Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 733

senting a quaternionic CR-structure from [2] in which we have introduced an integrable

nondegenerate quaternionic Carnot-Caratheodory (Q C-C) structure B on a (4n + 3)-

manifold M more generally rather than the global case In sect2 we define and study

the basic properties of (para) quaternionic CR-manifolds The real 1-forms ωαα=123

define a codimension 3-subbundle H of TM on which there exists a quaternionic or re-

spectively para-quaternionic structure Jγγ=123 In sect3 we describe relations between

quaternionic CR structure contact pseudo - (Riemannian) metric 3-structure pseudo-K-

contact 3-structures and pseudo-Sasakian 3-structures We associate with a quaternionic

CR structure three (integrable ) CR structures Jα α = 1 2 3 and a pseudo-Riemannian

metric g1 We prove that these objects are consistent and define a pseudo-Sasakian 3-

structure In particular the metric g1 is an Einstein metric More generally we prove

that quaternionic CR structures are equivalent to the pseudo-Sasakian 3-structure As

by-product we get the following result

Theorem 33 A contact pseudo-metric 3-structure is automatically a pseudo-Sasakian

3-structure of type (3 + 4p 4q)

This is a generalization of the results obtained by Tanno [17] Jelonek [11] and Ka-

shiwada [13] The key point is a generalization of the Hitchin Lemma (cf Proposition

25) which implies the integrability of the almost complex structures Jα (α = 1 2 3) on

corresponding codimension 1-contact subbundle

In sect4 we indicate another proof of the result that the natural metric associated

with a (para)-quaternionic CR structure is Einstein (cf Theorem 41) It is based on

the OrsquoNeillrsquos formulas for Riemannian submersion with totally geodesic fibers In sect5 we

assume that the action of 3-dimensional Lie algebra of vector fields associated with a

(para)-quaternionic CR manifold can be integrated to an almost free action of the corre-

sponding Lie group H1 We prove that in the case of proper free action the orbit space

MH1 carries a (para)-quaternionic Kahler geometry We gave some homogeneous exam-

ples of manifolds with (para)-quaternionic CR structure in sect6 In sect7 we show that the

reduction method works smoothly for manifolds with a (para)-quaternionic CR structure

and allows us to construct non homogeneous (para)-quaternionic CR manifolds starting

from (para)-quaternionic CR manifolds with a proper group of symmetries In sect8 we

consider ε-hyperKahler manifolds

We fix some notations and give a unified descriptions of two algebras of quaternions

We denote by Hε the associative 4-dimensional algebra over R with the standard basis

1 i1 = i i2 = j i3 = k and multiplication

i2 = minus1 j2 = k2 = ε jk = minuskj = minusεi

where ε = plusmn1

Then i j k anti-commute and ij = k ki = j For ε = minus1 we get the algebra H = Hminus1

of quaternions and for ε = +1 the algebra Hprime = H+1 is the split-quaternions We shall

refer to this last algebra also as algebra of paraquaternions The algebra Hε admits a

multiplicative ldquonormrdquo |x| =radic

(xx) such that |xy| = |x||y| where x = x0 minusx1iminusx2j minusx3k

734 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

is the quaternionic conjugation of x = x0 +x1i+x2j +x3k associated with the Hermitian

scalar product lt x y gt= xy which is positively defined for ε = minus1 and has split signature

(2 2) in the para case when ε = +1 The group H1ε of unit quaternions is isomorphic

to Sp1 for ε = minus1 and SO021 = Sl2(R) for ε = 1 (Here 0 indicates the connected

component of the group) The left multiplication defines an irreducible representation of

