Directory UMM :Journals:Journal_of_mathematics:BJGA:
Hyper-K¨ahler structures on the tangent bundle of a
K¨ahler manifold
Vasile Oproiu
Dedicated to the 70-th anniversary
of Professor Constantin Udriste
Abstract. We construct a family of almost hyper-complex structures on
the tangent bundle of a K¨ahlerian manifold by using two anti-commuting
almost complex structures obtained from the natural lifts of the Riemannian metric (see [11], [12], [13], [18]) and the integrable almost complex structure on the base manifold. Next we obtain an almost hyperHermitian metric obtained from the same natural lifts, related to the considered almost complex structures. We study the integrability conditions
for the almost complex structures, obtaining that the base manifold must
have constant holomorphic sectional curvature, and the conditions under
which the considered almost hyper-Hermitian metric leads to a hyperK¨ahlerian structure on the tangent bundle.
M.S.C. 2000: 53C55, 53C15, 53C05.
Key words: tangent bundle; natural lifts; almost hyper-Hermitian structures; hyperK¨ahlerian structures.
1
Introduction
Consider an m(= 2n)-dimensional Riemannian manifold (M, g) and denote by τ :
T M −→ M its tangent bundle. Several Riemannian and semi-Riemannian metrics
can be used in order to obtain geometric properties of the tangent bundle T M of
(M, g). They are induced from the Riemannian metric g on M by using some lifts
of g. Among these metrics, we may quote the Sasaki metric and the complete lift of
the metric g. On the other hand, the natural lifts of g to T M , induce some other
Riemannian and pseudo-Riemannian geometric structures with many nice geometric
properties (see [8], [7]). By similar methods one can get from g some natural almost
complex structures on T M . If (M, g) has a structure of K¨ahlerian manifold we can
find some other Riemannian metrics and almost complex structures on its tangent
bundle and from them we can get some almost hyper-Hermitian structures (see also
[19], [20]). Similar results are obtained in the case of the cotangent bundle (see e.g.
[3]).
∗
Balkan
Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 104-119.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
Hyper-K¨ahler structures on the tangent bundle
105
In the present paper we study a class of natural almost hyper-Hermitian structures
(G, J1 , J2 ), on the tangent bundle T M of a K¨ahlerian manifold (M, g, J), induced from
the Riemannian metric g and the integrable almost complex structure J. The metric
G and the anti-commuting almost complex structures J1 , J2 are obtained as natural
lifts of diagonal type from g and J.
The manifolds, tensor fields and other geometric objects we consider in this paper
are assumed to be differentiable of class C ∞ (i.e. smooth). We use the computations
in local coordinates in a fixed local chart but many results may be expressed in
an invariant form by using the vertical and horizontal lifts. Some quite complicate
computations have been made by using the Ricci package under Mathematica for
doing tensor computations. The well known summation convention is used throughout
this paper, the range of the indices h, i, j, k, l being always {1, . . . , m = 2n}.
1. Hyper-complex structures on T M .
Let (M, g) be a smooth m = (2n)-dimensional Riemannian manifold and denote
its tangent bundle by τ : T M −→ M . Recall that there is a structure of a smooth 2mdimensional manifold on T M , induced from the structure of smooth m-dimensional
manifold of M . From every local chart (U, ϕ) = (U, x1 , . . . , xm ) on M , it is induced
a local chart (τ −1 (U ), Φ) = (τ −1 (U ), x1 , . . . , xm , y 1 , . . . , y m ), on T M , as follows. For
a tangent vector y ∈ τ −1 (U ) ⊂ T M , the first m local coordinates x1 , . . . , xm are the
local coordinates x1 , . . . , xm of its base point x = τ (y) in the local chart (U, ϕ) (in fact
we made an abuse of notation, identifying xi with τ ∗ xi = xi ◦ τ, i = 1, . . . , m). The
last m local coordinates y 1 , . . . , y m of y ∈ τ −1 (U ) are the vector space coordinates of y
∂
∂
with respect to the natural basis (( ∂x
1 )τ (y) , . . . , ( ∂xm )τ (y) ), defined by the local chart
(U, ϕ). Due to this special structure of differentiable manifold for T M , it is possible
to introduce the concept of M -tensor field on it. An M -tensor field of type (p, q)
on T M is defined by sets of np+q components (functions depending on xi and y i ),
with p upper indices and q lower indices, assigned to induced local charts (τ −1 (U ), Φ)
on T M , such that the local coordinate change rule is that of the local coordinate
components of a tensor field of type (p, q) on the base manifold M , when a change
of local charts on M (and hence on T M ) is performed (see [10] for further details);
e.g., the components y i , i = 1, . . . , m, corresponding to the last m local coordinates
of a tangent vector y, assigned to the induced local chart (τ −1 (U ), Φ) define an M tensor field of type (1, 0) on T M . A usual tensor field of type (p, q) on M may be
thought of as an M -tensor field of type (p, q) on T M . If the considered tensor field
on M is covariant only, the corresponding M -tensor field on T M may be identified
with the induced (pullback by τ ) tensor field on T M . Some useful M -tensor fields on
T M may be obtained as follows. Let u : [0, ∞) −→ R be a smooth function and let
kyk2 = gτ (y) (y, y) be the square of the norm of the tangent vector y ∈ τ −1 (U ). If δji
are the Kronecker symbols (in fact, they are the local coordinate components of the
identity tensor field I on M ), then the components u(kyk2 )δji define an M -tensor field
of type (1, 1) on T M . Similarly, if gij (x) are the local coordinate components of the
metric tensor field g on M in the local chart (U, ϕ), then the components u(kyk2 )gij
define a symmetric M -tensor field of type (0, 2) on T M . The components g0i = y k gki ,
as well as u(kyk2 )g0i define M -tensor fields of type (0, 1) on T M . Of course, all the
components considered above are in the induced local chart (τ −1 (U ), Φ).
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Vasile Oproiu
We shall use the horizontal distribution HT M , defined by the Levi Civita connec˙ of g, in order to define some first order natural lifts to T M of the Riemannian
tion ∇
metric g on M . Denote by V T M = Ker τ∗ ⊂ T T M the vertical distribution on T M .
Then we have the direct sum decomposition
(1.1)
T T M = V T M ⊕ HT M.
If (τ −1 (U ), Φ) = (τ −1 (U ), x1 , . . . , xm , y 1 , . . . y m ) is a local chart on T M , induced from
the local chart (U, ϕ) = (U, x1 , . . . , xm ), the local vector fields ∂y∂ 1 , . . . , ∂y∂m define a
local frame for V T M over τ −1 (U ) and the local vector fields δxδ 1 , . . . , δxδm define a
local frame for HT M over τ −1 (U ), where
δ
∂
∂
=
− Γh0i h ,
δxi
∂xi
∂y
Γh0i = y k Γhki
and Γhki (x) are the Christoffel symbols of g.
The set of vector fields ( ∂y∂ 1 , . . . , ∂y∂m , δxδ 1 , . . . , δxδm ) defines a local frame on T M ,
adapted to the direct sum decomposition (1.1). Remark that
∂
δ
∂
∂
= ( i )V ,
= ( i )H ,
∂y i
∂x
δxi
∂x
where X V and X H denote the vertical and horizontal lifts of the vector field X on
M.
Now assume that (M, g, J) is a K¨ahlerian manifold. The Riemannian metric g
and the integrable almost complex structure J are related by
˙ = 0,
g(JX, JY ) = g(X, Y ), ∇J
˙ is the Levi Civita connection of g. Recall that we have too the following
where ∇
relations
N = 0, dφ = 0,
where N is the Nijehuis tensor field of J and φ is the associated 2-form, defined by
φ(X, Y ) = g(X, JY ).
Denote by gij , Jji the components of g, J in the local chart (U, ϕ) = (U, x1 , . . . , xm ).
Introduce the components Jij = gih Jjh , obtained from the components of J by lowering the contravariance index on the first place (in fact, Jij are the components of
the fundamental 2-form φ defined by the K¨ahlerian structure (g, J). Consider the
following M -tensor fields on τ −1 (U ), defined by the components
gi0 = gih y h , Ji0 = Jih y h = −J0i .
Lemma 1. If m > 1 and u1 , u2 , u3 , u4 , u5 , u6 are smooth functions on T M such
that
u1 gij + u2 gi0 gj0 + u3 Ji0 Jj0 + u4 gi0 Jj0 + u5 Ji0 gj0 + u6 Jij = 0, y ∈ τ −1 (U )
107
Hyper-K¨ahler structures on the tangent bundle
on the domain of any induced local chart on T M , then u1 = u2 = u3 = u4 = u5 =
u6 = 0.
The proof is obtained easily by transvecting the given relation with g ij , J ij =
= Jhj y h and y j (Recall that the functions g ij (x) are the components of the
inverse of the matrix (gij (x)), associated to g in the local chart (U, ϕ) on M ; moreover,
the components g ij (x) define a tensor field of type (2, 0) on M ).
Jhi g hj , J0j
Remark. From a relation of the type
u1 δji + u2 y i gj0 + u3 J0i Jj0 + u4 y i Jj0 + u5 J0i gj0 + u6 Jji = 0, y ∈ τ −1 (U )
it is obtained, in a similar way, u1 = u2 = u3 = u4 = u5 = u6 = 0.
Since we work in a fixed local chart (U, ϕ) on M and in the corresponding induced
local chart (τ −1 (U ), Φ) on T M , we shall use the following simpler notations
∂
δ
= ∂i ,
= δi .
i
∂y
δxi
Denote by
(1.2)
t=
1
1
1
kyk2 = gτ (y) (y, y) = gik (x)y i y k ,
2
2
2
y ∈ τ −1 (U )
the energy density defined by g in the tangent vector y. We have t ∈ [0, ∞) for
∂
V
all y ∈ T M . Let C = y i ∂y
be the Liouville vector field on T M and coni = y
e = y i δ i = y H on T M , obtained
sider the corresponding horizontal vector field C
δx
in a similar way. Consider the real valued smooth functions a1 , a2 , a3 , a4 , a5 , a6 , b1 ,
b2 , b3 , b4 , b5 , b6 , c1 , c2 , c3 , c4 , c5 , c6 , d1 , d2 , d3 , d4 , d5 , d6 defined on [0, ∞) ⊂ R and define two diagonal natural almost complex structures J1 , J2 on T M , by using these
coefficients, the Riemannian metric g and the integrable almost complex structure J
J1 XyH = a1 (t)XyV + a2 (t)gτ (y) (y, X)Cy + a3 (t)gτ (y) (Jy, X)(Jy)Vy +
+a4 (t)(JX)Vy + a5 (t)gτ (y) (X, y)(Jy)Vy + a6 (t)gτ (y) (Jy, X)Cy ,
(1.3)
ey + b3 (t)gτ (y) (Jy, X)(Jy)H
J1 XyV = −(b1 (t)XyH + b2 (t)gτ (y) (y, X)C
y +
H
e
+b4 (t)(JX)H
y + b5 (t)gτ (y) (X, y)(Jy)y + b6 (t)gτ (y) (Jy, X)Cy ),
(1.4)
J2 XyH = c1 (t)XyV + c2 (t)gτ (y) (y, X)Cy + c3 (t)gτ (y) (Jy, X)(Jy)Vy +
+c4 (t)(JX)Vy + c5 (t)gτ (y) (X, y)(Jy)Vy + c6 (t)gτ (y) (Jy, X)Cy ,
ey + d3 (t)gτ (y) (Jy, X)(Jy)H
J2 XyV = −(d1 (t)XyH + d2 (t)gτ (y) (y, X)C
y +
H
e
+d4 (t)(JX)H
y + d5 (t)gτ (y) (X, y)(Jy)y + d6 (t)gτ (y) (Jy, X)Cy ).
