.sub.2], [J.sub.3]} are parallel in respect to the Levi-Civita connection of g, then the manifold is called para-hyper-Kahler.

An almost para-hypercomplex manifold (M, H) is called a para-hypercomplex manifold if each [J.sub.[alpha]], [alpha] = 1,2,3, is integrable, that is, if the corresponding Nijenhuis tensors

[N.sub.[alpha]](X,Y) = [[J.sub.[alpha]]X, [J.sub.[alpha]]Y] - [J.sub.[alpha]][X, [J.sub.[alpha]]Y] - [J.sub.[alpha]][[J.sub.[alpha]X],Y] - [[epsilon].sub.[alpha]][X, Y], (2)

[alpha] = 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 TM be the tangent bundle, endowed with the Sasakian metric

G(X, Y) = (g(KX,KY)+ g([[pi].sub.*],X,[[pi].sub.*]Y)) [??] [pi]

for all vector fields X, Y on TM, where [pi] is the natural projection of TM onto M and K is the connection map (see [11]).

We remark that for each u [member of] [T.sub.x]M, x [member of] M, we have a direct sum decomposition

[T.sub.u]TM = [T.sup.h.sub.u]TM [direct sum] [T.sup.v.sub.u]TM,

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the horizontal subspace of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called the vertical subspace of [T.sub.u]TM. Moreover, the elements of [T.sup.h.sub.u]TM are called horizontal vectors at u and the elements of [T.sup.v.sub.u]TM are said to be vertical vectors at .

We can see that if u,X [member of] [T.sub.x]M and [X.sup.h.sub.u] (resp. [X.sup.v.sub.u]) denotes the horizontal lift (resp. vertical lift) of X to [T.sub.u]TM then

[[pi].sub.*][X.sup.h.sub.u] = X, [[pi].sub.*][X.sup.v.sub.u] = 0, K[X.sup.h.sub.u] = 0, K[X.sup.v.sub.u] = X.

Remark 2.2. If A is a vector field along [pi] (i.e. a map A : TM [right arrow] TM such that [pi] [??] A = [pi]) and [A.sup.h] (resp. [A.sup.v]) denotes the horizontal lift (resp. vertical lift) of A (that is: [A.sup.h] : u [??] [A.sup.h.sub.u] = A[(u).sup.h.sub.u] and [A.sup.v] : u [??] [A.sup.v.sub.u] = A[(u).sup.v.sub.u]), then any horizontal (vertical) vector field X on TM can be written as X = [A.sup.h] (X = [A.sup.v]) for a unique vector A along [pi]. If A, B are vector fields along [pi], 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where the covariant derivative of a vector field C along [pi] in the direction of [xi] [member of] [T.sub.u]TM, u [member of] TM, is defined as the tangent vector to M at x = [pi](u) given by

[[nabla].sub.[xi]]C = (K [??]dC)([xi]).

We can also remark that every tensor field T of type on M is a vector field along [pi]. Moreover, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

and, if T is parallel,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

We note that the identity map id : u [??] u on TM is a parallel tensor field of type on M. Moreover, if [[parallel] x [parallel].sup.2] is the function u [??] [[parallel]u[parallel].sup.2] = g(u,u) on TM, then we have

[A.sup.h][[parallel] x [parallel].sup.2] = 0, [A.sup.v] [[parallel] x [parallel].sup.2] = 2g(A,id). (8)

Remark 2.3. If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields [J.sub.1], [J.sub.2], [J.sub.3] on TM by the equalities:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to see that [J.sub.1] is an almost complex structure and [J.sub.2], [J.sub.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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] is an almost para-hypercomplex structure on TM 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-Kahler manifold.

In the next section, following the same techniques as in [3,21,22,24-26], we deform the almost para-hyperhermitian 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.

3. 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 Ji be a tensor field of type (i, i) on TM, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

for all vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t = [[parallel]u[parallel].sup.2] and a,b,c,m,n,p are differentiable real functions. Then [J.sub.1] defines an almost complex structure if and only if

am + 1 = 0, an(a + tb) - b = 0, ap(a - tc) - c = 0. (10)

Proof. The conditions follow from the property [J.sub.2.sub.1] = -Id. []

Lemma 3.2. Let (M, P, g) be an almost para-hermitian manifold and let [J.sub.2] be a tensor field of type (1, 1) on TM, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

for all vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t = [[parallel]u[parallel].sup.2] and a, b, c, q, r, s are differentiable real functions. Then [J.sub.2] defines an almost product structure if and only if

aq - i = 0, ar(a - tc) - c = 0, as(a + tb) - b = 0. (i2)

Proof. The conditions follow from the property [J.sub.2.sub.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 TM.

Proof. We define a tensor field [J.sub.3] of type on TM by [J.sub.3] = [J.sub.2][J.sub.1], where [J.sub.1], [J.sub.2] are given by (9) and (11), such that (10) and (12) are satisfied. We can easily see now that H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] is an almost para-hypercomplex structure on TM. []

Proposition 3.4. Let (M, P, g) be an almost para-hermitian manifold, let H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] be the almost para-hypercomplex structure on TM given above and let G be a semi-Riemannian metric on TM defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all vectors X, Y [member of] [T.sub.[pi](u)]M, u [member of] [T.sub.x]M, x [member of] M, where t = [[parallel]u[parallel].sup.2] and [alpha], [beta], [gamma], [delta], [epsilon], [theta] are smooth real functions such that [alpha], [alpha] + t[beta], [alpha] - t[gamma] or [delta], [delta] + t[epsilon], [delta] - t[theta] are nowhere null. Then [??] is adapted to the almost para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Proof. The conditions (i3) are obtained by direct computations using the property (1). []

Corollary 3.5. There exists an infinite class of almost para-hyperhermitian structures on the tangent bundle of an almost para-hermitian manifold.

