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