The expressions of J1 , J2 in adapted local frames are
J1 δi = J1 Hih ∂h , J1 ∂i = J1 Vih δh ,
J2 δi = J2 Hih ∂h , J2 ∂i = J2 Vih δh ,
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Vasile Oproiu
where the M -tensor fields J1 Hih , J1 Vih , J2 Hih , J2 Vih are given by
J1 Hih = a1 δih + a2 gi0 y h + a3 Ji0 J0h + a4 Jih + a5 gi0 J0h + a6 Ji0 y h ,
J1 Vih = −(b1 δih + b2 gi0 y h + b3 Ji0 J0h + b4 Jih + b5 gi0 J0h + b6 Ji0 y h ),
J2 Hih = c1 δih + c2 gi0 y h + c3 Ji0 J0h + c4 Jih + c5 gi0 J0h + c6 Ji0 y h ,
J2 Vih = −(d1 δih + d2 gi0 y h + d3 Ji0 J0h + d4 Jih + d5 gi0 J0h + d6 Ji0 y h ).
The matrices associated to J1 , J2 have a diagonal form
µ
¶
µ
0
J1 Hih
0
J1 =
, J2 =
0
J1 Vih
J2 Vih
J2 Hih
0
¶
.
Remark that, one can consider the case of the general natural tensor fields J1 , J2 on
T M , when J1 δi , J1 ∂i , J2 δi , J2 ∂i are expressed as combinations of ∂h , δh . In this case
we should have 48 coefficients and the computations would become really complicate.
However, the results obtained in the general case do not differ too much from that
obtained in the diagonal case.
We use the following notation:
α = (a1 + 2a2 t)(a1 + 2a3 t) + (a4 + 2a5 t)(a4 − 2a6 t).
Proposition 2. The operator J1 defines an almost complex structure on T M if and
only if the coefficients b1 , b2 , b3 , b4 , b5 , b6 are expressed as
1
4
b1 = a2a+a
b4 = a−a
2,
2 +a2 ,
1
4
1
4
1
b2 = α [b1 (−a1 a2 − 2a2 a3 t + 2a5 a6 t) + b4 (a1 a5 − a1 a6 − a3 a4 )],
b3 = α1 [b1 (−a1 a3 − 2a2 a3 t + 2a5 a6 t) + b4 (a1 a5 − a1 a6 − a2 a4 )],
(1.5)
b5 = α1 [b1 (−a1 a5 + a2 a4 + a3 a4 ) + b4 (a4 a6 − 2a2 a3 t + 2a5 a6 t)],
b6 = α1 [b1 (−a1 a6 − a2 a4 − a3 a4 ) + b4 (a4 a5 + 2a2 a3 t − 2a5 a6 t)].
Proof. The relations are obtained by some quite straightforward but long computations, from the property J12 = −I of J1 and Lemma 1.
Remark. Using the first two relations (1.5) we may find the expressions of
b2 , b3 , b5 , b6 as functions of a1 , a2 , a3 , a4 , a5 , a6 only. Remark that the parameters
a1 , a4 cannot vanish simultaneously and that α 6= 0. A similar result is obtained from
the condition for J2 to be an almost complex structure on T M . In this case we can
express the coefficients d1 , d2 , d3 , d4 , d5 , d6 as functions of c1 , c2 , c3 , c4 , c5 , c6 . We shall
use the following notation:
β = (c1 + 2c2 t)(c1 + 2c3 t) + (c4 + 2c5 t)(c4 − 2c6 t).
Then we get
d1
d2
d3
(1.6)
d5
d
6
=
c1
,
c21 +c24
=
1
β [d1 (−c1 c2
− 2c2 c3 t + 2c5 c6 t) + d4 (c1 c5 − c1 c6 − c3 c4 )],
=
1
β [d1 (−c1 c3
− 2c2 c3 t + 2c5 c6 t) + d4 (c1 c5 − c1 c6 − c2 c4 )],
=
1
β [d1 (−c1 c5
+ c2 c4 + c3 c4 ) + d4 (c4 c6 − 2c2 c3 t + 2c5 c6 t)],
=
1
β [d1 (−c1 c6
− c2 c4 − c3 c4 ) + d4 (c4 c5 + 2c2 c3 t − 2c5 c6 t)].
d4 =
−c4
,
c21 +c24
Hyper-K¨ahler structures on the tangent bundle
109
Now we shall study the conditions under which the almost complex structures J1 , J2
satisfy the relation J1 J2 + J2 J1 = 0, leading to the almost hyper-complex structure
on T M .
Theorem 3. The almost complex structures J1 , J2 define an almost hyper-complex
structure on T M if
(1.7)
c1 = a4 , c4 = −a1 ,
c3 = (a21 a5 + a24 a5 − a21 a6 − a24 a6 − a21 c2 − a24 c2 + 2a2 a3 a4 t − 2a1 a2 a6 t−
−2a1 a3 a6 t − 4a4 a5 a6 t + 2a4 a26 t − 2a1 a3 c2 t + 2a4 a6 c2 t − 2a2 a4 c6 t − 2a3 a4 c6 t+
+4a1 a5 c6 t − 2a1 c2 c6 t + 2a4 c26 t − 4a2 a3 a6 t2 + 4a5 a26 t2 − 4a3 c2 c6 t2 + 4a5 c26 t2 )/
((a1 + 2a2 t)(a1 + 2c6 t) + (a4 + 2c2 t)(a4 − 2a6 t)),
c5 =
−1
a1 +2c6 t (a1 a2
+ a1 a3 + a4 a5 − a4 a6 − a4 c2 − a4 c3 − a1 c6 +
+2a2 a3 t − 2a5 a6 t − 2c2 c3 t).
Proof. From the relation J1 Vhk J2 Hih + J2 Vhk J1 Hih = 0 we get
(−b1 c1 + b4 c4 − a1 d1 + a4 d4 )δik − (b4 c1 + b1 c4 + a4 d1 + a1 d4 )Jik −
−(b5 c1 + b4 c2 + b3 c4 + b1 c5 + a5 d1 + a4 d3 + a2 d4 + a1 d5 +
+2b5 c2 t + 2b3 c5 t + 2a5 d3 t + 2a2 d5 t)gi0 J0k +
(−b3 c1 − b1 c3 + b5 c4 − b4 c6 − a3 d1 − a1 d3 + a6 d4 + a4 d5 −
−2b3 c3 t − 2b5 c6 t − 2a3 d3 t − 2a6 d5 t)Ji0 J0k +
−b2 c1 − b1 c2 − b6 c4 + b4 c5 − a2 d1 − a1 d2 + a5 d4 − a4 d6 −
−2b2 c2 t − 2b6 c5 t − 2a2 d2 t − 2a5 d6 t)gi0 y k −
(b6 c1 − b4 c3 − b2 c4 + b1 c6 + a6 d1 − a4 d2 − a3 d4 + a1 d6 +
+2b6 c3 t + 2b2 c6 t + 2a6 d2 t + 2a3 d6 t)Ji0 y k .
Replacing bα , dα ; α = 1, . . . , 6 and using Lemma 1 we get the following relations
(from the vanishing of the first two coefficients)
(a1 c1 + a4 c4 )(a21 + a24 + c21 + c24 ) = 0,
(a4 c1 − a1 c4 )(a21 + a24 − c21 − c24 ) = 0.
Since a21 + a24 6= 0, c21 + c24 6= 0, we obtain the relations
c1 = ±a4 , c4 = ∓a1 .
From now on we shall consider only the case c1 = a4 , c4 = −a1 . The expressions
of c3 , c5 are obtained from the vanishing of the next 4 coefficients. Then the other
relations obtained from J1 J2 + J2 J1 = 0 are identically fulfilled.
Remark that the final expression of c5 is obtained after replacing the obtained
expression of c3 .
110
Vasile Oproiu
Hence an almost hyper-complex structure on T M , of the considered type depends
on 8 essential parameters a1 , a2 , a3 , a4 , a5 , a6 , c2 , c6 (real valued smooth functions depending on the density energy t ∈ [0, ∞). Remark that the functions aα ; α = 1, . . . , 6,
must fulfill some supplementary conditions which assure the existence of the expressions obtained above.
Now we shall study the integrability problem for the obtained almost hypercomplex structure. The integrability conditions for such a structure are expressed with
the help of various Nijenhuis tensor fields obtained from the tensor fields J1 , J2 , J3 =
J1 J2 . For a tensor field K of type (1, 1) on a given manifold, we can consider its
Nijenhuis tensor field NK defined by
NK (X, Y ) = [KX, KY ] − K[X, KY ] − K[KX, Y ] + K 2 [X, Y ],
where X, Y are vector fields on the given manifold. For two tensor fields K, L of type
(1, 1) on the given manifold, we can consider the corresponding Nijenhuis tensor field
NK,L defined by
NK,L (X, Y ) = [KX, LY ] + [LX, KY ] − K([X, LY ] + [LX, Y ])−
−L([KX, Y ] + [X, KY ]) + (KL + LK)[X, Y ].
The almost hyper-complex structure defined by J1 , J2 is integrable iff N1 =
0, N2 = 0, where N1 , N2 are the Nijenhuis tensor fields of J1 , J2 . Equivalently,
the structure is integrable iff N1 + N2 + N3 = 0, or iff N12 = 0, where N3 is the
Nijenhuis tensor field of J3 = J1 J2 and N12 = NJ1 ,J2 is the Nijenhuis tensor field of
J1 , J2 .
In the case of the almost hyper-complex structure defined on T M by the tensor
fields J1 , J2 the most convenient way to study its integrability is the using of the
Nijenhuis tensor fields N1 , N2 .
Proposition 4. If the almost hyper-complex structure defined by (J1 , J2 ) on T M
is integrable then the K¨
ahlerian manifold (M, g, J) has constant holomorphic sectional
curvature.
Proof. Recall the following formulas, useful in computing the expressions of N1 , N2
k
∂k ,
[∂i , ∂j ] = 0, [∂i , δj ] = −Γkij ∂k , [δi , δj ] = −R0ij
δi y h = −Γhi0 , δi gjk = Γhij ghk + Γhik gjh , δi gj0 = g0h Γhij ,
δi Jlk = −Γkih Jlh + Γhil Jhk , δi J0k = −Γkih J0h , δi Jj0 = Γhij Jh0 .
We have used the notations
k
k
, Γki0 = y h Γkih , gj0 = gjh y h , J0k = Jhk y h , Jj0 = Jjh y h .
R0ij
= y h Rhij
Then we get
h
)∂h .
N1 (δi , δj ) = (J1 Hik ∂k J1 Hjh − J1 Hjk ∂k J1 Hih + R0ij
Remark that all the terms containing the Christoffel symbols cancel. Doing the necessary replacements, we get a relation of the following type
α1 (Jih gj0 − Jjh gi0 ) + α2 (g0i δjh − g0j δih ) + 2α3 Jij y h + α4 (Ji0 Jjh − Jj0 Jih ) + 2α5 Jij J0h +
111
Hyper-K¨ahler structures on the tangent bundle
h
+α6 (δih Jj0 − δjh Ji0 ) + R0ij
+ α7 (gi0 Jj0 − gj0 Ji0 )y h + α8 (gi0 Jj0 − gj0 Ji0 )J0h = 0,
where the coefficients α1 , α2 , α3 , α4 , α5 , α6 , α7 , α8 , are functions of t, expressed with
the help of the coefficients a1 , a2 , a3 , a4 , a5 , a6 and their derivatives.
Differentiating this relation with respect to y k , then taking y = 0, one gets
α1 (0)(Jih gjk − Jjh gik ) + α2 (0)(gki δjh − gkj δih ) + 2α3 (0)Jij δkh + α4 (0)(Jik Jjh − Jjk Jih )+
h
= 0.
+2α5 (0)Jij Jkh + α6 (0)(δih Jjk − δjh Jik ) + Rkij
Then, using the well known (skew) symmetries of the components of R, as well as the
(first) Bianchi identity and the invariance properties of R with respect to J, one finds
(1.8)
h
Rkij
= c(gjk δih − gik δjh + Jih gkl Jjl − Jjh gkl Jil + 2Jkh gil Jjl ),
i.e. the K¨ahlerian manifold (M, g, J) has constant holomorphic sectional curvature
4c.
h
Replacing the obtained expression of Rkij
in the relation N1 (δi , δj ) = 0, and using
Lemma 1, one obtains some further relations
(1.9)
a3 =
c − a4 a5
,
a1
a6 = −
a2 a4
.
a1
Then, replacing these expressions of a3 , a6 in the remaining terms one gets
(1.10)
a2 =
a21
a1 (a1 a′1 + a4 a′4 − c)
−a2 a4 + a1 a′4 + 2a2 a′4 t
.
