On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

Applied Mathematics, 2018, 9, 114-129 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385

On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric Yoshinobu Kamishima Department of Mathematics, Josai University, Saitama, Japan

How to cite this paper: Kamishima, Y. (2018) On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric. Applied Mathematics, 9, 114-129. https://doi.org/10.4236/am.2018.92008 Received: January 15, 2018 Accepted: February 20, 2018 Published: February 23, 2018

A CR-structure on a 2n + 1 -manifold gives a conformal class of Lorentz metrics on the Fefferman S 1 -bundle. This analogy is carried out to the quarternionic conformal 3-CR structure (a generalization of quaternionic CR-structure) on a 4n + 3 -manifold M. This structure produces a conformal class Let

Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access

Abstract

[g]

of a pseudo-Riemannian metric g of type

( PSp ( n + 1,1) , S ) 4 n +3

( 4n + 3,3)

on M × S 3 .

be the geometric model obtained from the pro-

jective boundary of the complete simply connected quaternionic hyperbolic manifold. We shall prove that M is locally modeled on if and only if

( M × S , [ g ]) 3

( PSp ( n + 1,1) , S ) 4 n +3

is conformally flat (i.e. the Weyl conformal cur-

vature tensor vanishes).

Keywords Conformal Structure, Quaternionic CR-Structure, G-Structure, Conformally Flat Structure, Weyl Tensor, Integrability, Uniformization, Transformation Groups

1. Introduction This paper concerns a geometric structure on

( 4n + 3) -manifolds which is re-

lated with CR-structure and also quaternionic CR-structure (cf. [1] [2]). Given a quaternionic CR-structure

{ωα }α =1,2,3

on a 4n + 3 -manifold M, we have proved in [3] that the associated endomorphism Jα on the 4n-bundle D

naturally extends to a complex structure Jα on kerωα . So we obtain 3 CR-structures on M. Taking into account this fact, we study the following

( 4n + 3) -manifolds globally. A hypercomplex 3 CR-structure on a ( 4n + 3) -manifold M consists of (po-

geometric structure on

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sitive definite) 3 pseudo-Hermitian structures tisfies that

{ωα , Jα }α =1,2,3

on M which sa-

3

1) D =  kerωα is a 4n-dimensional subbundle of TM such that α =1

D + [ D, D ] = TM .

(

)

−1

2) Each J γ coincides with the endomorphism d ωβ | D  ( d ωα | D ) : D → D ( (α , β , γ ) ~ (1, 2,3) ) such that { J1 , J 2 , J 3} constitutes a hypercomplex structure on D .

, J 3 } ) also a hypercomplex 3 CR-structure if it is represented by such pseudo-Hermitian structures on M. A quaternionic CRWe call the pair

( D, { J , J 1

2

structure is an example of our hypercomplex 3 CR-structure. As Sasakian 3structure is equivalent with quaternionic CR-structure, Sasakian 3-structure is also an example. Especially the 4n + 3 -dimensional standard sphere S 4 n+3 is a

( PSp ( n + 1,1) , S ) 4 n +3

hypercomplex 3 CR-manifold. The pair

is the spherical

homogeneous model of hypercomplex 3 CR-structure in the sense of Cartan geometry (cf. [4]). First we study the properties of hypercomplex 3 CR-structure. Next we introduce a quaternionic 3 CR-structure on M in a local manner. In fact, let D be a 4n-dimensional subbundle endowed with a quaternionic structure Q on a

( 4n + 3) -manifold

M. The pair

( D,Q )

is called quaternionic 3

CR-structure if the following conditions hold: TM ; 1) D + [ D, D ] = 2) M has an open cover {U i }i∈Λ each U i of which admits a hypercomplex 3 CR-structure ωα( i ) , Jα( i ) such that:

(

)

3

(i )

α =1,2,3

a) D | U i =  kerωα ; α =1

b) Each hypercomplex structure quaternionic structure Q on D .

{J ( ) , J ( ) , J ( )} i

1

i 2

i

3

i∈Λ

on D | U i generates a

A 4n + 3 -manifold equipped with this structure is said to be a quaternionic 3

CR-manifold. A typical example of a quaternionic 3 CR-manifold but not a hypercomplex 3 CR-manifold is a quaterninic Heisenberg nilmanifold. In this paper, we shall study an invariant for quaternionic 3 CR-structure on manifolds. Theorem A. Let

( 4n + 3 ) -

( M , {D, Q})

be a quaternionic 3 CR-manifold. There exists a pseudo-Riemannian metric g of type ( 4n + 3,3) on M × S 3 . Then the conformal class [ g ] is an invariant for quaternionic 3 CR-structure. As well as the spherical quaternionic 3 CR homogeneous manifold S 4 n+3 , we have the pseudo-Riemannian homogeneous manifold S 4 n+3 × S 3 which is a two-fold covering of the pseudo-Riemannian homogeneous manifold

(S

)

(

)

×2 S 3 , g 0 . The pair PSp ( n + 1,1) × SO ( 3) , S 4 n+3 ×2 S 3 is a subgeometry of conformally flat pseudo-Riemannian homogeneous geometry PO ( 4n + 4, 4 ) , S 4 n+3 ×2 S 3 where PSp ( n + 1,1) × SO ( 3) ≤ PO ( 4n + 4, 4 ) . Theorem B. A quaternionic 3 CR-manifold M is spherical (i.e. locally modeled on PSp ( n + 1,1) , S 4 n+3 ) if and only if the pseudo-Riemannian 4 n +3

(

)

(

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manifold ( M × S 3 , g ) is conformally flat, more precisely it is locally modeled on ( PSp ( n + 1,1) × SO ( 3) , S 4 n+3 ×2 S 3 ) . We have constructed a conformal invariant on ( 4n + 3) -dimensional pseudoconformal quaternionic CR manifolds in [3]. We think that the Weyl conformal curvature of our new pseudo-Riemannian metric obtained in Theorem A is theoretically the same as this invariant in view of Uniformization Theorem B. But we do not know whether they coincide. Section 2 is a review of previous results and to give some definition of our notion. In Section 3 we prove the conformal equivalence of our pseudo-Riemannian metrics and prove Theorem A. In Section 4 first we relate our spherical 3 CR-homogeneous model PSp ( n + 1,1) , S 4 n+3 and the conformally flat PSp n pseudo-Riemannian homogeneous model ( + 1,1) × SO ( 3) , S 4 n+3,3 . We

(

(

)

)

study properties of 3-dimensional lightlike groups with respect to the pseudo0 Riemannian metric g of type

( 4n + 3,3)

on S 4 n+3 × S 3 . We apply these

results to prove Theorem B.