H1ε in Hε = R4 and the adjoint action Da x 7rarr axaminus1 = axa gives the 3-dimensional

representation of H1ε in the space ImHε = R

3 of imaginary quaternions Note that the

linear group DH1ε

is the connected group of automorphisms of Hε The Lie algebra h1ε of

H1ε is identified with the space ImHε of imaginary quaternions with respect to the Lie

bracket [x y] = xy minus yx The commutator relations with respect to the standard basis

i j k are

[i j] = 2k [j k] = minus2εi [k i] = 2j

Let π M rarr B is a principal bundle with the structure group H1ε = Sp1 or Sl2(R) A

connection in π is defined by a horizontal h1ε-valued 1-form ω on M which is an extension

of the natural vertical parallelism ienatural identification of vertical spaces T vx M with

h1ε With respect to the standard basis i j k of h1

ε the form ω = ω1i + ω2j + ω3k splits

into a triple (ω1 ω2 ω3) of scalar forms Similarly the curvature form ρ = dω + [ω and ω]

splits into a triple of scalar 2-forms (ρ1 ρ2 ρ3) where

ρ1 = dω1 minus 2εω2 and ω3 ρ2 = dω2 + 2ω3 and ω1 ρ3 = dω3 + 2ω1 and ω2

Assume that the curvature forms ρα α = 1 2 3 are non-degenerate on the horizontal

distribution H = Ker ω Then we can construct three fields Jα of endomorphisms of H

setting

J1 = minusε(ρ3|H)minus1 ρ2|H J2 = (ρ1|H)minus1 ρ3|H J3 = (ρ2|H)minus1 ρ1|H

We are interested in the case when these three endomorphisms Jα define a representation

of the quaternion algebra Hε that is they anti-commute and satisfy the quaternionic

relations

J21 = minusεJ2

2 = minusεJ23 = minus1 J2J3 = minusεJ1

We will show that a connection ω which satisfies these algebraic conditions for the cu-

rvature ρ is closely related with contact metric 3-structure (in particular Sasakian 3

structures) and determines Einstein metrics In the next section we give a more gene-

ral definition of such structure in terms of three 1-forms ωα which satisfy the structure

equations

2 Quaternionic and para-quaternionic CR structure definition

and the main properties

Let ω = (ω1 ω2 ω3) be a triple of scalar 1-forms on a (4n + 3)-dimensional manifold

M which are linearly independent ie ω1 and ω2 and ω3 6= 0 We associate with ω a triple

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 735

ρ = (ρ1 ρ2 ρ3) where

ρ1 = dω1 minus 2εω2 and ω3 ρ2 = dω2 + 2ω3 and ω1 ρ3 = dω3 + 2ω1 and ω2

and ε = +1 or minus1

Definition 21 A triple of linearly independent 1-forms ω = (ω1 ω2 ω3) is called a ε-

quaternionic CR structure (ε = +1 or minus1) if the associated 2-forms ρα α = 1 2 3 satisfy

the following conditions

(1) They are non degenerate on the codimension three distribution H = Ker ω1capKerω2cap

Ker ω3 and have the same 3-dimensional kernel V

(2) The three fields of endomorphisms Jα of the distribution H defined by

J1 = minusε(ρ3|H)minus1 ρ2|H J2 = (ρ1|H)minus1 ρ3|H J3 = (ρ2|H)minus1 ρ1|H

anti-commute and satisfy the ε-quaternionic relations

J21 = minusεJ2

2 = minusεJ23 = minus1 J2J3 = minusεJ1

For ε = minus1 the ε-quaternionic CR structure is called also a quaternionic CR struc-

ture and ε = +1 quaternionic CR structure is called also para-quaternionic CR struc-

ture The manifold M with an ε-quaternionic CR structure is called ε-quaternionic

CR manifold

We will see that there exists a big similarity between quaternionic and para-quaternionic

CR manifolds

The distribution H is called the horizontal distribution and V the vertical distribution

of ε-quaternionic CR manifold It follows from the definition that TM = V oplus H Using

this direct sum decomposition of the tangent bundle we define an one-parameter family

of pseudo-Riemannian metrics gt t isin R+ on a ε-quaternionic CR manifold M by

gt = gtV + gH (1)

where

gtV = t(ω1 otimes ω1 minus εω2 otimes ω2 minus εω3 otimes ω3)