, a5 =
2
′
′
+ a4 − 2a1 a1 t − 2a4 a4 t
a1
Next, computing the expressions N1 (∂i , ∂j ), N1 (δi , ∂j ), we get that they are identically
zero.
Similar results are obtained from the integrability conditions for J2 , but we should
prefer to present some other expressions (we shall assume that c4 6= 0)
(1.11)
c2 = − c1c4c6 ,
c3 =
c5 =
c−c1 c3
c4 ,
c1 c6 +c1′ c4 −2c′1 c6 t
,
c4
c6 =
c4 (c−c1 c′1 −c4 c′4 )
.
c21 +c24 −2c1 c′1 t−2c4 c′4 t
Finally, by using the relations obtained in Theorem 3, one gets the expressions of
c2 , c3 , c5 , c6 as functions a1 , a4 an their derivatives
(1.12)
c2 =
c5 =
a (c−a1 a′1 )+a′4 (a21 −2ct)
a4 (a1 a′1 +a4 a′4 −c)
, c3 = a42 +a2 −2a
,
′
′
a21 +a24 −2a1 a′1 t−2a4 a′4 t
1 a1 t−2a4 a4 t
4
1
a1 (a4 a′4 −c)−a′1 (a24 −2ct)
a1 (a1 a′1 +a4 a′4 −c)
, c6 = a2 +a2 −2a1 a′ t−2a4 a′ t
a21 +a24 −2a1 a′1 t−2a4 a′4 t
1
4
1
4
Remark that the values of c3 , c5 obtained in (1.12) do coincide with the values of
c3 , c5 obtained in Theorem 3 after replacing c2 , c6 obtained in (1.12) and a2 , a3 , a5 , a6
obtained in (1.9) and (1.10).
112
2
Vasile Oproiu
Hyper-K¨
ahler structures on T M
Consider a natural Riemannian metric G on T M of diagonal type induced from g and
J and given by
(2.1)
Gy (X H , Y H ) = p1 (t)gτ (y) (X, Y ) + p2 (t)gτ (y) (y, X)gτ (y) (y, Y )+
+p3 (t)gτ (y) (JX, y)gτ (y) (JY, y) + p4 (t)(gτ (y) (JX, y)gτ (y) (Y, y)+
+g
τ (y) (JY, y)gτ (y) (X, y)),
Gy (X V , Y V ) = q1 (t)gτ (y) (X, Y ) + q2 (t)gτ (y) (y, X)gτ (y) (y, Y )+
+q3 (t)gτ (y) (JX, y)gτ (y) (JY, y) + q4 (t)(gτ (y) (JX, y)gτ (y) (Y, y)+
+gτ (y) (JY, y)gτ (y) (X, y)),
Gy (X H , Y V ) = Gy (Y V , X H ) = Gy (X V , Y H ) = Gy (Y H , X V ) = 0,
where p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 are smooth real valued functions defined on [0, ∞).
Remark that we have to find the conditions under which G is real Riemannian metric.
The expression of G in local adapted frames is defined by the following M -tensor
fields
Gij = G(δi , δj ) = p1 gij + p2 g0i g0j + p3 Ji0 Jj0 + p4 (gi0 Jj0 + gj0 Ji0 ),
Hij = G(∂i , ∂j ) = q1 gij + q2 g0i g0j + q3 Ji0 Jj0 + q4 (gi0 Jj0 + gj0 Ji0 )
and the associated 2m × 2m-matrix with respect to the adapted local frame
µ
δ
δ
∂
∂
,..., m, 1,..., m
δx1
δx ∂y
∂y
¶
has two m × m-blocks on the first diagonal
G=
µ
Gij
0
0
Hij
¶
.
We shall be interested in the conditions under which the metric G is almost Hermitian with respect to the almost complex structures J1 , J2 , considered in the previous
section, i.e.
G(J1 X, J1 Y ) = G(X, Y ), G(J2 X, J2 Y ) = G(X, Y ),
for all vector fields X, Y on T M .
From the relation
G(J1 δi , J1 δj ) = G(δi , δj ),
we get
(2.2)
Hkl J1 Hik J1 Hjl = Gij ,
Hyper-K¨ahler structures on the tangent bundle
113
from which we obtain the following expressions for p1 , p2 , p3 , p4
p1 = (a21 + a24 )q1 ,
p2 = (2a1 a2 + 2a4 a5 + 2a22 t + 2a25 t)q1 +
(a1 + 2a2 t)2 q2 + (a4 + 2a5 t)2 q3 + 2(a1 + 2a2 t)(a4 + 2a5 t)q4 ,
p3 = (2a1 a3 − 2a4 a6 + 2a23 t + 2a26 t)q1 +
(2.3)
(a4 − 2a6 t)2 q2 + (a1 + 2a3 t)2 q3 − 2(a1 + 2a3 t)(a4 − 2a6 t)q4 ,
p4 = (−a2 a4 + a3 a4 + a1 a5 + a1 a6 + 2a3 a5 t + 2a2 a6 t)q1 +
+(−a1 a4 − 2a2 a4 t + 2a1 a6 t + 2a1 a6 t)q2 + (a1 + 2a3 t)(a4 + 2a5 t)q3 +
+(a21 − a24 + 2a1 a2 t + 2a1 a3 t − 2a4 a5 t + 2a4 a6 t + 4a2 a3 t2 + 4a5 a6 t2 )q4 .
Remark that from the conditions G(J1 ∂i , J1 ∂j ) = G(∂i , ∂j ), G(J1 δi , J1 ∂j ) =
G(δi , ∂j ), we do not obtain new essential relations fulfilled by p′ s and q ′ s.
Now we deal with the condition
G(J2 δi , J2 δj ) = G(δi , δj ),
from which we get
(2.4)
Hkl J2 Hik J2 Hjl = Gij .
We find the coefficients p1 , p2 , p3 , p4 expressed in function of the coefficients q1 , q2 , q3 , q4
by formulas similar to (2.3), where the parameters a1 , a2 , a4 , a5 , a6 are replaced by
c1 , c2 , c3 , c4 , c5 , c6 respectively. Next, we may write the system fulfilled by q1 , q2 , q3 , q4 ,
obtained by equalizing the obtained values for p1 , p2 , p3 , p4 . Remark that, due to the
formula (1.7), the first equation, corresponding to p1 , is trivial. So, we get a homogeneous system consisting of 3 equations
q1 (−2a1 a2 − 2a4 a5 + 2a4 c2 − 2a1 c5 − 2a22 t − 2a25 t + 2c22 t + 2c25 t)+
+q2 (−a21 + a24 − 4a1 a2 t + 4a4 c2 t − 4a22 t2 + 4c22 t2 )+
+q3 (a21 − a24 − 4a4 a5 t − 4a1 c5 t − 4a25 t2 + 4c25 t2 )+
+q4 (−4a1 a4 − 4a2 a4 t − 4a1 a5 t − 4a1 c2 t + 4a4 c5 t − 8a2 a5 t2 + 8c2 c5 t2 ) = 0,
q1 (−a2 a4 + a3 a4 + a1 a5 + a1 a6 − a1 c2 + a1 c3 − a4 c5 − a4 c6 + 2a3 a5 t + 2a2 a6 t − 2c3 c5 t−
−2c2 c6 t) + q2 (−2a1 a4 − 2a2 a4 t + 2a1 a6 t − 2a1 c2 t − 2a4 c6 t + 4a2 a6 t2 − 4c2 c6 t2 )+
+q3 (2a1 a4 + 2a3 a4 t + 2a1 a5 t + 2a1 c3 t − 2a4 c5 t + 4a3 a5 t2 − 4c3 c5 t2 ) + q4 (2a21 − 2a24 +
+2a1 a2 t + 2a1 a3 t − 2a4 a5 t + 2a4 a6 t − 2a4 c2 t − 2a4 c3 t − 2a1 c5 t + 2a1 c6 t + 4a2 a3 t2 +
+4a5 a6 t2 − 4c2 c3 t2 − 4c5 c6 t2 ) = 0,
q1 (2a1 a3 − 2a4 a6 − 2a4 c3 − 2a1 c6 + 2a23 t + 2a26 t − 2c23 t − 2c26 t)+
+q2 (−a21 + a24 − 4a4 a6 t − 4a1 c6 t + 4a26 t2 − 4c26 t2 )+
+q3(a21 − a24 + 4a1 a3 t − 4a4 c3 t + 4a23 t2 − 4c23 t2 )+
+q4 (−4a1 a4 − 4a3 a4 t + 4a1 a6 t − 4a1 c3 t − 4a4 c6 t + 8a3 a6 t2 − 8c3 c6 t2 ) = 0.
The matrix of this system has the rank 2 and we may obtain its general solution
depending on two parameters
(2.5)
q1 = λ,
q3 = µ,
114
Vasile Oproiu
q4 = ((a3 a4 − a1 a5 + a1 c2 − a4 c6 + 2a3 c2 t − 2a5 c6 t)λ+
+(2a3 a4 t − 2a1 a5 t + 2a1 c2 t − 2a4 c6 t + 4a3 c2 t2 − 4a5 c6 t2 )µ)/
(a21 + a24 + 2a1 a2 t − 2a4 a6 t + 2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 ),
q1 + 2tq2 =
= ((a41 + 2a21 a24 + a44 + 4a31 a2 t + 4a13 a3 t + 4a1 a2 a24 t + 4a1 a3 a24 t + 4a21 a4 a5 t+
+4a34 a5 t − 4a21 a4 a6 t − 4a34 a6 t + 4a21 a22 t2 + 16a21 a2 a3 t2 + 4a21 a23 t2 + 8a2 a3 a24 t2 +
+4a23 a24 t2 + 8a1 a2 a4 a5 t2 + 4a21 a25 t2 + 4a24 a25 t2 − 8a1 a2 a4 a6 t2 − 8a1 a3 a4 a6 t2 −
−8a21 a5 a6 t2 − 16a24 a5 a6 t2 + 4a24 a26 t2 + 8a1 a3 a4 c2 t2 − 8a21 a5 c2 t2 + 4a21 c22 t2 −
−8a3 a24 c6 t2 + 8a1 a4 a5 c6 t2 − 8a1 a4 c2 c6 t2 + 4a24 c26 t2 + 16a1 a22 a3 t3 + 16a1 a2 a23 t3 +
+16a2 a3 a4 a5 t3 − 16a2 a3 a4 a6 t3 − 16a1 a2 a5 a6 t3 − 16a1 a3 a5 a6 t3 − 16a4 a25 a6 t3 +
+16a4 a5 a26 t3 + 16a23 a4 c2 t3 − 16a1 a3 a5 c2 t3 + 16a1 a3 c22 t3 − 16a3 a4 a5 c6 t3 +
+16a1 a25 c6 t3 − 16a3 a4 c2 c6 t3 − 16a1 a5 c2 c6 t3 + 16a4 a5 c26 t3 + 16a22 a23 t4 −
−32a2 a3 a5 a6 t4 + 16a25 a26 t4 + 16a23 c22 t4 − 32a3 a5 c2 c6 t4 + 16a25 c26 t4 )(λ + 2tµ))/
(a21 + a24 + 2a1 a2 t − 2a4 a6 t + 2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 .