2. Preliminaries Let Put

( M ,{ω , J } α

α α =1,2,3

(ωα , Jα ) = (ω , J )

)

be a (4n + 3)-dimensional hypercomplex 3 CR-manifold. ( M , {ω , J }) is a

for one of α’s. By the definition,

CR-manifold. Let C 2 n+2,0 ( M ) be the canonical bundle over M (i.e. the  -line = C ( M ) C 2 n+2,0 ( M ) − {0} / * bundle of complex ( 2n + 2, 0 ) -forms). Put p 1 → M . Compare [[5], Section 2.2]. which is a principal bundle: S → C ( M )  Fefferman [6] has shown that C ( M ) admits a Lorentz metric g for which the Lorentz isometries S1 induce a lightlike vector field. We recognize the following definition from pseudo-Riemannian geometry. Definition 1. In general if S 1 induces a lightlike vector field with respect to a Lorentz metric of a Lorentz manifold, then S 1 is said to be a lightlike group acting as Lorentz isometries. Similarly if each generator S 1 of S 3 is chosen to be a lightlike group, then we call S 3 also a lightlike group. We recall a construction of the Fefferman-Lorentz metric from [5] (cf. [6]). Let ξ be the Reeb vector field for (ω , J ) . The circle S1 generates the vector field T on C ( M ) . Define dt to be a 1-form on C ( M ) such that

dt = ( T ) 1, dt= (V ) 0

(

∀

)

V ∈ TM .

(2.1)

In [[5], (3.4) Proposition] J. Lee has shown that there exists a unique real

1-form σ on C ( M ) . The explicit form of σ is obtained from [[5], (5.1) Theorem] in this case: = σ

Here 1-forms

 i αβ 1  1 α Rω  .  dt + iωα − h dhαβ −   2n + 3  2 2 ( 2n + 2 ) 

{ω

β α , β

τ

}

are connection forms of ω such that

= d ω ihαβ ω α ∧ ω β , d ω α = ω β ∧ ωβα + ω ∧ τ α . DOI: 10.4236/am.2018.92008

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(2.2)

(2.3)

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The function R is the Webster scalar curvature on M. Note from (2.2) dσ =

 1  1 1 α Rd ω − dR ∧ ω  .  id ωα −  2n + 3  2 ( 2n + 2 ) 2 ( 2n + 2 ) 

(2.4)

Normalize dt so that we may assume σ ( T ) = 1 . Let σ  ω denote the symmetric 2-form defined by σ ⋅ ω + ω ⋅ σ . Since ω ( T ) = 0 , it follows σ  ω ( T, T ) = 0 . The Fefferman-Lorentz metric for (ω , J ) on C ( M ) is defined by

g ( X , Y ) σ  ω ( X , Y ) + dω ( JX , Y ) . =

(2.5)

Here T ( C ( M ) ) = T ⊕ ξ ⊕ kerω . Since ξ is the Reeb field, dω ( JX , ξ ) = 0 . As [ kerω , T ] = 0 , dω ( JX , T ) = 0 ∀ X ∈ ker ω . On the other hand, J ({T, ξ } ) = 0 by the definition. We have

(

)

g (ξ , T ) 1,= g ( T, T ) 0. =

(2.6)

Thus g becomes a Lorentz metric on C ( M ) in which S1 is a lightlike group. Theorem 2 ([5]). If ω ′ = uω , then g ′ = ug .

3. Hypercomplex 3 CR-Structure Our strategy is as follows: first we construct a pseudo-Riemannian metric locally on each neighborhood of M × S 3 by Condition I below and then sew these metrics on each intersection to get a globally defined pseudo-Riemannian metric on M × S 3 using Theorem 4. (See the proof of Theorem A.) Suppose that dimension

( M ,{ω , J }

( 4n + 3 ) .

α

α α =1,2,3

)

is a hypercomplex 3 CR-manifold of

Put ω =ω1i + ω2 j + ω3 k . It is an Im -valued 1-form

annihilating D . In general, there is no canonical choice of ω annihilating D . In [[3], Lemma 1.3] we observed that if ω ′ is another Im -valued 1-form annihilating D , then

ω ′ = λωλ

(3.1)

for some  -valued function λ on M. (Here λ is the quaternion conjugate.) If we put λ = ua for a positive function u and a ∈ Sp (1) , then ω ′ = uaω a

such that the map z  aza ( z ∈  ) represents a matrix function A∈ SO ( 3) . If { Jα′ }α =1,2,3 is a hypercomplex structure on D for ω ′ , then they are related

as

[ J1′ J 2′ J 3′ ] = [ J1 J 2 J 3 ] A . For each (ωα , Jα ) , we

obtain a unique real 1-form σ α on C ( M ) from

Section 2 (cf. (2.2)). First of all we construct a pseudo-Riemannian metric on

M × S 3 . In general C ( M ) is a nontrivial principal S1 -bundle. It is the trivial

bundle when we restrict to a neighborhood. So for our use we assume: Condition I. C ( M ) is trivial as bundle, i.e. C ( M= ) M × S1 .

We construct a 1-form σ α on M × S 3 (α = 1, 2,3) as follows. Let of S 3 respectively. Obtained Tα , Tβ , Tγ generate {eiθ } , {e jθ } , {e kθ } θ ∈ θ ∈ θ ∈ as in (2.2), we have σ α ’s on each C ( M= ) M × S 1 such that DOI: 10.4236/am.2018.92008

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= σ α ( Tα ) 1,= σ β ( Tβ ) 1,= σ γ ( Tγ ) 1. We then extend σ α to M × S 3 by setting

σ= 0 α ( Tβ ) σ= α ( Tγ )

(3.2)

Since Tβ , Tγ  = 2Tα on TS 3 , 1 dσ α ( Tβ , Tγ ) =− σ α Tβ , Tγ  =−1 =−2σ β ∧ σ γ ( Tβ , Tγ ) . Note that for any 2 p∈M ,

(

)

dσ α + 2σ β ∧ = σ γ 0 on { p} × S 3

( (α , β , γ ) ~ (1, 2,3) ) .