= tsum

εαωα otimes ωα

(2)

gH = ρ1 J1 = ρ2 J2 = minusερ3 J3

and ε1 = 1 ε2 = ε3 = minusε Note that 1-forms ωα|x α = 1 2 3 at a point x isin M form

a basis of the dual space V lowastx sub T lowast

x M We denote by ξα|x the dual basis of Vx Then

ξα α = 1 2 3 are vertical vector fields such that ωβ(ξα) = δαβ We have

gt ξ1 = tω1 gt ξβ = minustεωβ for β = 2 3 (3)

We will denote by LX the Lie derivative with respect to a vector field X

736 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

Lemma 22 (1) The vector fields ξα preserve the decomposition TM = V oplusH and span

a 3-dimensional Lie algebra hε of Killing fields of the metric gt for t gt 0 which is

isomorphic to sp(1 R) for ε = 1 and sp(1) for ε = minus1 More precisely the following

commutator relations hold

[ξ1 ξ2] = 2ξ3 [ξ2 ξ3] = minus2εξ1 [ξ3 ξ1] = 2ξ2

(2) The vector field ξα preserves the forms ωα and ρα for α = 1 2 3 Moreover the

following relations hold

Lξ2ω3 = minusLξ3ω2 = ω1 Lξ3ω1 = εLξ1ω3 = minusεω2 Lξ1ω2 = εLξ2ω1 = ω3

and similar relations for ρα

Proof Using the formula LX = dιX + ιXd for the Lie derivative we have for any hori-

zontal vector field h isin Γ(H) and a vector field ξ isin ξ1 ξ2 ξ3

(Lξωα)(h) = ((dιξ + ιξd)ωα)(h)

= dωα(ξ h) = (ρα minus 2δαωβ and ωγ)(ξ h) = 0

(δ1 = minusε δ2 = δ3 = 1) Here (α β γ) is a cyclic permutation of (1 2 3) This shows that

ξα preserves the horizontal distribution H Now we prove that ξα preserves the vertical

distribution V Using the fact that ξα preserves the horizontal distribution we get for any

h isin Γ(H)

ρα([ξβ ξγ] h) = dωα([ξβ ξγ] h) + 2δαωβ and ωγ([ξβ ξγ] h)

= minus1

2ωα([[ξβ ξγ] h])

=1

2(ωα([[ξγ v] ξβ]) + ωα([[ξβ h] ξγ])) (by Jacobi identity)

= 0

This shows that [ξα ξβ] is a linear combination of ξ1 ξ2 ξ3 To determine the coefficients

we calculate for ε = minus1

0 = ρα(ξβ ξγ) = dωα(ξβ ξγ) + 2ωβ and ωγ(ξβ ξγ)

=1

2(minusωα([ξβ ξγ]) + 2)

that is ωα([ξβ ξγ]) = 2 Similarly we can check that ωα([ξα ξβ]) = 0 This proves the

relations of [ξα ξβ] = 2δγξγ The relations of (2) follow now immediately Since ξα

preserves V and ωα it preserves also the vertical part gVt of the metric gt due to the

equation (3) One can easily check that ξα preserves the field Jα considered as a field

of endomorphisms of TM which is zero on V This implies that ξα preserves also the

horizontal part gH = plusmnρα Jα of the metric gt

Using the Koszul formulas for the covariant derivative nablat of the metric gt we get

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 737

Corollary 23 (1) nablatξα

ξβ = minusnablatξα

ξβ = 12[ξα ξβ] for α β isin 1 2 3 that is

nablatξα

ξα = 0 nablatξ1

ξ2 = ξ3 nablatξ2

ξ3 = minusεξ1 nablatξ3

ξ1 = ξ2

(2)

nablatξ1

ω2 = ω3 nablatξ2

ω3 = minusεω1 nablatξ3

ω1 = ω2

(3)

nablatξα

ρα = 0

nablatξ1

ρ2 = minusnablatξ2

ρ1 = ρ3 nablatξ2

ρ3 = minusnablatξ3

ρ2 = minusερ1 nablatξ3

ρ1 = minusnablatξ1

ρ3 = ρ2

Proof (1) follows from the Koszul formulas (2) follows from (1) since gtξα = minusεδαtωα

and nablat preserves gt (δ1 = minusε δ2 = δ3 = 1) Now we calculate for example nablatξ1