The explicit expression of q2 is obtained from the expression of q1 + 2tq2 and is more
complicate. Next, the expressions of p1 , p2 , p3 , p4 are obtained from (2.3).
p1 = (a21 + a24 )λ,
p1 + 2tp2 = (a21 + a24 + 2a1 a2 t + 2a1 a3 t + 2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2
(a21 + a24 + 4a1 a2 t + 4a4 c2 t + 4a22 t2 + 4c22 t2 )(λ + 2tµ)/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 ,
p1 + 2tp3 = (a21 + a24 + 2a1 a2 t + 2a1 a3 t + 2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2
(a21 + a24 − 4a4 a6 t + 4a1 c6 t + 4a26 t2 + 4c26 t2 )(λ + 2tµ)/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 ,
p4 = (−a2 a4 + a1 a6 + a1 c2 + a4 c6 + 2a2 a6 t + 2c2 c6 t)(a21 + a24 + 2a1 a2 t + 2a1 a3 t+
+2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2 (λ + 2tµ))/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 .
If we assume that the almost hyper-complex structure defined by J1 , J2 is integrable,
the expressions of the coefficients in G are simpler.
For the almost hyper-Hermitian manifold (T M, G, J1 , J2 ) the fundamental 2-forms
φ1 , φ2 are defined by
φ1 (X, Y ) = G(X, J1 Y ), φ2 (X, Y ) = G(X, J2 Y ),
where X, Y are vector fields on T M .
Hyper-K¨ahler structures on the tangent bundle
115
Since we have a third almost complex structure J3 = J1 J2 which is almost Hermitian with respect to G, we can consider a third 2-form φ3 defined by φ3 (X, Y ) =
G(X, J3 Y ), next we have the fundamental 4 form Ω, defined by
Ω = φ1 ∧ φ1 + φ2 ∧ φ2 + φ3 ∧ φ3 .
The almost hyper-Hermitian manifold (T M, G, J1 , J2 ) is hyper-K¨ahlerian if the almost complex structures J1 , J2 are parallel with respect to the Levi Civita connection
∇ defined by G, i.e. ∇J1 = 0, ∇J2 = 0. Equivalently, (T M, G, J1 , J2 ) is hyperK¨ahlerian if and only if the almost hyper-complex structure (J1 , J2 ) is integrable and
the 4-form Ω is closed, i.e. N1 = 0, N2 = 0, dΩ = 0. The condition for Ω to be
closed is equivalent to the conditions for φ1 , φ2 (and hence for φ3 too) to be closed i.e.
dφ1 = 0, dφ2 = 0. In our case, it is more convenient to study the conditions under
which the 2-forms φ1 , φ2 are closed.
The expressions of φ1 , φ2 in adapted local frames are
φ1 = φ1,jk Dy j ∧ dxk , φ2 = φ2,jk Dy j ∧ dxk ,
where
Dy j = dy j + Γji0 dxi ,
φ1,jk = G(∂j , J1 δk ) = Hjh J1 Hkh = −J1 Vjh Ghk ,
φ2,jk = G(∂j , J2 δk ) = Hjh J2 Hkh = −J2 Vjh Ghk .
Replacing the expressions of Hjh and J1 Hkh , J2 Hkh , we find the following expressions
φ1,jk = a1 q1 gjk + a4 q1 Jjk + (a2 q1 + a1 q2 + a4 q4 + 2a2 q2 t + 2a5 q4 t)gj0 gk0 +
+(a6 q1 − a4 q2 + a1 q4 + 2a6 q2 t + 2a3 q4 t)gj0 Jk0 +
+(a5 q1 + a4 q3 + a1 q4 + 2a5 q3 t + 2a2 q4 t)Jj0 gk0 +
+(−a3 q1 − a1 q3 + a4 q4 − 2a3 q3 t − 2a6 q4 t)Jj0 Jk0 ,
φ2,jk = a4 q1 gjk − a1 q1 Jjk + (c2 q1 + c4 q2 − a1 4q4 + 2c2 q2 t + 2c5 q4 t)gj0 gk0 +
+(c6 q1 + a1 q2 + a4 q4 + 2c6 q2 t + 2c3 q4 t)gj0 Jk0 +
+(c5 q1 − a1 q3 + a4 q4 + 2c5 q3 t + 2c2 q4 t)Jj0 gk0 +
+(−c3 q1 − a4 q3 − a1 q4 − 2c3 q3 t − 2c6 q4 t)Jj0 Jk0 .
The final expressions of φ1 , φ2 are obtained by replacing the values of q1 , q2 , q3 , q4
obtained from (2.5), then the values of c3 , c5 , obtained in Theorem 3. We get for
φ1,jk an expression of the type
φ1,jk = α1 gjk + α2 Jjk + α3 Jjk + α4 J0j g0k + α5 g0j J0k + α6 J0j J0k ,
where α1 , α2 , α3 , α4 , α5 , α6 are functions of t, expressed with the help of the coefficients a1 , a2 , a3 , a4 , a5 , a6 , c2 , c6 . A similar expression is obtained for φ2,jk .
Now we shall compute the expression of dφ1 by using the following formulas
dαr = αr′ g0i Dy i , r = 1, 2, 3, 4, 5, 6,
dgjk = (Γhij ghk + Γhik gjh )dxi ,
116
Vasile Oproiu
dg0j = dgj0 = gji Dy i + g0h Γhji dxi , dJjk = (Γhij Jhk + Γhik Jjh )dxi ,
1 h
dDy h = Γhij Dy i ∧ dxj + R0ij
dxi ∧ dxj .
2
We get the cancellation of all terms containing dxi ∧ Dy j ∧ dxk in the expression
of dφ1 . Next the terms containing dxi ∧ dxj ∧ dxk are
dJj0 = −dJ0j = Jh0 Γhji dxi + Jji Dy i ,
1
h
h
h
(φ1,hk R0ij
+ φ1,hi R0jk
+ φ1,hj R0ki
)dxi ∧ dxj ∧ dxk .
6
From the vanishing of this term and under the assumption that the base manifold
(M, g, J) has constant holomorphic sectional curvature 4c, we get that the factors λ, µ
are related by
µ 2
¶
a1 + a24
(2.6)
λ=−
+ 2t µ.
c
Finally, assuming that this relation as, well as the integrability conditions for the
almost hyper-complex structure defined by J1 , J2 are fulfilled, we get that α5 = α6 = 0
and the expression of dφ1 becomes
dφ1 = (α1′ g0i gjk + α2′ g0i Jjk + α3 g0j gki + α4 g0j Jik )Dy i ∧ Dy j ∧ dxk ,
where the coefficients α1 , α2 , α3 , α4 are given by
α1 = a1 λ, α2 = a4 λ,
(2.7)
α3 =
λ(−a21 a′1 + a′1 a24 − 2a1 a4 a′4 + 2a1 c − 2a′1 ct)
,
a21 + a24 − 2ct
α4 =
λ(−2a1 a′1 a4 + a21 a′4 − a′4 a24 + 2a4 c − 2a′4 ct)
.
a21 + a24 − 2ct
Doing the necessary alternation in the relation dφ1 = 0, we get the equations
α1′ = α3 , α2′ = α4 .
Then, after some simple computations, we obtain that the coefficients λ, µ are given
by
k
−ck
λ= 2
, µ= 2
,
a1 + a24 − 2ct
(a1 + a24 )2 − 4c2 t2
where k is a constant.
Under the same assumptions that the relation (2.6) is true and that the integrability conditions for the almost hyper-complex structure defined by J1 , J2 are fulfilled,
we get the following expression expression for the 2-form φ2
φ2 = (α2 gjk − α1 Jjk + α4 g0j g0k − α3 g0j J0k )Dy j ∧ dxk .
If dφ1 = 0 we have that dφ2 = 0 too, so that the structure (G, J1 , J2 ) on T M becomes
K¨ahlerian. We write down the explicit expressions of the coefficients involved in the
expressions of (G, J1 , J2 ). First of all, from the integrability condition N1 = 0, we get
(2.8)
a2 =
a21
a1 (a1 a′1 + a4 a′4 − c)
a′ a2 − a1 a4 a′4 + a1 c − 2a′1 ct
, a3 = 1 2 4 2
,
2
′
′
+ a4 − 2a1 a1 t − 2a4 a4 t
a1 + a4 − 2a1 a′1 t − 2a4 a′4 t
117
Hyper-K¨ahler structures on the tangent bundle
a5 =
−a4 (a1 a′1 + a4 a′4 − c)
−a1 a′1 a4 + a21 a′4 + a4 c − 2a′4 ct
, a6 = 2
.
2
2
′
′
a1 + a4 − 2a1 a1 t − 2a4 a4 t
a1 + a24 − 2a1 a′1 t − 2a4 a′4 t
The values of c2 , c3 , c5 , c6 are given by (1.12). Next we have
(2.9)
q1 = λ =
p1 =
ck
k(a21 + a24 )
, p 2 = p3 = 2
, p4 = 0,
+ a24 − 2ct
a1 + a24 − 2ct
a21
k
−ck
k(a′1 a4 − a1 a′4 )
,
q
=
µ
=
,
q
=
.
3
4
a21 + a24 − 2ct
(a21 + a24 )2 − 4c2 t2
(a21 + a24 )2 − 4c2 t2
The expression of q2 is quite complicate and can be obtained from (2.5). Hence we
may state
Theorem 5. Consider the almost hyper-Hermitian structure (G, J1 , J2 ) defined
as above on the tangent bundle T M of the K¨
ahlerian manifold M . This structure
is hyper-K¨
ahlerian if and only if the almost complex structures J1 , J2 are integrable
(hence the Proposition 4 and the relations (1.9), (1.10), (1.11), (1.12) are fulfilled)
and and the relations (2.6), (2.7), (2.8), (2.9) are fulfilled by the hyper-Hermitian
metric G.
The case where a4 = 0
We shall study a special case when a4 = 0. In this case we shall obtain some much
more simple formulas and results and many of them are related to those obtained in
[19], [20]. However, our parametrization is quite different.
Since the integrability conditions for the almost complex structure J1 we get that
the condition a4 = 0 implies the conditions a5 = 0, a6 = 0. We are interested in
the integrable case, so that we shall assume from the beginning a4 = a5 = a6 = 0.
According to the result obtained in Proposition 2, the tensor field J1 defines almost
complex structure on T M if and only if
½
2
3
, b3 = − a1 (a1a+2a
,
b1 = a11 , b2 = − a1 (a1a+2a
2 t)
3 t)
(2.10)
b4 = 0, b5 = 0, b6 = 0.
he integrability condition for J1 gives
(2.11)
a2 =
c
a1 a′1 − c
, a3 = .
a1 − 2a′1 t
a1
Next, from (1.5), (1.6), the Theorem 3 and the integrability conditions for J2 , we get
c1 = 0, c2 = 0, c3 = 0, c4 = −a1 , c5 = −c , c6 = a1 a′1 −c
a1
a1 −2a′1 t ,
(2.12)
′
c−a
a
1
1
1
d1 = 0, d2 = 0, d3 = 0, d4 = , d5 =
, d6 = a1 (a2c+2ct).
a1
a1 (a2 −2ct)
1
1
Finally, in the case where (T M, G, J1 , J2 ) is hyper-K¨ahler, we have
2
p1 = 2ka1 , p2 = 2 ck , p3 = 2 ck , p4 = 0,
a1 −2ct
a1 −2ct
a1 −2ct
(2.13)
′
2
2
q1 = 2 k , q1 + 2q2 t = k(a1 −2a21 t) (a31 +2ct) , q3 = 4 −ck2 2 , q4 = 0.
a −2ct
(a −2ct)
a −4c t
1
1
1
118
Vasile Oproiu
Hence we may state
Theorem 6. In the case where a4 = 0, the almost hyper-Hermitian structure
(G, J1 , J2 ) defined on the tangent bundle T M becomes hyper-K¨
ahlerian if the conditions (2.10)-(2.13) are fulfilled.
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Author’s address:
V.Oproiu
Faculty of Mathematics, University ”Al.I.Cuza”, Ia¸si, Romania.
Institute of Mathematics ”O.Mayer”,
Ia¸si Branch of the Romanian Academy, Romˆania.