(3.3)

On the other hand, we recall the following from [[3], Lemma 4.1]. Proposition 3. The following hold:

d= ω1 ( J1 X , Y ) d= ω2 ( J 2 X , Y ) dω3 ( J 3 X , Y ) D In particular g = d ωα  Jα bilinear form on D ;

(

∀

)

X ,Y ∈ D .

is a positive definite invariant symmetric

g D ( X , Y ) = g D ( Jα X , Jα Y ) . Choose a frame field { X 1 , , X 4 n } on D such that Jα X j = X α n+ j j ( = 1, , n ) with dωα ( Jα X j , X k ) = δ jk . Let θ i be the dual frame to X i ( i = 1, , 4n ) such that 4n

d ωα ( Jα X , Y ) = ∑θ i ( X ) ⋅ θ i (Y ) i =1

(

∀

)

X ,Y ∈ D .

(3.4)

Let ξα be the Reeb field for ωα respectively. There is a decomposition T M × S 3 = TM ⊕ {Tα , Tβ , Tγ }= {ξ1 , ξ 2 , ξ3 } ⊕ D ⊕ {Tα , Tβ , Tγ } . 3  ω ∑ α =1 (σ α ⋅ ωα + ωα ⋅ σ α ) be a symmetric 2-form. Define As before let σ = a pseudo-Riemannian metric on M × S 3 by

(

)

= g ( X ,Y )

3

∑ (σ α ( X ) ⋅ ωα (Y ) + ωα ( X ) ⋅ σ α (Y ) ) + dωα ( Jα X , Y )

α =1

4n

= σ  ω ( X , Y ) + ∑θ ⋅ θ i =1

i

i

( X ,Y ).

(3.5)

As in (2.6) it follows that g (ξα , Tα ) = 1 , g ( Tα , Tα ) = 0 . If we note ξα − σ α (ξα ) Tα , it follows g (ηα ,ηα ) = 0 . So σ α (ξα ) ≠ 0 , letting η= α

 g (ηα ,ηα ) g (ηα , Tα )  0 1   =   g ( Tα ,ηα ) g ( Tα , Tα )  1 0 

(α = 1, 2,3) .

As g | D = g is positive definite from Proposition 3, g is a pseudo-Riemannian metric of type ( 4n + 4,3) on M × S 3 . Theorem 4. Let g ′ be the pseudo-Riemannian metric on M × S 3 corresponding to another Im -valued 1-form ω ′ on M representing ( D,Q ) , i.e. ω ′ = uaω a ( a ∈ Sp (1) , u > 0 ) , then g ′= u ⋅ g . We divide a proof according to whether ω ′ = uω or ω ′ = aω a . Proposition 5. If ω ′ = uω , then g ′= u ⋅ g . Proof. (Existence.) Suppose ω ′ = uω . We show the existence of such a 1-form DOI: 10.4236/am.2018.92008

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σ ′ for ω ′ . Let {Tα , ξα , X 1 , , X 4 n }α =1,2,3 be the frame on M × S 3 for ω . Then ω ′ determines another frame {Tα′ , ξα′ , X 1′, , X 4′ n } . Since each Tα′ generates the same S 1 as that of Tα , note ′ = Tα T= (3.6) α (α 1, 2,3 ) . Let

{ X i }i=1,,4 n

be the frame on D . Then the Reeb field ξα′ for each ωα′ is

described as

(

ξα =u ⋅ ξα′ + x1(α ) u X 1′ +  + x4(αn ) u X 4′ n (α =1, 2,3) .

(3.7)

)

(α )

xi ∈ , i =1, , n . As u ⋅ d ω = d ω ′ on D and D g ( X , Y ) = g ( Jα X , Jα Y ) from Proposition 3, there exists a matrix = B bik ∈ Sp ( n ) such that ∃

D

( )

4n

X i = u ∑bik X k′ .

(3.8)

k =1

Two frames {Tα , ξα , X 1 , , X 4 n } , {Tα′ , ξα′ , X 1′, , X 4′ n } give the coframes ωα ,θ 1 , ,θ 4 n , σ α , ωα′ ,θ ′1 , ,θ ′4 n , σ α′ on M × S 3 respectively. Then the

{

} {

}

above Equations (3.6), (3.7), (3.8) determine the relations between coframes: ωα′ = u ⋅ ωα (α = 1, 2,3) , 4n

(3.9)

u ∑bijθ j + u xi( ) ⋅ ω1 + u xi( ) ⋅ ω2 + u xi( ) ⋅ ω3 ,

θ ′i =

1

2

3

j =1

Moreover if we put



4n



i1 =j 1 =



4n



4n

1

σ α′ = σ α −  ∑  ∑bij xi(α )  θ j + ∑ xi( β ) xi(α ) ⋅ ωβ 2 i =1

(3.10)

2 1  1 γ α α + ∑ xi( ) xi( ) ⋅ ωγ  − ∑ xi( ) ωα , 2 i 1= =  2i1 4n

4n

then (3.15) and (3.10) show that

(ω′, ω′ , ω′ ,θ ′ ,,θ ′ 1

1

2

4n

3

) (

)

, σ 1′, σ 2′ , σ 3′ = ω1 , ω2 , ω3 , θ 1 , , θ 4 n , σ 1 , σ 2 , σ 3 P

for which

      uI 3 P=      0  0 

− x( ) 1

ux

(1)

u x( u x(

2

− x( ) ⋅ x( 2 1

2 2)

3)

−x

( 2)

1

−x

( 3)

(1)

uB 0

⋅ x( ) 2

⋅x 2 1 − B t x( )

− x(

2)

2)

2

2 ( 3)

− x ⋅ x( ) 2 2 − B t x( ) I3 2

1 3  − x( ) ⋅ x( )   2  2 3 − x( ) ⋅ x( )   2 . ( 3) 2  −x   2 t ( 3)  −B x   

3 If I 4 n is a symmetric matrix defined by

I34 n

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0  0 =   0 I  3

0  I4n 0 

0 I3   0  ,  0 0 0 

(3.11)

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Y. Kamishima 3 t 3 it is easily checked that PI 4 n P= u ⋅ I 4 n .

Letting

ω ′ = (ω1′, ω2′ , ω3′ ) and σ ′ = (σ 1′, σ 2′ , σ 3′ ) , we define a pseudo-

Riemannian metric 4n

g ′= σ ′  ω ′ + ∑θ ′i ⋅ θ ′i .