ρ2 as

follows

nablatξ1

ρ2 = nablatξ1

(dω2 + 2ω3 and ω1) = dω3 + 2nablaξ1ω3 and ω1 = dω3 minus 2ω2 and ω1 = ρ3

The proof of other identities is similar

Now we extend endomorphisms Jα α = 1 2 3 of the horizontal distribution H to

endomorphisms Jα of the all tangent bundle by the following conditions

Jαξα = 0 Jα|H = Jα

J1ξ2 = minusεξ3 J1ξ3 = εξ2

J2ξ3 = ξ1 J2ξ1 = εξ3

J3ξ1 = ξ2 J3ξ2 = εξ1

(4)

Note that the endomorphisms Jα α = 1 2 3 at a point x constitute the standard basis

of the Lie algebra h1ε sub End(TxM) Using Lemma 22 and Corollary 23 we can prove

Lemma 24 The vector filed ξα preserves the field of endomorphism Jα for α = 1 2 3

and the Lie derivatives of Jα with respect to ξβ are given by

Lξ1 J2 = minusLξ2 J1 = minusεJ3 Lξ2J3 = εLξ3J2 = minusJ1 Lξ3 J1 = εLξ1J3 = J2

The following proposition shows that the restriction of the field of endomorphisms

Jα α = 1 2 3 to the (non-holonomic) codimension one distribution Tα = Kerωα are

integrable This means that the Nijenhuis tensor N(Jα Jα)Tα= 0 or equivalently the

eigendistributions Tplusmnα of Jα|Tα

are involutive We remark that J1 J2 J3 are gt-skew sym-

metric anticommuting endomorphisms with one-dimensional kernel Tperpα Moreover in the

case ε = minus1 Jα|Tαis a complex structure in Tα and in the case ε = 1 J1|T1

is a complex

structure and J2|T2and J3|T3

are involutive endomorphisms (iehas square +1)

Proposition 25 Let (M ω1 ω2 ω3) be a ε-quaternionic CR manifold Then the above

defined field of endomorphisms J1|T1 J2|T2

J3|T3are integrable

738 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

Proof First of all note the following formula (cf [15])

LX(ιY dωα) = ι(LXY )dωα + ιY LXdωα

= ι[XY ]dωα + ιY LXdωα (forallX Y isin TM)(5)

Secondly we remark that if X isin H then

ιXdω2 = minusειJ1Xdω3

ιXdω3 = ιJ2Xdω1

ιXdω1 = ιJ3Xdω2

(6)

Then the proof is based on the following Lemma which is a generalization of Lemma by

Hitchin

Lemma 26 (i) Let T1 otimes C = T+1 + Tminus

1 be the eigenspace decomposition of the com-

plexified distribution T1 otimes C = Kerω1 otimes C with respect to the endomorphism J1 with

eigenvalues +i minusi Then

ι[XY ]dω2 = minusεiι[XY ]dω3 forallX Y isin T+1

(ii) For ε = 1 let Tα = T+α + Tminus

α be the eigenspace decomposition of Tα α = 2 3 with

respect to the endomorphism Jα with eigenvalues +1 and minus1 Then

ι[XY ]dω3 = ι[XY ]dω1 forallX Y isin T+2

ι[XY ]dω2 = ι[XY ]dω1 forall X Y isin T+3

Proof We prove (i) Let X isin T+1 such that J1X = iX Then

LXdω2 = (dιX + ιXd)dω2 = d(ιXdω2)

= minusεd(ιJ1Xdω3) (by (6))

= minusεi(dιX)dω3

= minusεi(LX minus ιXd)dω3 = minusεiLXdω3

(7)

Applying Y isin T+1 to the equation (5) we get

LX(ιY dω2) = LX(minusειJ1Y dω3) = minusεiLX(ιY dω3)

= minusεi(ι[XY ]dω3 + ιY LXdω3) ((5))

= minusεiι[XY ]dω3 + ιY LXdω2 ((7))

(8)