E-mail: voproiu@uaic.ro
K¨ahler manifold
Vasile Oproiu
Dedicated to the 70-th anniversary
of Professor Constantin Udriste
Abstract. We construct a family of almost hyper-complex structures on
the tangent bundle of a K¨ahlerian manifold by using two anti-commuting
almost complex structures obtained from the natural lifts of the Riemannian metric (see [11], [12], [13], [18]) and the integrable almost complex structure on the base manifold. Next we obtain an almost hyperHermitian metric obtained from the same natural lifts, related to the considered almost complex structures. We study the integrability conditions
for the almost complex structures, obtaining that the base manifold must
have constant holomorphic sectional curvature, and the conditions under
which the considered almost hyper-Hermitian metric leads to a hyperK¨ahlerian structure on the tangent bundle.
M.S.C. 2000: 53C55, 53C15, 53C05.
Key words: tangent bundle; natural lifts; almost hyper-Hermitian structures; hyperK¨ahlerian structures.
1
Introduction
Consider an m(= 2n)-dimensional Riemannian manifold (M, g) and denote by τ :
T M −→ M its tangent bundle. Several Riemannian and semi-Riemannian metrics
can be used in order to obtain geometric properties of the tangent bundle T M of
(M, g). They are induced from the Riemannian metric g on M by using some lifts
of g. Among these metrics, we may quote the Sasaki metric and the complete lift of
the metric g. On the other hand, the natural lifts of g to T M , induce some other
Riemannian and pseudo-Riemannian geometric structures with many nice geometric
properties (see [8], [7]). By similar methods one can get from g some natural almost
complex structures on T M . If (M, g) has a structure of K¨ahlerian manifold we can
find some other Riemannian metrics and almost complex structures on its tangent
bundle and from them we can get some almost hyper-Hermitian structures (see also
[19], [20]). Similar results are obtained in the case of the cotangent bundle (see e.g.
[3]).
∗
Balkan
Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 104-119.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
Hyper-K¨ahler structures on the tangent bundle
105
In the present paper we study a class of natural almost hyper-Hermitian structures
(G, J1 , J2 ), on the tangent bundle T M of a K¨ahlerian manifold (M, g, J), induced from
the Riemannian metric g and the integrable almost complex structure J. The metric
G and the anti-commuting almost complex structures J1 , J2 are obtained as natural
lifts of diagonal type from g and J.
The manifolds, tensor fields and other geometric objects we consider in this paper
are assumed to be differentiable of class C ∞ (i.e. smooth). We use the computations
in local coordinates in a fixed local chart but many results may be expressed in
an invariant form by using the vertical and horizontal lifts. Some quite complicate
computations have been made by using the Ricci package under Mathematica for
doing tensor computations. The well known summation convention is used throughout
this paper, the range of the indices h, i, j, k, l being always {1, . . . , m = 2n}.
1. Hyper-complex structures on T M .
Let (M, g) be a smooth m = (2n)-dimensional Riemannian manifold and denote
its tangent bundle by τ : T M −→ M . Recall that there is a structure of a smooth 2mdimensional manifold on T M , induced from the structure of smooth m-dimensional
manifold of M . From every local chart (U, ϕ) = (U, x1 , . . . , xm ) on M , it is induced
a local chart (τ −1 (U ), Φ) = (τ −1 (U ), x1 , . . . , xm , y 1 , . . . , y m ), on T M , as follows. For
a tangent vector y ∈ τ −1 (U ) ⊂ T M , the first m local coordinates x1 , . . . , xm are the
local coordinates x1 , . . . , xm of its base point x = τ (y) in the local chart (U, ϕ) (in fact
we made an abuse of notation, identifying xi with τ ∗ xi = xi ◦ τ, i = 1, . . . , m). The
last m local coordinates y 1 , . . . , y m of y ∈ τ −1 (U ) are the vector space coordinates of y
∂
∂
with respect to the natural basis (( ∂x
1 )τ (y) , . . . , ( ∂xm )τ (y) ), defined by the local chart
(U, ϕ). Due to this special structure of differentiable manifold for T M , it is possible
to introduce the concept of M -tensor field on it. An M -tensor field of type (p, q)
on T M is defined by sets of np+q components (functions depending on xi and y i ),
with p upper indices and q lower indices, assigned to induced local charts (τ −1 (U ), Φ)
on T M , such that the local coordinate change rule is that of the local coordinate
components of a tensor field of type (p, q) on the base manifold M , when a change
of local charts on M (and hence on T M ) is performed (see [10] for further details);
e.g., the components y i , i = 1, . . . , m, corresponding to the last m local coordinates
of a tangent vector y, assigned to the induced local chart (τ −1 (U ), Φ) define an M tensor field of type (1, 0) on T M . A usual tensor field of type (p, q) on M may be
thought of as an M -tensor field of type (p, q) on T M . If the considered tensor field
on M is covariant only, the corresponding M -tensor field on T M may be identified
with the induced (pullback by τ ) tensor field on T M . Some useful M -tensor fields on
T M may be obtained as follows. Let u : [0, ∞) −→ R be a smooth function and let
kyk2 = gτ (y) (y, y) be the square of the norm of the tangent vector y ∈ τ −1 (U ). If δji
are the Kronecker symbols (in fact, they are the local coordinate components of the
identity tensor field I on M ), then the components u(kyk2 )δji define an M -tensor field
of type (1, 1) on T M . Similarly, if gij (x) are the local coordinate components of the
metric tensor field g on M in the local chart (U, ϕ), then the components u(kyk2 )gij
define a symmetric M -tensor field of type (0, 2) on T M . The components g0i = y k gki ,
as well as u(kyk2 )g0i define M -tensor fields of type (0, 1) on T M . Of course, all the
components considered above are in the induced local chart (τ −1 (U ), Φ).
106
Vasile Oproiu
We shall use the horizontal distribution HT M , defined by the Levi Civita connec˙ of g, in order to define some first order natural lifts to T M of the Riemannian
tion ∇
metric g on M . Denote by V T M = Ker τ∗ ⊂ T T M the vertical distribution on T M .
Then we have the direct sum decomposition
(1.1)
T T M = V T M ⊕ HT M.
If (τ −1 (U ), Φ) = (τ −1 (U ), x1 , . . . , xm , y 1 , . . . y m ) is a local chart on T M , induced from
the local chart (U, ϕ) = (U, x1 , . . . , xm ), the local vector fields ∂y∂ 1 , . . . , ∂y∂m define a
local frame for V T M over τ −1 (U ) and the local vector fields δxδ 1 , . . . , δxδm define a
local frame for HT M over τ −1 (U ), where
δ
∂
∂
=
− Γh0i h ,
δxi
∂xi
∂y
Γh0i = y k Γhki
and Γhki (x) are the Christoffel symbols of g.
The set of vector fields ( ∂y∂ 1 , . . . , ∂y∂m , δxδ 1 , . . . , δxδm ) defines a local frame on T M ,
adapted to the direct sum decomposition (1.1). Remark that
∂
δ
∂
∂
= ( i )V ,
= ( i )H ,
∂y i
∂x
δxi
∂x
where X V and X H denote the vertical and horizontal lifts of the vector field X on
M.
Now assume that (M, g, J) is a K¨ahlerian manifold. The Riemannian metric g
and the integrable almost complex structure J are related by
˙ = 0,
g(JX, JY ) = g(X, Y ), ∇J
˙ is the Levi Civita connection of g. Recall that we have too the following
where ∇
relations
N = 0, dφ = 0,
where N is the Nijehuis tensor field of J and φ is the associated 2-form, defined by
φ(X, Y ) = g(X, JY ).
Denote by gij , Jji the components of g, J in the local chart (U, ϕ) = (U, x1 , . . . , xm ).
Introduce the components Jij = gih Jjh , obtained from the components of J by lowering the contravariance index on the first place (in fact, Jij are the components of
the fundamental 2-form φ defined by the K¨ahlerian structure (g, J). Consider the
following M -tensor fields on τ −1 (U ), defined by the components
gi0 = gih y h , Ji0 = Jih y h = −J0i .
Lemma 1. If m > 1 and u1 , u2 , u3 , u4 , u5 , u6 are smooth functions on T M such
that
u1 gij + u2 gi0 gj0 + u3 Ji0 Jj0 + u4 gi0 Jj0 + u5 Ji0 gj0 + u6 Jij = 0, y ∈ τ −1 (U )
107
Hyper-K¨ahler structures on the tangent bundle
on the domain of any induced local chart on T M , then u1 = u2 = u3 = u4 = u5 =
u6 = 0.
The proof is obtained easily by transvecting the given relation with g ij , J ij =
= Jhj y h and y j (Recall that the functions g ij (x) are the components of the
inverse of the matrix (gij (x)), associated to g in the local chart (U, ϕ) on M ; moreover,
the components g ij (x) define a tensor field of type (2, 0) on M ).
Jhi g hj , J0j
Remark. From a relation of the type
u1 δji + u2 y i gj0 + u3 J0i Jj0 + u4 y i Jj0 + u5 J0i gj0 + u6 Jji = 0, y ∈ τ −1 (U )
it is obtained, in a similar way, u1 = u2 = u3 = u4 = u5 = u6 = 0.
Since we work in a fixed local chart (U, ϕ) on M and in the corresponding induced
local chart (τ −1 (U ), Φ) on T M , we shall use the following simpler notations
∂
δ
= ∂i ,
= δi .
i
∂y
δxi
Denote by
(1.2)
t=
1
1
1
kyk2 = gτ (y) (y, y) = gik (x)y i y k ,
2
2
2
y ∈ τ −1 (U )
the energy density defined by g in the tangent vector y. We have t ∈ [0, ∞) for
∂
V
all y ∈ T M . Let C = y i ∂y
be the Liouville vector field on T M and coni = y
e = y i δ i = y H on T M , obtained
sider the corresponding horizontal vector field C
δx
in a similar way. Consider the real valued smooth functions a1 , a2 , a3 , a4 , a5 , a6 , b1 ,
b2 , b3 , b4 , b5 , b6 , c1 , c2 , c3 , c4 , c5 , c6 , d1 , d2 , d3 , d4 , d5 , d6 defined on [0, ∞) ⊂ R and define two diagonal natural almost complex structures J1 , J2 on T M , by using these
coefficients, the Riemannian metric g and the integrable almost complex structure J
J1 XyH = a1 (t)XyV + a2 (t)gτ (y) (y, X)Cy + a3 (t)gτ (y) (Jy, X)(Jy)Vy +
+a4 (t)(JX)Vy + a5 (t)gτ (y) (X, y)(Jy)Vy + a6 (t)gτ (y) (Jy, X)Cy ,
(1.3)
ey + b3 (t)gτ (y) (Jy, X)(Jy)H
J1 XyV = −(b1 (t)XyH + b2 (t)gτ (y) (y, X)C
y +
H
e
+b4 (t)(JX)H
y + b5 (t)gτ (y) (X, y)(Jy)y + b6 (t)gτ (y) (Jy, X)Cy ),
(1.4)
J2 XyH = c1 (t)XyV + c2 (t)gτ (y) (y, X)Cy + c3 (t)gτ (y) (Jy, X)(Jy)Vy +
+c4 (t)(JX)Vy + c5 (t)gτ (y) (X, y)(Jy)Vy + c6 (t)gτ (y) (Jy, X)Cy ,
ey + d3 (t)gτ (y) (Jy, X)(Jy)H
J2 XyV = −(d1 (t)XyH + d2 (t)gτ (y) (y, X)C
y +
H
e
+d4 (t)(JX)H
y + d5 (t)gτ (y) (X, y)(Jy)y + d6 (t)gτ (y) (Jy, X)Cy ).