(3.12)

i =1

Then a calculation shows 3

4n

∑ (σ α′ ⋅ ωα′ + ωα′ ⋅ σ α′ ) + ∑θ ′i ⋅θ ′i

= g′

= α 1 =i 1 1 4n 3 t 4n

( ) (ω ′,θ ′ ,,θ ′ ,σ ′) = (ω , θ , , θ , σ ) PI P (ω , θ , , θ , σ ) = u ⋅ ( ω , θ , , θ , σ ) I ( ω , θ , , θ , σ ) = ω ′, θ ′ , , θ ′ , σ ′ I 1

2n

1

3 t 4n

2n

1

t

1

3 t 4n

1

4n

(3.13)

2n

2n

4n  3  =u  ∑ (σ α ⋅ ωα + ωα ⋅ σ α ) + ∑θ i ⋅ θ i  =u ⋅ g . =  α 1 =i 1 

(Uniqueness.) We prove the above σ ′ is uniquely determined with respect to ω ′ . Let  = ωα , θ 1 , , θ 4 n , θ 4 n+1 , θ 4 n+2 be the coframe for ωα where

{

}

= θ ω= ωγ . We have a Fefferman-Lorentz metric on M × S 1 from β ,θ (3.5) and (3.4) under Condition I: 1 = gα σ α  ωα + d ωα  Jα 3 (3.14) 1  4n i i  = σ α  ωα +  ∑θ ⋅ θ + ωβ ⋅ ωβ + ωγ ⋅ ωγ  . 3  i=1  4 n +1

4 n+2

1 for our use.) When ωα′ = uωα , the coframe  3 will be transformed into a coframe  ′ = ωα′ , θα′1 , , θα′4 n , θα′4 n+1 , θα′4 n+2 such as (We take the coefficient

{

= θα′ i

}

u ∑ cα jθ + u yα ωα , i

j

i

j

= θα′

= uθ 4 n+1

uω β ,

= θα′

= uθ

uωγ ,

4 n+1

4 n+ 2

(

∃

4 n+ 2

( )

(3.15)

)

yαi ∈ , ∃ cαi j ∈ Sp ( n ) , i, j = 1, , n .

If gα′ is the corresponding metric on M × S 1 , then gα′ = ugα by Theorem 2 and there exists a unique 1-form σα such that

1  4n  = gα′ σα  ωα′ +  ∑θα′i ⋅ θα′i + θα′4 n+1 ⋅ θα′4 n+1 + θα′4 n+2 ⋅ θα′4 n+2  3  i=1  4n 1  = σα  ωα′ +  ∑θα′i ⋅ θα′i + uωβ ⋅ ωβ + uωγ ⋅ ωγ  . 3  i=1 

(3.16)

If we sum up this equality for α = 1, 2,3 ; 1 2 θα′i ⋅ θα′i + u (ωα ⋅ ωα + ωβ ⋅ ωβ + ωγ ⋅ ωγ ) ∑ 3 α ,i 3 = ug1 + ug 2 + ug3

g1′ + g 2′ + = g3′ σ  ω ′ +

4n 2   = u  σ  ω + ∑θ i ⋅ θ i + (ωα ⋅ ωα + ωβ ⋅ ωβ + ωγ ⋅ ωγ )  , 3 i =1  

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which yields 4n 1 3 4n i i   θα′ = ⋅ θα′ u  σ  ω + ∑θ i = ⋅ θ i  ug. ∑∑ 3 α= 1 =i 1 =i 1  

σ  ω ′ +

(3.17)

Compared this with (3.13) it follows

. σ α′ σ= σ ′ σ , i.e= = α (α 1, 2,3 ) .

(3.18)

By uniqueness of σα , σ α′ defined by (3.10) is a unique real 1-form with respect to ω ′ . Next put ω = a ⋅ ω ⋅ a . The conjugate z  aza  a11 a12 a13  = matrix A  a21 a22 a23  ∈ SO ( 3) . Then it follows  a31 a32 a33 

(

∀

z∈

)

represents a

i   ω = [ω1 , ω2 , ω3 ] A  j   k  By our definition, a hypercomplex structure

(3.19)

{ J1 , J 2 , J 3 }

on D satisfies that

( dωβ | D )  ( dωα | D ) = Jγ (α , β , γ ) ~ (1, 2,3) . A new hypercomplex structure −1

on D is described as

 J1   J1    t    J2  = A J2 . J   J   3  3

(3.20)

Differentiate (3.19) and restrict to D (in fact, d ω =a ⋅ d ω ⋅ a on D ), using Proposition 3, a calculation shows

−a1α g D ( J1 X , Y ) + a2α g D ( J 2 X , Y ) + a3α g D ( J 3 X , Y ) d ωα ( X , Y ) =

(

)

= − g D ( ( a1α J1 + a2α J 2 + a3α J 3 ) X , Y ) = − g D Jα X , Y ,

( ) ( dω | D )  ( dω

D d= ωα Jα X , Y g= ( X , Y ) (α 1, 2,3) .

(3.21)

−1  (α , β , γ ) ~ (1, 2,3) . In particular, we have β α | D ) = Jγ Proposition 6. If ω = aω a , then g = g . = g ( X , Y ) σ  ω ( X , Y ) + d ωα Jα X , Y . Since σα is uniquely Proof. Let = determined by = ωα and ω [ω 1 , ω2 , ω3 ] A ω A from (3.19), it implies that

(

)

= σ [σ= 1 , σ 2 , σ 3 ] A σ A.

(3.22)

Note that 3

∑ (σα ⋅ ωα + ωα ⋅ σα=)

σ  = ω

σ A t A t ω + ω A t A tσ

α =1

(3.23)

= σ t ω + ω tσ = σ  ω . By (3.21),

 σ  ω + dωα  Jα= σ  ω + g D= g. g= Proof of Theorem 4. Suppose= ω ′ λωλ = uω where ω = aω a . It follows DOI: 10.4236/am.2018.92008

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from Proposition 5 that g ′ = ug . By Proposition 6, we have g = g and hence g ′ = ug . This finishes the proof under Condition I.

Proof of Theorem A Proof. Let ( M , {D, Q} ) be a quaternionic 3 CR-manifold. Then M has an open cover {U i }i∈Λ where each U i admits a hypercomplex 3 CR-structure i i i i . Put ω ( ) =ω1( )i + ω2( ) j + ω3( ) k which is an Im -valued 1-form ωα( i ) , Jα( i ) α =1,2,3 on U i . Since we may assume that U i is homeomorphic to a ball (i.e. contractible), Condition I is satisfied for each U i , i.e. C (U i= ) U i × S 1 . Then we 3 i i i i i have a pseudo-Riemannian = metric g ( ) ∑α =1σ α( )  ωα( ) + d ωα( )  Jα( ) on U i × S 3 for ω ( i ) by Theorem 4. Suppose U i  U j ≠ ∅ . By condition a) of 2) = D | U i  U j ker = ω (i ) | U i  U j kerω ( j ) | U i  U j . Then by the (cf. Introduction), equivalence (3.1) there exists a function λ = ua defined on U i  U j such that

(

)

ω ( j ) = λ ⋅ ω (i ) ⋅ λ = uaω (i ) a on U i  U j .