Since LX(ιY dω2) = ι[XY ]dω2 + ιY LXdω2 by (5) comparing this with (8) we obtain

minusεiι[XY ]dω3 = ι[XY ]dω2

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 739

3 Quaternionic CR structure and contact pseudo-metric

3-structure

We will show here that the quaternionic CR structure (that is (ε = minus1)-quaternionic

CR structure) is equivalent to the contact (pseudo-Riemannian) metric 3-structure and

moreover any (pseudo-Riemannian) metric 3-structure is in fact Sasakian 3-structure

The last statement is a generalization of results obtained by Tanno [17] [18] Jelonek

[11] and Kashiwada [13] We recall the classical definitions of contact metric 3-structure

normal contact metric structure and 3-Sasakian structure

Definition 31 (Tanno [17][18] Blair [5])A contact pseudo-metric 3-structure

g (ηα ξα φα) α = 1 2 3 on a (4n + 3)-manifold M consists of a pseudo-Riemannian

metric g of signature (3 + 4p 4q) p + q = n contact forms ηα the dual vector fields

ξα = gminus1(ηα) and endomorphisms φα which satisfy the following conditions

(1) g(ξα ξβ) = δαβ

(2) φ2α(X) = minusX + ηα(X)ξα φα(ξα) = 0 dηα(X Y ) = g(X φαY )

(3) g(φαX φαY ) = g(X Y ) minus ηα(X)ηα(Y )

(4) φα = φβφγ minus ξβ otimes ηγ = minusφγφβ + ξγ otimes ηβ where (α β γ) is a cyclic permutation of

(1 2 3)

A contact pseudo-metric 3-structure g (ηα ξα φα) is called a pseudo-K-contact

3-structure if

(5) the vector fields ξα are Killing fields with respect to g

A pseudo-K-contact 3-structure is called pseudo-Sasakian 3-structure if

(7) it is normal ie if the following tensors Nηα(middot middot) (α = 1 2 3) vanish

Nηα(X Y ) = Nφα(X Y ) + (Xηα(Y ) minus Y ηα(X))ξα (9)

(forall X Y isin TM) Here

Nφα(X Y ) = [φαX φαY ] minus [X Y ] minus φα[φαX Y ] minus φα[X φαY ]

is the usual Nijenhuis tensor of a field of endomorphisms φα

Remark 32 (1) Some historical explanation may be needed When g is a Riemannian

metric (the case q = 0) the above set (g ηα ξα φα) is called a contact metric 3-structure

If in addition each ξα is Killing with respect to g it is called a K-contact 3-structure A

K-contact 3-structure with normality condition is called a Sasakian 3-structure Tanno

had proved that a K-contact 3-structure on 7-dimensional manifold is always a Sasakian

3-structure Later this result was generalized by Jelonek who proved that any pseudo-

Riemannian K-contact 3-structure in the case when it comes from a quaternionic Kahler

metric of positive or negative scalar curvature [11] is a pseudo-Sasakian 3-structure Re-

cently Kashiwada [13] has shown this result for contact (positive) metric 3-structures

(not necessarily K-contact) It is natural to ask whether this will be true also for any

contact pseudo-Riemannian metric 3-structures

740 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

The following theorem gives an affirmative answer on this question

Theorem 33 The following three structures on a (4n+3)-dimensional manifold M are

equivalent contact pseudo-metric 3-structures quaternionic CR structures and pseudo-

Sasakian 3-structures

Proof Since any pseudo-Sasakian 3-structure is a contact metric 3-structure we have

to prove that

i) if (g ηα ξα φα) is a contact pseudo-metric 3-structure then ωα = ηα is a quaternionic

CR structure and

ii) if (ωα) is a quaternionic CR structure then the structure (g = g1 ηα = ωα ξα φα = Jα)

defined by the equations (1) (3) (4) is a pseudo-Sasakian 3-structure

i) Let (g ηα ξα φα) be a contact pseudo-metric 3-structure We have to prove that

1-forms ωα = ηα satisfy conditions (1) (2) of Definition 21 It follows from the definition

that 2-forms dωα = dηα are non-degenerate on the codimension three distribution H =3cap