The expressions of J1 , J2 in adapted local frames are
J1 δi = J1 Hih ∂h , J1 ∂i = J1 Vih δh ,
J2 δi = J2 Hih ∂h , J2 ∂i = J2 Vih δh ,
108
Vasile Oproiu
where the M -tensor fields J1 Hih , J1 Vih , J2 Hih , J2 Vih are given by
J1 Hih = a1 δih + a2 gi0 y h + a3 Ji0 J0h + a4 Jih + a5 gi0 J0h + a6 Ji0 y h ,
J1 Vih = −(b1 δih + b2 gi0 y h + b3 Ji0 J0h + b4 Jih + b5 gi0 J0h + b6 Ji0 y h ),
J2 Hih = c1 δih + c2 gi0 y h + c3 Ji0 J0h + c4 Jih + c5 gi0 J0h + c6 Ji0 y h ,
J2 Vih = −(d1 δih + d2 gi0 y h + d3 Ji0 J0h + d4 Jih + d5 gi0 J0h + d6 Ji0 y h ).
The matrices associated to J1 , J2 have a diagonal form
µ
¶
µ
0
J1 Hih
0
J1 =
, J2 =
0
J1 Vih
J2 Vih
J2 Hih
0
¶
.
Remark that, one can consider the case of the general natural tensor fields J1 , J2 on
T M , when J1 δi , J1 ∂i , J2 δi , J2 ∂i are expressed as combinations of ∂h , δh . In this case
we should have 48 coefficients and the computations would become really complicate.
However, the results obtained in the general case do not differ too much from that
obtained in the diagonal case.
We use the following notation:
α = (a1 + 2a2 t)(a1 + 2a3 t) + (a4 + 2a5 t)(a4 − 2a6 t).
Proposition 2. The operator J1 defines an almost complex structure on T M if and
only if the coefficients b1 , b2 , b3 , b4 , b5 , b6 are expressed as
1
4
b1 = a2a+a
b4 = a−a
2,
2 +a2 ,
1
4
1
4
1
b2 = α [b1 (−a1 a2 − 2a2 a3 t + 2a5 a6 t) + b4 (a1 a5 − a1 a6 − a3 a4 )],
b3 = α1 [b1 (−a1 a3 − 2a2 a3 t + 2a5 a6 t) + b4 (a1 a5 − a1 a6 − a2 a4 )],
(1.5)
b5 = α1 [b1 (−a1 a5 + a2 a4 + a3 a4 ) + b4 (a4 a6 − 2a2 a3 t + 2a5 a6 t)],
b6 = α1 [b1 (−a1 a6 − a2 a4 − a3 a4 ) + b4 (a4 a5 + 2a2 a3 t − 2a5 a6 t)].
Proof. The relations are obtained by some quite straightforward but long computations, from the property J12 = −I of J1 and Lemma 1.
Remark. Using the first two relations (1.5) we may find the expressions of
b2 , b3 , b5 , b6 as functions of a1 , a2 , a3 , a4 , a5 , a6 only. Remark that the parameters
a1 , a4 cannot vanish simultaneously and that α 6= 0. A similar result is obtained from
the condition for J2 to be an almost complex structure on T M . In this case we can
express the coefficients d1 , d2 , d3 , d4 , d5 , d6 as functions of c1 , c2 , c3 , c4 , c5 , c6 . We shall
use the following notation:
β = (c1 + 2c2 t)(c1 + 2c3 t) + (c4 + 2c5 t)(c4 − 2c6 t).
Then we get
d1
d2
d3
(1.6)
d5
d
6
=
c1
,
c21 +c24
=
1
β [d1 (−c1 c2
− 2c2 c3 t + 2c5 c6 t) + d4 (c1 c5 − c1 c6 − c3 c4 )],
=
1
β [d1 (−c1 c3
− 2c2 c3 t + 2c5 c6 t) + d4 (c1 c5 − c1 c6 − c2 c4 )],
=
1
β [d1 (−c1 c5
+ c2 c4 + c3 c4 ) + d4 (c4 c6 − 2c2 c3 t + 2c5 c6 t)],
=
1
β [d1 (−c1 c6
− c2 c4 − c3 c4 ) + d4 (c4 c5 + 2c2 c3 t − 2c5 c6 t)].
d4 =
−c4
,
c21 +c24
Hyper-K¨ahler structures on the tangent bundle
109
Now we shall study the conditions under which the almost complex structures J1 , J2
satisfy the relation J1 J2 + J2 J1 = 0, leading to the almost hyper-complex structure
on T M .
Theorem 3. The almost complex structures J1 , J2 define an almost hyper-complex
structure on T M if
(1.7)
c1 = a4 , c4 = −a1 ,
c3 = (a21 a5 + a24 a5 − a21 a6 − a24 a6 − a21 c2 − a24 c2 + 2a2 a3 a4 t − 2a1 a2 a6 t−
−2a1 a3 a6 t − 4a4 a5 a6 t + 2a4 a26 t − 2a1 a3 c2 t + 2a4 a6 c2 t − 2a2 a4 c6 t − 2a3 a4 c6 t+
+4a1 a5 c6 t − 2a1 c2 c6 t + 2a4 c26 t − 4a2 a3 a6 t2 + 4a5 a26 t2 − 4a3 c2 c6 t2 + 4a5 c26 t2 )/
((a1 + 2a2 t)(a1 + 2c6 t) + (a4 + 2c2 t)(a4 − 2a6 t)),
c5 =
−1
a1 +2c6 t (a1 a2
+ a1 a3 + a4 a5 − a4 a6 − a4 c2 − a4 c3 − a1 c6 +
+2a2 a3 t − 2a5 a6 t − 2c2 c3 t).
Proof. From the relation J1 Vhk J2 Hih + J2 Vhk J1 Hih = 0 we get
(−b1 c1 + b4 c4 − a1 d1 + a4 d4 )δik − (b4 c1 + b1 c4 + a4 d1 + a1 d4 )Jik −
−(b5 c1 + b4 c2 + b3 c4 + b1 c5 + a5 d1 + a4 d3 + a2 d4 + a1 d5 +
+2b5 c2 t + 2b3 c5 t + 2a5 d3 t + 2a2 d5 t)gi0 J0k +
(−b3 c1 − b1 c3 + b5 c4 − b4 c6 − a3 d1 − a1 d3 + a6 d4 + a4 d5 −
−2b3 c3 t − 2b5 c6 t − 2a3 d3 t − 2a6 d5 t)Ji0 J0k +
−b2 c1 − b1 c2 − b6 c4 + b4 c5 − a2 d1 − a1 d2 + a5 d4 − a4 d6 −
−2b2 c2 t − 2b6 c5 t − 2a2 d2 t − 2a5 d6 t)gi0 y k −
(b6 c1 − b4 c3 − b2 c4 + b1 c6 + a6 d1 − a4 d2 − a3 d4 + a1 d6 +
+2b6 c3 t + 2b2 c6 t + 2a6 d2 t + 2a3 d6 t)Ji0 y k .
Replacing bα , dα ; α = 1, . . . , 6 and using Lemma 1 we get the following relations
(from the vanishing of the first two coefficients)
(a1 c1 + a4 c4 )(a21 + a24 + c21 + c24 ) = 0,
(a4 c1 − a1 c4 )(a21 + a24 − c21 − c24 ) = 0.
Since a21 + a24 6= 0, c21 + c24 6= 0, we obtain the relations
c1 = ±a4 , c4 = ∓a1 .
From now on we shall consider only the case c1 = a4 , c4 = −a1 . The expressions
of c3 , c5 are obtained from the vanishing of the next 4 coefficients. Then the other
relations obtained from J1 J2 + J2 J1 = 0 are identically fulfilled.
Remark that the final expression of c5 is obtained after replacing the obtained
expression of c3 .
110
Vasile Oproiu
Hence an almost hyper-complex structure on T M , of the considered type depends
on 8 essential parameters a1 , a2 , a3 , a4 , a5 , a6 , c2 , c6 (real valued smooth functions depending on the density energy t ∈ [0, ∞). Remark that the functions aα ; α = 1, . . . , 6,
must fulfill some supplementary conditions which assure the existence of the expressions obtained above.
Now we shall study the integrability problem for the obtained almost hypercomplex structure. The integrability conditions for such a structure are expressed with
the help of various Nijenhuis tensor fields obtained from the tensor fields J1 , J2 , J3 =
J1 J2 . For a tensor field K of type (1, 1) on a given manifold, we can consider its
Nijenhuis tensor field NK defined by
NK (X, Y ) = [KX, KY ] − K[X, KY ] − K[KX, Y ] + K 2 [X, Y ],
where X, Y are vector fields on the given manifold. For two tensor fields K, L of type
(1, 1) on the given manifold, we can consider the corresponding Nijenhuis tensor field
NK,L defined by
NK,L (X, Y ) = [KX, LY ] + [LX, KY ] − K([X, LY ] + [LX, Y ])−
−L([KX, Y ] + [X, KY ]) + (KL + LK)[X, Y ].
The almost hyper-complex structure defined by J1 , J2 is integrable iff N1 =
0, N2 = 0, where N1 , N2 are the Nijenhuis tensor fields of J1 , J2 . Equivalently,
the structure is integrable iff N1 + N2 + N3 = 0, or iff N12 = 0, where N3 is the
Nijenhuis tensor field of J3 = J1 J2 and N12 = NJ1 ,J2 is the Nijenhuis tensor field of
J1 , J2 .
In the case of the almost hyper-complex structure defined on T M by the tensor
fields J1 , J2 the most convenient way to study its integrability is the using of the
Nijenhuis tensor fields N1 , N2 .
Proposition 4. If the almost hyper-complex structure defined by (J1 , J2 ) on T M
is integrable then the K¨
ahlerian manifold (M, g, J) has constant holomorphic sectional
curvature.
Proof. Recall the following formulas, useful in computing the expressions of N1 , N2
k
∂k ,
[∂i , ∂j ] = 0, [∂i , δj ] = −Γkij ∂k , [δi , δj ] = −R0ij
δi y h = −Γhi0 , δi gjk = Γhij ghk + Γhik gjh , δi gj0 = g0h Γhij ,
δi Jlk = −Γkih Jlh + Γhil Jhk , δi J0k = −Γkih J0h , δi Jj0 = Γhij Jh0 .
We have used the notations
k
k
, Γki0 = y h Γkih , gj0 = gjh y h , J0k = Jhk y h , Jj0 = Jjh y h .
R0ij
= y h Rhij
Then we get
h
)∂h .
N1 (δi , δj ) = (J1 Hik ∂k J1 Hjh − J1 Hjk ∂k J1 Hih + R0ij
Remark that all the terms containing the Christoffel symbols cancel. Doing the necessary replacements, we get a relation of the following type
α1 (Jih gj0 − Jjh gi0 ) + α2 (g0i δjh − g0j δih ) + 2α3 Jij y h + α4 (Ji0 Jjh − Jj0 Jih ) + 2α5 Jij J0h +
111
Hyper-K¨ahler structures on the tangent bundle
h
+α6 (δih Jj0 − δjh Ji0 ) + R0ij
+ α7 (gi0 Jj0 − gj0 Ji0 )y h + α8 (gi0 Jj0 − gj0 Ji0 )J0h = 0,
where the coefficients α1 , α2 , α3 , α4 , α5 , α6 , α7 , α8 , are functions of t, expressed with
the help of the coefficients a1 , a2 , a3 , a4 , a5 , a6 and their derivatives.
Differentiating this relation with respect to y k , then taking y = 0, one gets
α1 (0)(Jih gjk − Jjh gik ) + α2 (0)(gki δjh − gkj δih ) + 2α3 (0)Jij δkh + α4 (0)(Jik Jjh − Jjk Jih )+
h
= 0.
+2α5 (0)Jij Jkh + α6 (0)(δih Jjk − δjh Jik ) + Rkij
Then, using the well known (skew) symmetries of the components of R, as well as the
(first) Bianchi identity and the invariance properties of R with respect to J, one finds
(1.8)
h
Rkij
= c(gjk δih − gik δjh + Jih gkl Jjl − Jjh gkl Jil + 2Jkh gil Jjl ),
i.e. the K¨ahlerian manifold (M, g, J) has constant holomorphic sectional curvature
4c.
h
Replacing the obtained expression of Rkij
in the relation N1 (δi , δj ) = 0, and using
Lemma 1, one obtains some further relations
(1.9)
a3 =
c − a4 a5
,
a1
a6 = −
a2 a4
.
a1
Then, replacing these expressions of a3 , a6 in the remaining terms one gets
(1.10)
a2 =
a21
a1 (a1 a′1 + a4 a′4 − c)
−a2 a4 + a1 a′4 + 2a2 a′4 t
.