(3.24)

It follows from Theorem 4 that g ( ) = ug ( ) on U i  U j . We may put u = u ji which is a positive function defined on U i  U j . By construction, it is j

i

easy to see that u ki = u kj u ji on U i  U j  U k ≠ ∅ . This implies that {u}i , j∈Λ defines a 1-cocycle on M. Since  + is a fine sheaf as the germ of local continuous functions, note that the first cohomology H 1  ,  + = 0 . (Here 

(

)

is a chain complex of covers running over all open covers of M.) Therefore there

{ f }i , j∈Λ

defined on each U i such that δ f ( j , i ) = u ji , u ji on U i  U j . We obtain that i.e. fi ⋅ f j−1 = exists a local function

f j ⋅ g ( ) =fi ⋅ g ( ) on (U i  U j ) × S 3 . j

i

Then we may define

g | U i × S 3 = fi ⋅ g ( ) .

(3.25)

i

so that g is a globally defined pseudo-Riemannian metric on M × S 3 . If another family {ω ′i } represents the same quaternionic 3 CR-structure ( D,Q ) , then i∈Λ the same argument shows that g ′ = ug on M × S 3 for some positive function. Hence the conformal class

[g]

is an invariant for quaternionic 3 CR-structure.

In particular, the Weyl curvature tensor W ( g ) is also an invariant. This

completes the proof of Theorem A.

4. Model Geometry and Transformations We introduce spherical 3 CR-homogeneous model

( PSp ( n + 1,1) , S ) 4 n +3

and

conformally flat pseudo-Riemannian homogeneous model ( PSp ( n + 1,1) × SO ( 3) , S 4n+3,3 ) equipped with pseudo-Riemannian metric g 0 of type ( 4n + 3,3) and then characterize the lightlike subgroup in PSp ( n + 1,1) × SO ( 3) .

4.1. Pseudo-Riemannian Metric g0 Let us start with the quaternionic vector space  n+2 endowed with the HerDOI: 10.4236/am.2018.92008

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mitian form:

z, w = z1w1 +  + zn+1wn+1 − zn+2 wn+2

( z, w ∈  ) . n+2

(4.1)

The q-cone is defined by

{

}

V0 =z ∈  n+2 − {0} | z, z 〉 = 0 .

(4.2)

When  n+2 is viewed as the real vector space  4 n+8 , O ( 4n + 4, 4 ) denotes

the full subgroup of GL ( 4n + 8,  ) preserving the bilinear form Re , .

Consider the commutative diagrams below. The image of the pair ( O ( 4n + 4, 4 ) ,V0 ) by the projection P is the homogeneous model of conformally flat pseudo-Riemannian geometry PO ( 4n + 4, 4 ) , S 4 n+3,3 in

(

)

which S 4 n+3,3 = P (V0 ) is diffeomorphic to a quotient manifold S 4 n+3 ×2 S 3 . The identification  n+2 =  4 n+8 gives a natural embedding

Sp ( n + 1,1) ⋅ Sp (1) → O ( 4n + 4, 4 ) which results a special geometry

( PSp ( n + 1,1) × SO ( 3) , S

)

(

)

from PO ( 4n + 4, 4 ) , S 4 n+3,3 . As usual, the image of ( Sp ( n + 1,1) ⋅ Sp (1) , V0 ) by P is spherical quarternionic 3 CR-geometry PSp ( n + 1,1) , S 4 n+3 . 4 n +3,3

(

)

(4.3)

0 n +3,3 S 4 n+3 ×2 S 3 . Let We describe a pseudo-Riemannian metric g on S 4=

( z , w ) ∈ S 4 n +3 × S 3 , P (V0 ) = S 4 n+3,3 induces a 2-fold

S 4 n+3 × S 3 be the product of unit spheres. For

z − w = 1 − 1 = 0 so S 2

2

covering P : S

4 n +3

4 n +3

×S → S 3

× S ⊂ V0 . Then 3

4 n+3,3

(

)

for which P* : T S 4 n+3 × S 3 → TS 4 n+3,3 is an

isomorphism.

4 n +3 × S 3 such Let x ∈ S 4 n+3 × S 3 where we put P ( x ) = [ x ] . Choose y ∈ S

x, y = 1 . Denote by { x, y} the orthogonal complement in  n+2 with 0} , it follows respect to , . As TxV0 = {Z ∈  n+2 | Re x, Z = ⊥

that

TxV= y Im  ⊕ x ⊕ { x, y} ⊂  n+2 such that 0 ⊥

(

)

Tx S 4 n+3 × S 3= y Im  ⊕ x Im  ⊕ { x, y} .

(

⊥

)

x ⊕ Tx S 4 n+3 × S 3 . Note that this decomposition does In particular, TxV0 = not depend on the choice of points x′ ∈ [ x ] and y′ with x′, y′ = 1 . (see [3], Theorem 6.1]). We define a pseudo-Riemannian metric on S 4 n+3,3 to be

g[0x] ( P* X= , P*Y ) Re X , Y DOI: 10.4236/am.2018.92008

123

(

∀

(

))

X , Y ∈ Tx S 4 n+3 × S 3 .

(4.4)

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Noting Re , |

(

is positive definite, g { x , y} ( 4n + 3,3) at each [ x ] ∈ S 4n+3,3 . ⊥

0 [ x]

)

a ∈ Sp (1) and is a pseudo-Riemannian metric of type

Re = ya, ya Re = xa, xa 0 , Re xa, ya = 1

∀

4.2. Conformal Group O ( 4n + 4,4 ) It is known more or less but we need to check that O ( 4n + 4, 4 ) acts on

S 4 n+3 × S 3 as conformal transformations with respect to Re , PO ( 4n + 4, 4 ) on S 4 n+3,3 , g 0 .