4. THE STUDY OF INTEGRABILITY

Let (M, P, g) be a para-Kahler manifold. A plane [PI] [subset] [T.sub.p]M, p [member of] M, is called para-holomorphic if it is left invariant by the action of P, that is P[PI] [subset] [PI]. The para-holomorphic sectional curvature is defined as the restriction of the sectional curvature to para-holomorphic non- degenerate planes. A para-Kahler 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-Kahler manifold (M, P, g) is a paracomplex space form, denoted M(k), if and only if its curvature tensor is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where a, b, c are differentiable real functions such that 1/a, b/(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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable if and only if

4b(a - 2ta') - 8aa' + k = 0, 4ac + k = 0. (18)

Proof. First of all we remark that if two of the structures [J.sub.1], [J.sub.2], [J.sub.3] are integrable, then the third structure is also integrable because the corresponding Nijenhuis tensors are related by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any even permutation ([alpha], [beta], [gamma]) of (1,2,3), where [[epsilon].sub.1] = 1, [[epsilon].sub.2] = [[epsilon].sub.3] = -1.

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 (i,0), denoted by [K.sup.(1,0)], is involutive, i.e. it satisfies [[K.sup.(1,0)], [K.sup.(1,0)]] [subset] [K.sup.(1,0)]. Now, using (2), (3)-(8), and (i4), we obtain for any vector field A, B along [pi], satisfying g(A, id) = g(B, id) = g(A, P [??] id) = g(B, P [??] id) = 0 on TM

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, [J.sub.1] is integrable if and only if the next three relations are satisfied:

k/4 t + a(a + tb) - 2a't(a + tb)/a = a, (19)

-k/4 t + a(a - tc)/a = a, (20)

kt + (a + tb)(a - tc) - 2t(a + tb)(a - tc)'/a - tc = a - tc. (21)

Thirdly, an almost product structure on a manifold is integrable if and only if the eigendistributions [K.sup.+] and [K.sup.-] corresponding to the eigenvalues i and -1, respectively, are integrable. We similarly obtain that [J.sub.2] is integrable if and only if the same three relations hold.

Finally, we obtain the conclusion because the relation (2i) is involved by the relations (i9) and (20). []

Example 4.2. If M is a flat paracomplex space form, then we set

a = A, b = 0, c = 0,

where A is an arbitrary non-zero real constant, and we can easily see that the conditions (i8) are satisfied and a, b, c, 1/a, b/a(a+tb), c/(a-tc) are clearly differentiable, being constants. Consequently, the almost para-hypercomplex structure H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable.

Example 4.3. If M(k) is a non-flat paracomplex space form, then we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

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.sub.[alpha]]).sub.[alpha]=[bar.1,3]] given above is integrable.

Remark 4.4. From Proposition 3.4 we have a class of compatible semi-Riemannian metric on the tangent bundle of a paracomplex space form M(k), given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where [alpha], [beta] are differentiable real functions such that [alpha] and [alpha] + t[beta] are nowhere null. From Theorem 4.i 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.

Theorem 4.6. Let M(k) be a paracomplex space form. Then the almost para- hyperhermitian structure ([??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) on TM defined by (15), (16), (17), and (22) is para-hyper-Kahler if and only if M is flat and

a = [C.sub.1], [alpha] = [C.sub.2], b = c = [beta] = 0, (23)

where [C.sub.1], [C.sub.2] are non-null real constants.

Proof. If M is flat and the relations (23) hold, then it is clear that (TM, [??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) is a para-hyper-Kahler manifold.

Conversely, if (TM, [??], H = [([J.sub.[alpha]]).sub.[alpha]=[bar.1,3]]) is a para-hyper-Kahler manifold, then each Ja is integrable and the fundamental 2- forms [[omega].sub.[alpha]], given by

[[omega].sub.[alpha]](X, Y)= [??]([J.sub.[alpha]]X, Y),

for all vector fields X, Y on TM, are closed for all [alpha] [member of] {1, 2, 3}.

For any vector field A and B along [pi], satisfying g(A, id) = g(B, id) = g(A, P [??] id) = g(B, P [??] id) = 0 on TM, using (3)-(8), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

But, because [[omega].sub.1] is closed, from (24) and (25) we obtain

[alpha]/a = [alpha] + t[beta]/a + tb = [alpha] + t[beta]/a - tc = C, (26)

where C is a real constant.

On the another hand, [J.sub.1], [J.sub.2], [J.sub.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, [alpha], [beta], 1/a, b/a(a+tb), c/a(a-tc) must be differentiable functions satisfying (18) and (26). []

5. 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-Kahler. These results can have important applications both in differential geometry and theoretical physics, since the existence of para-hyper-Kahler 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 Kahler structures on the tangent bundle of a paracomplex space form.

doi: 10.3176/proc.2011.3.04

ACKNOWLEDGEMENTS

I would like to express my deepest appreciation to Professor Liviu Ornea for carefully reading this paper and offering me helpful suggestions. This work was partially supported by CNCSIS--UEFISCSU, project PNII--IDEI code 8/2008, contract No. 525/2009.

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Gabriel Eduard Vilcu

University of Bucharest, Faculty of Mathematics and Computer Science, Research Center in Geometry, Topology and Algebra, Academiei Str. 14, Sector 1, Bucharest 70109, Romania; and Petroleum-Gas University of Ploiesti, Department of Mathematics and Computer Science, Bulevardul Bucuresti 39, Ploiesti 100680, Romania; gvilcu@upg-ploiesti.ro

Received 22 June 2010, revised 22 February 2011, accepted 23 February 2011

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