, a5 =
2
′
′
+ a4 − 2a1 a1 t − 2a4 a4 t
a1
Next, computing the expressions N1 (∂i , ∂j ), N1 (δi , ∂j ), we get that they are identically
zero.
Similar results are obtained from the integrability conditions for J2 , but we should
prefer to present some other expressions (we shall assume that c4 6= 0)
(1.11)
c2 = − c1c4c6 ,
c3 =
c5 =
c−c1 c3
c4 ,
c1 c6 +c1′ c4 −2c′1 c6 t
,
c4
c6 =
c4 (c−c1 c′1 −c4 c′4 )
.
c21 +c24 −2c1 c′1 t−2c4 c′4 t
Finally, by using the relations obtained in Theorem 3, one gets the expressions of
c2 , c3 , c5 , c6 as functions a1 , a4 an their derivatives
(1.12)
c2 =
c5 =
a (c−a1 a′1 )+a′4 (a21 −2ct)
a4 (a1 a′1 +a4 a′4 −c)
, c3 = a42 +a2 −2a
,
′
′
a21 +a24 −2a1 a′1 t−2a4 a′4 t
1 a1 t−2a4 a4 t
4
1
a1 (a4 a′4 −c)−a′1 (a24 −2ct)
a1 (a1 a′1 +a4 a′4 −c)
, c6 = a2 +a2 −2a1 a′ t−2a4 a′ t
a21 +a24 −2a1 a′1 t−2a4 a′4 t
1
4
1
4
Remark that the values of c3 , c5 obtained in (1.12) do coincide with the values of
c3 , c5 obtained in Theorem 3 after replacing c2 , c6 obtained in (1.12) and a2 , a3 , a5 , a6
obtained in (1.9) and (1.10).
112
2
Vasile Oproiu
Hyper-K¨
ahler structures on T M
Consider a natural Riemannian metric G on T M of diagonal type induced from g and
J and given by
(2.1)
Gy (X H , Y H ) = p1 (t)gτ (y) (X, Y ) + p2 (t)gτ (y) (y, X)gτ (y) (y, Y )+
+p3 (t)gτ (y) (JX, y)gτ (y) (JY, y) + p4 (t)(gτ (y) (JX, y)gτ (y) (Y, y)+
+g
τ (y) (JY, y)gτ (y) (X, y)),
Gy (X V , Y V ) = q1 (t)gτ (y) (X, Y ) + q2 (t)gτ (y) (y, X)gτ (y) (y, Y )+
+q3 (t)gτ (y) (JX, y)gτ (y) (JY, y) + q4 (t)(gτ (y) (JX, y)gτ (y) (Y, y)+
+gτ (y) (JY, y)gτ (y) (X, y)),
Gy (X H , Y V ) = Gy (Y V , X H ) = Gy (X V , Y H ) = Gy (Y H , X V ) = 0,
where p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 are smooth real valued functions defined on [0, ∞).
Remark that we have to find the conditions under which G is real Riemannian metric.
The expression of G in local adapted frames is defined by the following M -tensor
fields
Gij = G(δi , δj ) = p1 gij + p2 g0i g0j + p3 Ji0 Jj0 + p4 (gi0 Jj0 + gj0 Ji0 ),
Hij = G(∂i , ∂j ) = q1 gij + q2 g0i g0j + q3 Ji0 Jj0 + q4 (gi0 Jj0 + gj0 Ji0 )
and the associated 2m × 2m-matrix with respect to the adapted local frame
µ
δ
δ
∂
∂
,..., m, 1,..., m
δx1
δx ∂y
∂y
¶
has two m × m-blocks on the first diagonal
G=
µ
Gij
0
0
Hij
¶
.
We shall be interested in the conditions under which the metric G is almost Hermitian with respect to the almost complex structures J1 , J2 , considered in the previous
section, i.e.
G(J1 X, J1 Y ) = G(X, Y ), G(J2 X, J2 Y ) = G(X, Y ),
for all vector fields X, Y on T M .
From the relation
G(J1 δi , J1 δj ) = G(δi , δj ),
we get
(2.2)
Hkl J1 Hik J1 Hjl = Gij ,
Hyper-K¨ahler structures on the tangent bundle
113
from which we obtain the following expressions for p1 , p2 , p3 , p4
p1 = (a21 + a24 )q1 ,
p2 = (2a1 a2 + 2a4 a5 + 2a22 t + 2a25 t)q1 +
(a1 + 2a2 t)2 q2 + (a4 + 2a5 t)2 q3 + 2(a1 + 2a2 t)(a4 + 2a5 t)q4 ,
p3 = (2a1 a3 − 2a4 a6 + 2a23 t + 2a26 t)q1 +
(2.3)
(a4 − 2a6 t)2 q2 + (a1 + 2a3 t)2 q3 − 2(a1 + 2a3 t)(a4 − 2a6 t)q4 ,
p4 = (−a2 a4 + a3 a4 + a1 a5 + a1 a6 + 2a3 a5 t + 2a2 a6 t)q1 +
+(−a1 a4 − 2a2 a4 t + 2a1 a6 t + 2a1 a6 t)q2 + (a1 + 2a3 t)(a4 + 2a5 t)q3 +
+(a21 − a24 + 2a1 a2 t + 2a1 a3 t − 2a4 a5 t + 2a4 a6 t + 4a2 a3 t2 + 4a5 a6 t2 )q4 .
Remark that from the conditions G(J1 ∂i , J1 ∂j ) = G(∂i , ∂j ), G(J1 δi , J1 ∂j ) =
G(δi , ∂j ), we do not obtain new essential relations fulfilled by p′ s and q ′ s.
Now we deal with the condition
G(J2 δi , J2 δj ) = G(δi , δj ),
from which we get
(2.4)
Hkl J2 Hik J2 Hjl = Gij .
We find the coefficients p1 , p2 , p3 , p4 expressed in function of the coefficients q1 , q2 , q3 , q4
by formulas similar to (2.3), where the parameters a1 , a2 , a4 , a5 , a6 are replaced by
c1 , c2 , c3 , c4 , c5 , c6 respectively. Next, we may write the system fulfilled by q1 , q2 , q3 , q4 ,
obtained by equalizing the obtained values for p1 , p2 , p3 , p4 . Remark that, due to the
formula (1.7), the first equation, corresponding to p1 , is trivial. So, we get a homogeneous system consisting of 3 equations
q1 (−2a1 a2 − 2a4 a5 + 2a4 c2 − 2a1 c5 − 2a22 t − 2a25 t + 2c22 t + 2c25 t)+
+q2 (−a21 + a24 − 4a1 a2 t + 4a4 c2 t − 4a22 t2 + 4c22 t2 )+
+q3 (a21 − a24 − 4a4 a5 t − 4a1 c5 t − 4a25 t2 + 4c25 t2 )+
+q4 (−4a1 a4 − 4a2 a4 t − 4a1 a5 t − 4a1 c2 t + 4a4 c5 t − 8a2 a5 t2 + 8c2 c5 t2 ) = 0,
q1 (−a2 a4 + a3 a4 + a1 a5 + a1 a6 − a1 c2 + a1 c3 − a4 c5 − a4 c6 + 2a3 a5 t + 2a2 a6 t − 2c3 c5 t−
−2c2 c6 t) + q2 (−2a1 a4 − 2a2 a4 t + 2a1 a6 t − 2a1 c2 t − 2a4 c6 t + 4a2 a6 t2 − 4c2 c6 t2 )+
+q3 (2a1 a4 + 2a3 a4 t + 2a1 a5 t + 2a1 c3 t − 2a4 c5 t + 4a3 a5 t2 − 4c3 c5 t2 ) + q4 (2a21 − 2a24 +
+2a1 a2 t + 2a1 a3 t − 2a4 a5 t + 2a4 a6 t − 2a4 c2 t − 2a4 c3 t − 2a1 c5 t + 2a1 c6 t + 4a2 a3 t2 +
+4a5 a6 t2 − 4c2 c3 t2 − 4c5 c6 t2 ) = 0,
q1 (2a1 a3 − 2a4 a6 − 2a4 c3 − 2a1 c6 + 2a23 t + 2a26 t − 2c23 t − 2c26 t)+
+q2 (−a21 + a24 − 4a4 a6 t − 4a1 c6 t + 4a26 t2 − 4c26 t2 )+
+q3(a21 − a24 + 4a1 a3 t − 4a4 c3 t + 4a23 t2 − 4c23 t2 )+
+q4 (−4a1 a4 − 4a3 a4 t + 4a1 a6 t − 4a1 c3 t − 4a4 c6 t + 8a3 a6 t2 − 8c3 c6 t2 ) = 0.
The matrix of this system has the rank 2 and we may obtain its general solution
depending on two parameters
(2.5)
q1 = λ,
q3 = µ,
114
Vasile Oproiu
q4 = ((a3 a4 − a1 a5 + a1 c2 − a4 c6 + 2a3 c2 t − 2a5 c6 t)λ+
+(2a3 a4 t − 2a1 a5 t + 2a1 c2 t − 2a4 c6 t + 4a3 c2 t2 − 4a5 c6 t2 )µ)/
(a21 + a24 + 2a1 a2 t − 2a4 a6 t + 2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 ),
q1 + 2tq2 =
= ((a41 + 2a21 a24 + a44 + 4a31 a2 t + 4a13 a3 t + 4a1 a2 a24 t + 4a1 a3 a24 t + 4a21 a4 a5 t+
+4a34 a5 t − 4a21 a4 a6 t − 4a34 a6 t + 4a21 a22 t2 + 16a21 a2 a3 t2 + 4a21 a23 t2 + 8a2 a3 a24 t2 +
+4a23 a24 t2 + 8a1 a2 a4 a5 t2 + 4a21 a25 t2 + 4a24 a25 t2 − 8a1 a2 a4 a6 t2 − 8a1 a3 a4 a6 t2 −
−8a21 a5 a6 t2 − 16a24 a5 a6 t2 + 4a24 a26 t2 + 8a1 a3 a4 c2 t2 − 8a21 a5 c2 t2 + 4a21 c22 t2 −
−8a3 a24 c6 t2 + 8a1 a4 a5 c6 t2 − 8a1 a4 c2 c6 t2 + 4a24 c26 t2 + 16a1 a22 a3 t3 + 16a1 a2 a23 t3 +
+16a2 a3 a4 a5 t3 − 16a2 a3 a4 a6 t3 − 16a1 a2 a5 a6 t3 − 16a1 a3 a5 a6 t3 − 16a4 a25 a6 t3 +
+16a4 a5 a26 t3 + 16a23 a4 c2 t3 − 16a1 a3 a5 c2 t3 + 16a1 a3 c22 t3 − 16a3 a4 a5 c6 t3 +
+16a1 a25 c6 t3 − 16a3 a4 c2 c6 t3 − 16a1 a5 c2 c6 t3 + 16a4 a5 c26 t3 + 16a22 a23 t4 −
−32a2 a3 a5 a6 t4 + 16a25 a26 t4 + 16a23 c22 t4 − 32a3 a5 c2 c6 t4 + 16a25 c26 t4 )(λ + 2tµ))/
(a21 + a24 + 2a1 a2 t − 2a4 a6 t + 2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 .