(

For any h ∈ O ( 4n + 4, 4 ) ,

)

hx= , hx

and so does

= x, x 0 so hx ∈ V0 . However hx

4 n +3

does not necessarily belong to S × S 3 . Normalized hx , there is x′ ∈ S 4 n+3 × S 3 such that ( hx ) λ = x′ for some λ ∈  + . Note = P ( x′ ) . If Rλ :  n+2 →  n+2 is the right multiplication defined [ hx ] P=  ( hx ) by Rλ ( z ) = zλ , then there is the commutative diagram:

in which R*= ( h* X )

( h* X ) λ ∈ Tx′V0 .

(

)

x′ Tx′ S 4 n+3 × S 3 , we have As Tx′V0 =⊕ λ x′µ + X ′ for some µ ∈  , X ′ ∈ Tx′ S 4 n+3 × S 3 . Since ( h* X )= P* = Tx * P= 0 and P : ( O ( 4n + 4, 4 ) , V0 ) → PO ( 4n + 4, 4 ) , S 4 n+3,3 is * ( x )

(

)

(

(

)

)

equivariant, it follows

′ ) P* ( X ′ ) . h* P* ( X= ) P* ( h* X=) P* ( ( h* X ) λ=) P* ( x′µ + X=

λ x′ν + Y ′ for some ν ∈  , Similarly h* P* (Y ) = P* (Y ′ ) for ( h*Y )= 4 n +3 3 ′ ′ ′ ′ ′ Re = x , X Re = x , Y 0 , a calculation shows Y ∈ Tx′ S × S . As

(

)

g[0hx] ( h* P* ( X ) , h* P* (Y ) )

0 ′ ′ ′ ′ = g= [ hx ] ( P* ( X ) , P* (Y ) ) Re X , Y

′ Re ( h* X ) λ , ( h*Y ) λ = Re x′µ + X ′, x′ν + Y= 2 = λ= Re h* X , h*Y λ 2 Re X , Y

= λ 2 g[0x] ( P* ( X ) , P* (Y ) ) .

Hence h ∈ O ( 4n + 4, 4 ) acts as conformal transformation with respect to

g . 0

4.3. Conformal Subgroup Sp ( n + 1,1) ⋅ Sp (1) Let

(I, J, K )

be the standard hypercomplex structure on  n+2 defined by

Iz = − zi, Jz = − zj , Kz = − zk . Put Q = span ( I , J , K ) as the associated quaternionic structure. Then Re ,

leaves invariant Q. The full subgroup of O ( 4n + 4, 4 ) preserving Q is

isomorphic to Sp ( n + 1,1) ⋅ Sp (1) , i.e. the intersection of O ( 4n + 4, 4 ) with

GL ( n + 2,  ) ⋅ GL (1,  ) .

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Let ρ : S 3 → O ( 4n + 4, 4 ) be a faithful representation. Then the subgroup ρ S 3 preserves Q so it is contained in

( )

+2 n +2     n  Sp (1) ×  × Sp (1)  ⋅ Sp (1) ≤ SO ( 4 ) ×  × SO ( 4 )    

which is a subgroup of SO ( 4n + 4 ) × SO ( 4 ) .

4.4. Three Dimensional Lightlike Group Choose S 1 ≤ S 3 and consider a representation restricted to S 1 . As we may assume that the semisimple group ρ S 3 belongs to ( Sp (1) ×  × Sp (1) ) ⋅ Sp (1) ,

( )

1 n+2 1 this reduces to a faithful representation: ρ : S → T ⋅ S such that

((e

= ρ (t )

ia1t

)

)

, , eian+2t ⋅ eibt .

(4.5)

= Here we may assume that ai ≥ 0 are relatively prime ( i 1, , n + 2 ) without loss of generality, and either b = 0 or 1. The element ρ ( t ) acts on

S 4 n+3 × S 3 ⊂ V0 as

(e = (e

= ρ ( t ) ( z1 , , zn+1 , w )

ia1t ia1t

)

z1 , , eian+1t zn+1 , eian+2t w ⋅ e −ibt z1e

− ibt

, , e

ian +1t

zn+1e

− ibt

, eian+2t we −ibt

(4.6)

)

0= z1 +  + zn+1 − w = for ( z, w ) ( z1 , , zn+1 , w ) ∈ V0 . If X is the 1 vector field induced by ρ S at ( z, w ) , then it follows where

2

2

2

( )

= X

( ia1 z1 , , ian+1 zn+1 , ian+2 w ) − ( z1ib, , zn+1ib, wib ) .

(4.7)

1 n+2 1 Proposition 7. If ρ : S → T ⋅ S is a faithful lightlike 1-parameter group,

then it has either one of the forms: +2   n , , eit ≤  Sp (1) ×  × Sp (1)  ≤ Sp ( n + 1,1) ⋅ {1} ,     it ρ= ( t ) (1, ,1) ⋅ e ≤ {1} ⋅ Sp (1) ≤ Sp ( n + 1,1) ⋅ Sp (1) .

= ρ (t )

(e

)

it

(4.8)

Proof. Case (i) b = 0 . X = ( ia1 z1 , , ian+1 zn+1 , ian+2 w ) from (4.7) so that 2

X , X = a12 z1 +  + an2+1 zn+1 − an2+2 w = 2

2

(a

2 1

)

(

)

− an2+2 z1 +  + an2+1 − an2+2 zn+1 2

Since Re X , X = 0 and we assume ai ≥ 0 , it follows

2

.

= a1 a= an+2 . n + 2 , , an +1 As ai ’s are relatively prime, this implies

a=  = an+= an+= 1. 1 1 2

ρ (t ) As a consequence =

(

)

( e ,, e ) ≤ Sp ( n + 1,1) ⋅ {1} . In this case note that it

it

Tx S 4 n+3 × S 3 = Im y ⊕ Im x ⊕ { x, y}

⊥

such that

Case (ii) b = 1 . It follows from (4.7) that

= X

x, y ∈  * .

( ia1 z1 , , ian+1 zn+1 , ian+2 w ) − ( z1i, , zn+1i, wi ) .