α=1Ker ηα and TM = ξ1 ξ2 ξ3 oplusH The conditions (2)(3) of Definition 31 show that

2-forms

ρα = dηα + 2ηβ and ηγ

have the kernel V = span(ξ1 ξ2 ξ3) This proves (1) To prove (2) it is sufficient to check

that Jα = (ργ|H)minus1 (ρβ|H) = φα|H From (2) of Definition 31 we have

ρα(X Y ) = dηα(X Y ) = g(X φaY ) for X Y isin H (10)

The left hand side is equal to ρβ(JγX Y ) The right hand side can be rewritten as

g(X φαY ) = g(φγX φγ(φαY )) (by (4) of Definition 31)

= g(φγX φβY ) = ρβ(φγX Y )(11)

Since ρα|H is non-degenerate we conclude that Jγ = φγ on H Since φα|H satisfies the

quaternionic relations this proves (2)

ii) Let now (ωα) be a quaternionic CR structure We have to check that the associated

structure (g = g1 ηα = ωα ξα φα = Jα) satisfy conditions (1)-(7) of Definition 31 The

conditions (1)-(5) follow directly from the definition of a quaternionic CR structure The

condition (6) is proved in (2) of Lemma 22 Now we check the normality condition

Nηα(X Y ) = Nφα(X Y ) + (Xηα(Y ) minus Y ηα(X))ξα = 0

for φα = Jα By Proposition 25 Nφα(X Y ) = 0 forallX Y isin Ker ηα = ξβ ξγ oplus H This

shows that Nηα = 0 on Ker ηα Since TM = ξα oplus Ker ηα it remains to check that

Nηα(ξα X) = 0 for a local vector field X isin Ker ηα We have

Nηα(ξα X) = minus([ξα X] + Jα[ξα JαX])

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 741

Since LξαJα = 0 by Lemma 24 we have

Nηα(ξα X) = minus([ξα X] + Jα[ξα JαX])

= minus(LξαX + JαLξα

JαX) = 0

Hence the normality condition Nηα = 0 holds

4 Einstein metric associated with ε-quaternionic CR structure

We have proved that (ε = minus1)-quaternionic CR structure ω = (ωα) on a (4n + 3)-

dimensional manifold M defines a pseudo-Sasakian 3-structure (g ηα ξα φα) where g =

g1 =sum

ωα otimes ωα + ρ1 J1 ηα = ωα φα = Jα It is known that the metric g of a Sasakian

3-structure is Einstein with the Einstein constant 2(2n + 1) Tanno [18] remarked that

this result remains true also for pseudo-Sasakian structure It is natural to expect that

the result can be generalized also for the metric g1 associated with para-quaternionic CR

structure The following theorem shows that this is true

Theorem 41 Let (M ω = (ωα)) be a ε-quaternionic CR manifold Then the metric

g = g1 is an Einstein metric

Proof Lemma 22 implies that the orbits of the Lie algebra h1ε (that is maximal integrable

submanifolds of the vertical distribution V ) are totally geodesic submanifolds of (M gt)

for t gt 0 To simplify the notations we will assume that h1ε consists of complete vector

fields Then it defines an isometric action of the Lie group H1ε with a discrete stabilizer

We will assume that the action of H1ε is proper Then the orbit space MH1

ε is an

orbifold Deleting the singular points we get a smooth fibration π Mreg rarr B sub MH1ε

which is a Riemannian submersion with respect to the induced metric on B For brevity

we will assume that π M rarr B = MH1ε is a Riemannian submersion (with totally

geodesic fibers) Then we can use OrsquoNeillrsquos formulas which relate the Ricci curvature rict

of (M gt) with the Ricci curvature rictV of the fiber and the Ricci curvature ricB of the

base manifold B Since OrsquoNeillrsquos formulas are purely local without loss of generality it

can be written as in [3]

rict(ξ ξprime) = rictV (ξ ξprime) + t2g(Aξ Aξprime)

rict(X Y ) = ricB(X Y ) minus 2tg(AX AY )

rict(X ξ) = tg(δA)X ξ)

for any vectors ξ ξprime isin Vx X Y isin Hx where

AXY =1

2[X Y ]v = (nablaXY )v

g(Aξ Aξprime) =sum

g(AXiξ AXi

ξprime) g(AX AY ) =sum

(AXξα AY ξα)