The explicit expression of q2 is obtained from the expression of q1 + 2tq2 and is more
complicate. Next, the expressions of p1 , p2 , p3 , p4 are obtained from (2.3).
p1 = (a21 + a24 )λ,
p1 + 2tp2 = (a21 + a24 + 2a1 a2 t + 2a1 a3 t + 2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2
(a21 + a24 + 4a1 a2 t + 4a4 c2 t + 4a22 t2 + 4c22 t2 )(λ + 2tµ)/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 ,
p1 + 2tp3 = (a21 + a24 + 2a1 a2 t + 2a1 a3 t + 2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2
(a21 + a24 − 4a4 a6 t + 4a1 c6 t + 4a26 t2 + 4c26 t2 )(λ + 2tµ)/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 ,
p4 = (−a2 a4 + a1 a6 + a1 c2 + a4 c6 + 2a2 a6 t + 2c2 c6 t)(a21 + a24 + 2a1 a2 t + 2a1 a3 t+
+2a4 a5 t − 2a4 a6 t + 4a2 a3 t2 − 4a5 a6 t2 )2 (λ + 2tµ))/(a21 + a24 + 2a1 a2 t − 2a4 a6 t+
+2a4 c2 t + 2a1 c6 t − 4a6 c2 t2 + 4a2 c6 t2 )2 .
If we assume that the almost hyper-complex structure defined by J1 , J2 is integrable,
the expressions of the coefficients in G are simpler.
For the almost hyper-Hermitian manifold (T M, G, J1 , J2 ) the fundamental 2-forms
φ1 , φ2 are defined by
φ1 (X, Y ) = G(X, J1 Y ), φ2 (X, Y ) = G(X, J2 Y ),
where X, Y are vector fields on T M .
Hyper-K¨ahler structures on the tangent bundle
115
Since we have a third almost complex structure J3 = J1 J2 which is almost Hermitian with respect to G, we can consider a third 2-form φ3 defined by φ3 (X, Y ) =
G(X, J3 Y ), next we have the fundamental 4 form Ω, defined by
Ω = φ1 ∧ φ1 + φ2 ∧ φ2 + φ3 ∧ φ3 .
The almost hyper-Hermitian manifold (T M, G, J1 , J2 ) is hyper-K¨ahlerian if the almost complex structures J1 , J2 are parallel with respect to the Levi Civita connection
∇ defined by G, i.e. ∇J1 = 0, ∇J2 = 0. Equivalently, (T M, G, J1 , J2 ) is hyperK¨ahlerian if and only if the almost hyper-complex structure (J1 , J2 ) is integrable and
the 4-form Ω is closed, i.e. N1 = 0, N2 = 0, dΩ = 0. The condition for Ω to be
closed is equivalent to the conditions for φ1 , φ2 (and hence for φ3 too) to be closed i.e.
dφ1 = 0, dφ2 = 0. In our case, it is more convenient to study the conditions under
which the 2-forms φ1 , φ2 are closed.
The expressions of φ1 , φ2 in adapted local frames are
φ1 = φ1,jk Dy j ∧ dxk , φ2 = φ2,jk Dy j ∧ dxk ,
where
Dy j = dy j + Γji0 dxi ,
φ1,jk = G(∂j , J1 δk ) = Hjh J1 Hkh = −J1 Vjh Ghk ,
φ2,jk = G(∂j , J2 δk ) = Hjh J2 Hkh = −J2 Vjh Ghk .
Replacing the expressions of Hjh and J1 Hkh , J2 Hkh , we find the following expressions
φ1,jk = a1 q1 gjk + a4 q1 Jjk + (a2 q1 + a1 q2 + a4 q4 + 2a2 q2 t + 2a5 q4 t)gj0 gk0 +
+(a6 q1 − a4 q2 + a1 q4 + 2a6 q2 t + 2a3 q4 t)gj0 Jk0 +
+(a5 q1 + a4 q3 + a1 q4 + 2a5 q3 t + 2a2 q4 t)Jj0 gk0 +
+(−a3 q1 − a1 q3 + a4 q4 − 2a3 q3 t − 2a6 q4 t)Jj0 Jk0 ,
φ2,jk = a4 q1 gjk − a1 q1 Jjk + (c2 q1 + c4 q2 − a1 4q4 + 2c2 q2 t + 2c5 q4 t)gj0 gk0 +
+(c6 q1 + a1 q2 + a4 q4 + 2c6 q2 t + 2c3 q4 t)gj0 Jk0 +
+(c5 q1 − a1 q3 + a4 q4 + 2c5 q3 t + 2c2 q4 t)Jj0 gk0 +
+(−c3 q1 − a4 q3 − a1 q4 − 2c3 q3 t − 2c6 q4 t)Jj0 Jk0 .
The final expressions of φ1 , φ2 are obtained by replacing the values of q1 , q2 , q3 , q4
obtained from (2.5), then the values of c3 , c5 , obtained in Theorem 3. We get for
φ1,jk an expression of the type
φ1,jk = α1 gjk + α2 Jjk + α3 Jjk + α4 J0j g0k + α5 g0j J0k + α6 J0j J0k ,
where α1 , α2 , α3 , α4 , α5 , α6 are functions of t, expressed with the help of the coefficients a1 , a2 , a3 , a4 , a5 , a6 , c2 , c6 . A similar expression is obtained for φ2,jk .
Now we shall compute the expression of dφ1 by using the following formulas
dαr = αr′ g0i Dy i , r = 1, 2, 3, 4, 5, 6,
dgjk = (Γhij ghk + Γhik gjh )dxi ,
116
Vasile Oproiu
dg0j = dgj0 = gji Dy i + g0h Γhji dxi , dJjk = (Γhij Jhk + Γhik Jjh )dxi ,
1 h
dDy h = Γhij Dy i ∧ dxj + R0ij
dxi ∧ dxj .
2
We get the cancellation of all terms containing dxi ∧ Dy j ∧ dxk in the expression
of dφ1 . Next the terms containing dxi ∧ dxj ∧ dxk are
dJj0 = −dJ0j = Jh0 Γhji dxi + Jji Dy i ,
1
h
h
h
(φ1,hk R0ij
+ φ1,hi R0jk
+ φ1,hj R0ki
)dxi ∧ dxj ∧ dxk .
6
From the vanishing of this term and under the assumption that the base manifold
(M, g, J) has constant holomorphic sectional curvature 4c, we get that the factors λ, µ
are related by
µ 2
¶
a1 + a24
(2.6)
λ=−
+ 2t µ.
c
Finally, assuming that this relation as, well as the integrability conditions for the
almost hyper-complex structure defined by J1 , J2 are fulfilled, we get that α5 = α6 = 0
and the expression of dφ1 becomes
dφ1 = (α1′ g0i gjk + α2′ g0i Jjk + α3 g0j gki + α4 g0j Jik )Dy i ∧ Dy j ∧ dxk ,
where the coefficients α1 , α2 , α3 , α4 are given by
α1 = a1 λ, α2 = a4 λ,
(2.7)
α3 =
λ(−a21 a′1 + a′1 a24 − 2a1 a4 a′4 + 2a1 c − 2a′1 ct)
,
a21 + a24 − 2ct
α4 =
λ(−2a1 a′1 a4 + a21 a′4 − a′4 a24 + 2a4 c − 2a′4 ct)
.
a21 + a24 − 2ct
Doing the necessary alternation in the relation dφ1 = 0, we get the equations
α1′ = α3 , α2′ = α4 .
Then, after some simple computations, we obtain that the coefficients λ, µ are given
by
k
−ck
λ= 2
, µ= 2
,
a1 + a24 − 2ct
(a1 + a24 )2 − 4c2 t2
where k is a constant.
Under the same assumptions that the relation (2.6) is true and that the integrability conditions for the almost hyper-complex structure defined by J1 , J2 are fulfilled,
we get the following expression expression for the 2-form φ2
φ2 = (α2 gjk − α1 Jjk + α4 g0j g0k − α3 g0j J0k )Dy j ∧ dxk .
If dφ1 = 0 we have that dφ2 = 0 too, so that the structure (G, J1 , J2 ) on T M becomes
K¨ahlerian. We write down the explicit expressions of the coefficients involved in the
expressions of (G, J1 , J2 ). First of all, from the integrability condition N1 = 0, we get
(2.8)
a2 =
a21
a1 (a1 a′1 + a4 a′4 − c)
a′ a2 − a1 a4 a′4 + a1 c − 2a′1 ct
, a3 = 1 2 4 2
,
2
′
′
+ a4 − 2a1 a1 t − 2a4 a4 t
a1 + a4 − 2a1 a′1 t − 2a4 a′4 t
117
Hyper-K¨ahler structures on the tangent bundle
a5 =
−a4 (a1 a′1 + a4 a′4 − c)
−a1 a′1 a4 + a21 a′4 + a4 c − 2a′4 ct
, a6 = 2
.
2
2
′
′
a1 + a4 − 2a1 a1 t − 2a4 a4 t
a1 + a24 − 2a1 a′1 t − 2a4 a′4 t
The values of c2 , c3 , c5 , c6 are given by (1.12). Next we have
(2.9)
q1 = λ =
p1 =
ck
k(a21 + a24 )
, p 2 = p3 = 2
, p4 = 0,
+ a24 − 2ct
a1 + a24 − 2ct
a21
k
−ck
k(a′1 a4 − a1 a′4 )
,
q
=
µ
=
,
q
=
.
3
4
a21 + a24 − 2ct
(a21 + a24 )2 − 4c2 t2
(a21 + a24 )2 − 4c2 t2
The expression of q2 is quite complicate and can be obtained from (2.5). Hence we
may state
Theorem 5. Consider the almost hyper-Hermitian structure (G, J1 , J2 ) defined
as above on the tangent bundle T M of the K¨
ahlerian manifold M . This structure
is hyper-K¨
ahlerian if and only if the almost complex structures J1 , J2 are integrable
(hence the Proposition 4 and the relations (1.9), (1.10), (1.11), (1.12) are fulfilled)
and and the relations (2.6), (2.7), (2.8), (2.9) are fulfilled by the hyper-Hermitian
metric G.
The case where a4 = 0
We shall study a special case when a4 = 0. In this case we shall obtain some much
more simple formulas and results and many of them are related to those obtained in
[19], [20]. However, our parametrization is quite different.
Since the integrability conditions for the almost complex structure J1 we get that
the condition a4 = 0 implies the conditions a5 = 0, a6 = 0. We are interested in
the integrable case, so that we shall assume from the beginning a4 = a5 = a6 = 0.
According to the result obtained in Proposition 2, the tensor field J1 defines almost
complex structure on T M if and only if
½
2
3
, b3 = − a1 (a1a+2a
,
b1 = a11 , b2 = − a1 (a1a+2a
2 t)
3 t)
(2.10)
b4 = 0, b5 = 0, b6 = 0.
he integrability condition for J1 gives
(2.11)
a2 =
c
a1 a′1 − c
, a3 = .
a1 − 2a′1 t
a1
Next, from (1.5), (1.6), the Theorem 3 and the integrability conditions for J2 , we get
c1 = 0, c2 = 0, c3 = 0, c4 = −a1 , c5 = −c , c6 = a1 a′1 −c
a1
a1 −2a′1 t ,
(2.12)
′
c−a
a
1
1
1
d1 = 0, d2 = 0, d3 = 0, d4 = , d5 =
, d6 = a1 (a2c+2ct).
a1
a1 (a2 −2ct)
1
1
Finally, in the case where (T M, G, J1 , J2 ) is hyper-K¨ahler, we have
2
p1 = 2ka1 , p2 = 2 ck , p3 = 2 ck , p4 = 0,
a1 −2ct
a1 −2ct
a1 −2ct
(2.13)
′
2
2
q1 = 2 k , q1 + 2q2 t = k(a1 −2a21 t) (a31 +2ct) , q3 = 4 −ck2 2 , q4 = 0.
a −2ct
(a −2ct)
a −4c t
1
1
1
118
Vasile Oproiu
Hence we may state
Theorem 6. In the case where a4 = 0, the almost hyper-Hermitian structure
(G, J1 , J2 ) defined on the tangent bundle T M becomes hyper-K¨
ahlerian if the conditions (2.10)-(2.13) are fulfilled.
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Author’s address:
V.Oproiu
Faculty of Mathematics, University ”Al.I.Cuza”, Ia¸si, Romania.
Institute of Mathematics ”O.Mayer”,
Ia¸si Branch of the Romanian Academy, Romˆania.
E-mail: voproiu@uaic.ro