Put Y = ( ia1 z1 , , ian+1 z= n+1 , ian+ 2 w ) , W

z1i, , zn+1i, wi ) (=

xi such that

W , W i= x, x i 0 . Calculate X= Y − W and = DOI: 10.4236/am.2018.92008

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Y ,= Y a12 z1 +  + an +1 zn +1 − an2+ 2 w , 2

2

Y ,W = a1 z1 iz1i +  + an +1 zn +1 izn +1i − an + 2 wiwi,

(4.9)

2

Re Y , W= a1 z1 +  + an +1 zn +1 − an + 2 = w Re W , Y . 2

This shows Re X , X = Re Y − W , Y − W Re Y , Y − 2 Re Y , W + Re W , W = R Y , Y − 2 Re Y , W =

( a − 2a ) z +  + ( a − 2a ) z − ( a − 2a ) w =( ( a − 2 a ) − ( a − 2 a ) ) z +  + ( ( a − 2 a ) − ( a

=

2

2 1

1

2 1

=

1

2

n +1

2 n+2

n +1

2

2 n+2

1

(( a − 1)

2

2 n +1

1

n+2

− ( an + 2 − 1)

)z

2

2 n +1

1

(

2 n+2

n +1

+  + ( an +1 − 1) − ( an + 2 − 1)

2

1

2

n+2

2

2

)z

− 2an + 2 2

n +1

)) z

2 n +1

.

Thus

( a1 − 1) = ( an+2 − 1) 2

2

, , ( an+1 − 1) = 2

( an+2 − 1)

2

.

(4.10)

On the other hand, we may assume in general  0. a= = a= 1 k ak +1 − 1 ≤ 0, , al − 1 ≤ 0. al +1 − 1 ≥ 0, , an +1 − 1 ≥ 0.

(ii-1). Suppose an+2 − 1 ≥ 0 . As 0 < a j ≤ 1 for k + 1 ≤ j ≤ l , it implies

ak +=  = a= 1 . Since ( ak +1 − 1) = ( an+2 − 1) from (4.10), it follows an+2 = 1 . 1 l 2 0 and so a j = 1 ( l + 1 ≤ j ≤ n + 1) . Note that Again from (4.10), a j − 1 = 2

(

ai ≠ 0 because

= ρ (t ) implies

2

)

( ai − 1) = ( an+2 − 1) = 2

( e , , e ) ⋅ e it

it

2

it

0 . Thus a= 1 . This a=  = an+= 1 2 2

.

(ii-2). Suppose an+2 − 1 < 0 . In this case an+2 = 0 . By (4.10), it follows that

ai ≠ 0 and a= 1 , ai = 2 ( l + 1 ≤ i ≤ n + 1) . Thus  = a= 1 l i 2t i 2t = ρ ( t ) (1, ,1, e , , e ,1) ⋅ eit . This contradicts that nonzero ai ’s (1 ≤ i ≤ n + 1) are relatively prime. ∀

(ii-3). Suppose an+2 − 1 < 0 and a= a=  = an+= 0 . Again an+2 = 0 and 1 2 1

ρ (t ) so=

(1, ,1) ⋅ eit .

To complete the proof of the proposition we prove the following. Put

= x

( z, w=) ( z1 , , zn+1 , w ) ∈ S 4 n+3 × S 3 ⊂ V0

such that

x, x = 0 .

Lemma 8. Case (ii-1) does not occur.

Proof. It follows from (4.7) that

X= ix − xi. ( iz1 , , izn+1 , iw ) − ( z1i, , zn+1i, wi ) = Put

x = p + jq (p, q ∈ Cn+2 ).

Then

X = 2kq.

As

X, X = 0

(4.11) implies

q, q = 0 . On the other hand, the equation

0 = x, x =( p, p + q, q ) − 2 j p, q p, p + q, q = 0, p, q = 0. Note that if S 2n+1 × S 1 is the canonical subset in S 4n+3 × S 3 , then p, p = 0 if and only if p ∈ S 2n+1 × S 1 . Since X is a shows

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nontrivial vector field on S 4n+3 × S 3 , there is a point x in the open subset

X , X ≠ 0 on S,

S= S 4n+3 × S 3 \ S 2n+1 × S 1 such that p, p ≠ 0 and thus which contradicts that X is a lightlike vector field.

4.5. Proof of Theorem B Applying Proposition 7 to a lightlike group S 3 we obtain: Corollary 9. Let ρ : S 3 → O ( 4n + 4, 4 ) be a faithful representation which preserves the metric Re , on V0 . If ρ S 3 is a lightlike group on

( )

S 4 n+3 × S 3 , then either one of the following holds.

( ) ρ ( S ) ={1} ⋅ Sp (1) ≤ Sp ( n + 1,1) ⋅ Sp (1) . ( diag (Sp (1) × × Sp (1) ) ⋅ Sp (1) , S × S ) be as in (4.13). If = ρ S 3 diag ( Sp (1) ×  × Sp (1) ) ≤ Sp ( n + 1,1) ⋅ {1} , 3

Let

f :S f

4 n +3

4 n +3

×S → S 3

4 n +3

×S

3

(4.13)

3

is a map defined by

( ( z1 ,, zn+1 , w) ) = ( wz1 ,, wzn+1 , w) , then for a ∈ Sp (1) , b ∈ Sp (1) , f

(( az ,, az 1

n +1

, awb

) ) = ( bwz ,, bwz

So the equivariant diffeomorphism diffeomorphism

(

1

f

n +1

, bwa ) .

induces a quotient equivariant

( ) ) → ( diag (Sp (1) × × Sp (1) ) , S ) .

fˆ : Sp (1) , S 4 n+3 × S 3 ρ S 3

4 n +3

(4.14)

We prove Theorem B of Introduction.

Proof. Suppose that the pseudo-Riemannian manifold ( M × S 3 , g ) is conformally flat. Let π = π 1 ( M ) be the fundamental group and M the universal covering of M. By the developing argument (cf. [7]), there is a developing pair:

( ρ , Dev ) : (π × S 3 , M × S 3 , g ) → ( O ( 4n + 4, 4 ) , S 4 n+3 × S 3 , g 0 ) where Dev is a conformal immersion such that Dev g = ug for some positive function u on M × S 3 and ρ : π × S 3 → O ( 4n + 4, 4 ) is a holonomy *

0

homomorphism for which Dev is equivariant with respect to ρ .