742 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

δA = minussum

(nablaXiA)Xi

Here nabla is the covariant derivative of the metric g1 Xi (respectively ξα) is an orthonormal

basis of Hx (respectively Vx ) for x isin M and Xv stands for the vertical part of a vector

X

To apply these formulas we calculate the Nomuzu operator Ltξα

= minusnablatξα isin End(TM)

associated with the Killing field ξα where nablat is the covariant derivative of the metric gt

Lemma 42

Ltξ1|H = minusnablatξ1 = tJ1 Lt

ξ2|H= tJ2 Lt

ξ3|H = minusnablatξ1 = minusεJ3

Ltξ1|V = J1 |V Lt

ξ2|V = J2 |V Lt

ξ3|V = minusεJ3 |V

In particular for t = 1

Lξα= L1

ξα= εαJα (α = 1 2 3 ε1 = 1 ε2 = ε3 = minusε)

Proof We recall the following Koszul formula for the covariant derivative

2g(nablaXY Z) = g([X Y ] Z) minus g([X Z] Y ) minus g(X [Y Z])

+ X middot g(Y Z) + Y middot g(X Z) minus Z middot g(X Y )(12)

where X Y Z are vector fields on a Riemannian manifold (M g) Applying this formula

to the metric gt for Y = ξα and horizontal vector fields Y Z and using the formula

gt ξα = tεαωα where ε1 = 1 ε2 = ε3 = minusε we get

2gt(nablaXξα Z) = minusgt(ξα [X Z]) = minustεαωα([X Z])

= 2εαdωα(X Z) = 2εαρα(X Z)(13)

Now the result follows from the identities

gH = ρ1 J1 = ρ2 J2 = minusερ3 J3

Corollary 43 For X Y isin H the following formulas hold

(i)

g1(AXY ξα) = g1(LξαX Y ) = g1(JαX Y )

(ii)

g1(AX AX) = 3g1(X X) g1(Aξα Aξβ) = 4nεαδαβ

(iii)

g1((δA)X ξα) = 0

Proof (i)

D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753 743

g1(AXY ξα) = g1(nablaXY ξα) = minusg1(YnablaXξα) = g1(JαX Y )

(ii)

g1(AX AX) =sum

εα(Axξα AXξα) =sum

εαg1(JαX JαX) = 3g1(X X)

g1(Aξα Aξβ) =sum

g1(AXiξα AXi

ξβ) =sum

g1(JαXi JβXi) = 4nεαδαβ

The last equation can be checked using the formulas for the covariant derivatives of ρα

Lemma 44 The Ricci curvature of the restriction gtV of the metric gt to the fiber F is

given by rictV =

2

tgt

V

Proof We remark that the restriction gtV of the metric gt to the fiber F ≃ H1

ε is pro-

portional to the standard metric B on H1ε associated with Killing form more precisely

gtV = minust

1

8B It is known that the Ricci curvature of a bi-invariant metric on a semisimple

Lie group is equal to minus14B This implies Lemma

To finish the proof of Theorem we have to calculate the Ricci curvature ricB of the

base manifold B = MH1ε The answer is given in the following proposition which is

known in the case ε = minus1 [18] The proof in the case ε = 1 is similar

Proposition 45 The metric gB induces by the metric gt on the base manifold B =

MH1ε is Einstein with Einstein constant λ = 4(n + 2) that is

ricB = 4(n + 2)gB

Collecting all these results we conclude that the for any t gt 0 the Ricci operator of

the metric gt act as a scalar on the vertical space V and horizontal space H Moreover

for t = 1 it is a scalar operator and hence g1 is an Einstein metric This finishes the

proof of Theorem

5 ε-quaternionic CR manifolds and ε-quaternionic Kahler ma-

nifolds

Recall that a (pseudo-Riemannian) quaternionic Kahler manifold (respectively para-

quaternionic Kahler manifold) is defined as a 4n-dimensional pseudo-Riemannian ma-

nifold (M g) with the holonomy group H sub Sp(1)Sp(p q) (respectively H sub Sp(1 R)