( )

By Corollary 9, if ρ S 3 ={1} ⋅ Sp (1) ≤ Sp ( n + 1,1) ⋅ Sp (1) , then the normalizer

of Sp (1) in O ( 4n + 4, 4 ) is isomorphic to Sp ( n + 1,1) ⋅ Sp (1) . In particular,

ρ (π × S 3 )= ρ (π ) × Sp (1) ≤ Sp ( n + 1,1) ⋅ Sp (1) where ρ ( S 3= )

{1} ⋅ Sp (1)

. We

have the commutative diagram:

(4.15)

where DOI: 10.4236/am.2018.92008

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equivariant. ρ S 3 diag ( Sp (1) ×  × Sp (1) ) ≤ Sp ( n + 1,1) ⋅ {1} from (4.13), then If = 3 ρ π × S= ρ S 3 ⋅ ρ (π ) ≤ diag ( Sp (1) ×  × Sp (1) ) ⋅ Sp (1) . Composed f with Dev , we have an equivariant diffeomorphism fˆ  dev : (π , M ) → ( ρ (π ) , S 4 n+3 )

( ) ) ( )

(

where ρ (π ) ≤ diag ( Sp (1) ×  × Sp (1) ) ≤ PSp ( n + 1,1) . In each case taking the developing map either dev of (4.15) or fˆ  dev , a quaternionic 3 CR-manifold M is spherical, i.e. uniformized with respect to PSp ( n + 1,1) , S 4 n+3 . is the standard quaternionic 3 CR-structure Conversely recall ω 0 , Jα0 α =1,2,3 4 n +3 equipped with the standard hypercomplex structure Q 0 = Jα0 on S α =1,2,3 0 on D . Suppose that ω , { Jα }α =1,2,3 is a spherical quaternionic 3 CR-structure on M with a quaternionic structure Q, then there exists a developing map dev : M → S 4 n+3 such that

(

(

)

{ }

(

)

{ }

)

 dev∗ω 0 = λωλ for some  -valued function λ on M with a lift of quaternionic 3 CR-structure ω . In particular, dev*D = D0 and dev*Q = Q 0 . Let g be a pseudo-Riemannian metric on M × S 3 for ω which is a lift of g and ω to M × S 3 respectively. Put ω ′ = dev∗ω 0 . Let λ = ua be a function for u > 0 and a ∈ Sp (1) such that

ω ′ = uaω a .

(

)

(

)

∀ V , W ∈ D0 . The By the definition, recall d ωβ J γ0V , W = d ωα0 (V , W ) induced quaternionic structure { Jα′ }α =1,2,3 for ω ′ = dev∗ω 0 is obtained as d dev*ωβ0 ( J γ′ X , Y ) = d dev*ωα0 ( X , Y ) . Since 0

(

)

(

)

d ωβ ( dev* J γ′ X , dev*Y ) = d ωα ( dev* X , dev*Y ) , taking V = dev* X , we obtain 0

0

= dev* J γ′ X J γ0 dev* X

(

(

∀

)

X ∈D .

(4.16)

)

0 α 1, 2,3 , note that { Jα′ }α =1,2,3 ∈ Q . Q Q= span Jα0 ,= As dev*= ′ On the other hand, let g be the pseudo-Riemannian metric on M × S 3 for ω ′ , it follows from Theorem 4

g ′ = ug .

(4.17)

3 3 Take the above element a ∈ S 3 and let ρ : S → S be a homomorphism 3 4 n +3 × S3 defined by ρ ( s ) = asa ∀ s ∈ S 3 . Define a map dev × ρ : M × S → S

(

)

which makes the diagram commutative. (Here p is the projection onto the left summand.)

(4.18)

p* : ( D, {Jα′ } ) → ( D, {Jα′ } ) isomorphisms such that

where both

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and

( { }) → ( D ,{J })

p* : D0 , Jα0

0

0 α

are

Applied Mathematics

Y. Kamishima 0 = = p*  Jα′ Jα′  p* and p*  Jα0 J= 1, 2,3) . α  p* (α

(4.19)

0 0 * 0 * 0 0 Recall from (3.5)= that g σ  p ω + dp ωα  Jα . (We write p more pre-

cisely.) Consider the pull-back metric

( dev × ρ )

*

g 0 ( X ,= Y ) σ 0  p*ω 0 ( ( dev × ρ )* X , ( dev × ρ )* Y )

(

)

+ dp*ωα0 Jα0 ( dev × ρ )* X , ( dev × ρ )* Y .

(4.20)

Calculate the first and the second summand of (4.20) respectively.

* * * ( dev × ρ ) σ 0  ( dev × ρ ) p*ω 0 ( dev × ρ ) (σ 0  p*ω 0 ) =

(4.21)

= ρ *dev*σ 0  p*dev*ω 0 .

( ) ( J p ( dev × ρ ) X , p ( dev × ρ ) Y ) ( J dev p X , dev p Y )

dp*ωα0 Jα0 ( dev × ρ )* X , ( dev × ρ )* Y = d ωα

0

= d ωα

0

0 α

*

*

*

*

0

α

* *

* *

= d ωα ( dev* Jα′ p* X , dev* p*Y ) (4.16) 0

= d ωα0 ( dev* p* Jα′ X , dev* p*Y ) (4.19)

(

)

* = dp = dev*ωα0 ( Jα′ X , Y ) d p*dev*ωα0  Jα′ ( X , Y ) .

(4.22) Thus

(

)

= ( dev × ρ ) g 0 Ra* dev*σ 0  p*dev*ω 0 + d p*dev*ωα0  Jα′ . *

Then it follows by the construction of (3.5) that ( dev × ρ ) g 0 is the corresponding pseudo-Riemannian metric for dev*ω 0 = ω ′ and so * g′ = ug by (4.17). Therefore M × S 3 , g is conformally flat ( dev × ρ ) g 0 = and so is M × S 3 , g . *

(

)

(

)

References [1]

Biquard, O. (2001) Quaternionic Contact Structures, Quaternionic Structures in Mathematics and Physics, Rome, World Sci. 1999 Publishing, River Edge, NJ. 23-30.

[2]

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[3]

Alekseevsky, D.V. and Kamishima, Y. (2008) Pseudo-Conformal Quaternionic CR Structure on ( 4n + 3) -Dimensional Manifolds. Annali di Matematica Pura ed Ap-

[4]

Kobayashi, S. (1970) Transformation Groups in Differential Geometry. Ergebnisse Math., 70.

plicata, 187, 487-529. https://doi.org/10.1007/s10231-007-0053-2

[5]

Lee, J. (1986) The Fefferman Metric and Pseudo Hermitian Invariants. Transactions

of the American Mathematical Society, 296, 411-429. https://doi.org/10.1090/S0002-9947-1986-0837820-2

DOI: 10.4236/am.2018.92008

[6]

Fefferman, C. (1976) Monge-Ampère Equations, the Bergman Kernel, and Geometry of Pseudo-Convex Domains. Annals of Mathematics, 103, 395-416. https://doi.org/10.2307/1970945

[7]

Kulkarni, R. (1978) On the Principle of Uniformization. Journal of Differential Geometry, 13, 109-138. https://doi.org/10.4310/jdg/1214434351 129

Applied Mathematics