Sp(n R)) This means that the manifold M admits a parallel 3-dimensional subbundle

Q (quaternionic subbundle) of the bundle of endomorphisms which is locally generated

by three skew-symmetric endomorphisms J1 J2 J3 which satisfy the quaternionic rela-

tions (respectively para-quaternionic relations) To unify the notations we shall call a

quaternionic Kahler manifold also a (ε = minus1)-quaternionic Kahler manifold and a para-

quaternionic Kahler manifold a (ε = 1)-quaternionic Kahler manifold

744 D Alekseevsky Y Kamishima Central European Journal of Mathematics 2(5) 2004 732ndash753

It is known that any ε-quaternionic Kahler manifold is Einstein and its curvature

tensor has the form R = νR1 + W where ν is a constant (proportional to the scalar

curvature) R1 is the curvature tensor of the canonical metric on the (para) quaternionic

projective space and W is a quaternionic Weyl tensor such that curvature operators

W (X Y ) commute with Q

Let (M ω) be a ε-quaternionic CR manifold We will assume that the Lie algebra

h1ε = span(ξα) of vector fields is complete and generates a free action of the group H1

ε

(or its quotient by a discrete central subgroup) on M Then the orbit space B = MH1ε

is a smooth manifold and π M rarr B is a principal bundle Moreover the pseudo-

Riemannian metric g1 of (M ω) induces a pseudo-Riemannian metric gB on B such that

π M rarr B is a Riemannian submersion with totally geodesic fibers

We will give a sketch of the proof of the following theorem

Theorem 51 Let (M ω) be a ε-quaternionic CR manifold ε = plusmn1 Assume that the

Lie algebra h1ε generates a free action of the corresponding Lie group H1

ε on M Then the

orbit space B = MH1ε has the natural structure of ε-quaternionic Kahler manifold Co-

nversely let (B gB Q) be a ε-quaternionic Kahler manifold and SB the Sasakian bundle

of frames (J1 J2 J3) of the quaternionic bundle Q which satisfy ε-quaternionic relations

Then SB has the natural structure of a ε-quaternionic CR manifold

Proof Let (M ω) be a ε-quaternionic CR manifold and π M rarr B the projection on

the orbit space The projection π induces an isometry πlowast of a horizontal space Hx at any

point x isin M onto the tangent space Tπ(x)B The images πlowast Jα of the endomorphisms

Jα α = 1 2 3 define a quaternionic subbundle Q of the bundle of endomorphisms of TB

It remains to check that this bundle is parallel with respect to the Levi-Civita connection

of the induces metric gB In the case when ε = minus1 and the metric g1 on M is positively

defined (hence defines a Sasakian 3-structure ) this is a classical result by Ishihara [8]

His approach can be generalized to the case of pseudo-Sasakian 3-structure and also to

the case ε = 1 see [2] Here we indicate another approach which is based on Swann

characterization of a quaternionic Kahler manifolds in terms of fundamental 4-form It

works when n gt 2

Swann [16] proved that a Riemannian manifold (M4n g) n gt 2 with a quaternionic

subbundle Q (ie a three-dimensional subbundle of End(TM) locally generated by three

skew-symmetric almost complex structures (J1 J2 J3) which satisfy the quaternionic re-

lations) is a quaternionic Kahler manifold if the globally defined fundamental 4-form

Ω =sum

g Jα and g Jα is closed This result remains valid for pseudo-Riemannian case

and can be generalized to the case of para-quaternionic Kahler manifolds

We can apply this result and finish the proof of the first part of the theorem as follows

Consider 4-form Ω =sum

ρα and ρα on M One can easily check that it is H1ε -invariant and

horizontal Hence it is a pull-back of a 4-form ΩB on B It is obvious that ΩB is the

fundamental 4-form of the quaternionic subbundle Q It remains to check that the form