Journal of Science Generic Riemannian Submersions ... - DergiPark

GU J Sci 30(3): 89-100 (2017) Gazi University

Journal of Science http://dergipark.gov.tr/gujs

Generic Riemannian Submersions from Almost Product Riemannian Manifolds Mehmet Akif AKYOL1 1

Bingöl University, Faculty of Arts and Sciences, Department of Mathematics, 12000, Bingöl, TURKEY.

Article Info

Abstract

Received: 29/09/2016 Revised: 06/06/2017 Accepted: 01/08/2017

In this paper, we define and study generic Riemannian submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion. We also find necessary and sufficient conditions for a generic Riemannian submersion to be totally geodesic.

Keywords Almost Product Riemannian Manifold, Riemannian Submersion, Generic Riemannian Submersion.

1. INTRODUCTION The theory of smooth maps between Riemannian manifolds has been widely studied in Riemannian geometry. Such maps are useful for comparing geometric structures between two manifolds. In this point of view, the study of Riemannian submersions between Riemannian manifolds was initiated by O’Neill [12] and Gray [6], see also [5] and [19]. Riemannian submersions have several applications in mathematical physics. Indeed, Riemannian submersions have their applications in the Yang-Mills theory ([3], [18]), Kaluza-Klein theory ([4], [7]), supergravity and superstring theories ([8], [11]), etc. Later such submersions were considered between manifolds with differentiable structures, see [5]. Furthermore, we have the following submersions: semi-Riemannian submersion and Lorentzian submersion [5], Riemannian submersion [6], almost Hermitian submersion [17], contact-complex submersion [10], quaternionic submersion [9], etc. Recently, B. Şahin [14] introduced the notion of anti-invariant Riemannian submersions which are Riemannian submersions from almost Hermitian manifolds such that the vertical distributions are antiinvariant under the almost complex structure of the total manifold and as a generalization of anti-invariant Riemannian submersions and almost Hermitian submersions, B. Şahin [15] introduced the notion of semiinvariant Riemannian submersions when the base manifold is an almost Hermitian manifold. (For recent developments on the geometry of almost Hermitian manifolds, see also:[16]). He showed that such submersions have rich geometric properties and they are useful for investigating the geometry of the total space. On the other hand, as a generalization of semi-invariant submersions, Ali and Fatima [1] introduced the notion of generic Riemannian submersions. They showed that such submersions have rich geometric properties and they are useful for investigating the geometry of the total space. The present work, we define and study the notion of generic Riemannian submersion from almost product Riemannian manifolds. The paper is organized as follows. In Section 2 we recall some notions needed for this paper. In section 3 we define generic Riemannian submersions from an almost product Riemannian manifold onto a Riemannian manifold. We also investigate the geometry of leaves of the distributions. Finally, we give necessary and sufficient conditions for such submersions to be totally geodesic.

*Corresponding author, e-mail: [email protected]

90

Mehmet Akif AKYOL / GU J Sci, 30(3):89-100(2017)

2. PRELIMINARIES In this section, we define almost product Riemannian manifolds, recall the notion of Riemannian submersions between Riemannian manifolds and give a brief review of basic facts of Riemannian submersions. Let M be a m-dimensional manifold with a tensor F of a type (1,1) such that

F 2 = I , ( F  I ). Then, we say that M is an almost product manifold with almost product structure F. We put

P=

1 1 ( I  F ), Q = ( I  F ). 2 2

Then we get

P  Q = I , P 2 = P, Q 2 = Q, PQ = QP = 0, F = P  Q. Thus P and Q define two complementary distributions P and Q. We easily see that the eigenvalues of F are +1 or -1. If an almost product manifold M admits a Riemannian metric g such that

g ( FX , FY ) = g ( X , Y )

(1)

for any vector fields X and Y on M. Then M is called an almost product Riemannian manifold, denoted by M

(M , g , F ). Denote the Levi-Civita connection on M with respect to g by  . Then, M is called a locally M

product Riemannian manifold [19] if F is parallel with respect to  , i.e., M

 X F = 0, X  (TM )

(2)

Let (M , g ) and ( N , g ' ) be two Riemannian manifolds. A surjective C   map  : M  N is a C   submersion if it has maximal rank at any point of M . Putting Vx = ker x , for any x  M , we obtain an integrable distribution V, which is called vertical distribution and corresponds to the foliation of M determined by the fibres of  . The complementary distribution H of V, determined by the Riemannian metric g , is called horizontal distribution. A C   submersion  : M  N between two Riemannian manifolds (M , g ) and ( N , g ' ) is called a Riemannian submersion if, at each point x of M ,   x preserves the length of the horizontal vectors. A horizontal vector field X on M is said to be basic if X is   related to a vector field X ' on N . It is clear that every vector field X ' on N has a unique horizontal lift X to M and X is basic. We recall that the sections of V, respectively H, are called the vertical vector fields, respectively horizontal vector fields. A Riemannian submersion  : M  N determines two (1,2) tensor fields T and A on M , by the formulas: M

M

T ( E, F ) = TE F = H VE VF  V VEHF

(3)

and M

M

A( E, F ) = AE F = VHEHF  HHEHF

(4)

91


for any E, F  (TM ), where V and H are the vertical and horizontal projections (see [5]). From (3) and (4), one can obtain M ˆ W; V W = TV W   V M

(5)

M

V X = TV X  H(V X ); M

(6)

M

 X V = V( X V )  AX V ; M

(7)

M

 X Y = AX Y  H( X Y ),

(8)

for any X , Y  ((ker  ) ), V ,W  (ker  ). Moreover, if X is basic then M

M

H(V X ) = V ( X V ) = AX V .

(9)

We note that for U , V  (ker  ), TU V coincides with the second fundamental form of the immersion of the fibre submanifolds and for X , Y  ((ker  ) ),

AX Y =

1 V[ X , Y ] reflecting the complete 2

integrability of the horizontal distribution H. It is known that A is alternating on the horizontal distribution: AX Y =  AY X , for X , Y  ((ker  ) ) and T is symmetric on the vertical distribution: TUV = TVU , for U ,V  (ker  ). We now recall the following result which will be useful for later. Lemma 2.1 (see [5],[12]). If  : M  N is a Riemannian submersion and X , Y basic vector fields on

M ,   related to X ' and Y ' on N , then we have the following properties 1. H[ X , Y ] is a basic vector field and  H[ X , Y ] = [ X ' , Y ' ]   ; M

2. H( X Y ) is a basic vector field

  related to ( X ' Y ' ), where  N

M

and 

N

are the Levi-Civita

connection on M and N ; 3. [ E ,U ]  (ker  ), for any U  (ker  ) and for any basic vector field E. Let ( M , g M ) and ( N , g N ) be Riemannian manifolds and  : M  N is a smooth map. Then the second fundamental form of  is given by

(  )( X , Y ) =  X  Y    ( X Y )

(10)



for X , Y  (TM ), where we denote conveniently by  the Levi-Civita connections of the metrics g M and g N . Recall that

 is said to be harmonic if trace(  ) = 0 and  is called a totally geodesic map

if (  )( X , Y ) = 0 for X , Y  (TM ) [2]. It is known that the second fundamental form is symmetric. Let g be a Riemannian metric tensor on the manifold M = M 1  M 2 and assume that the canonical foliations DM and DM 1

2

intersect perpendicularly everywhere. Then g is the metric tensor of a usual

product of Riemannian manifolds if and only if DM and DM are totally geodesic foliations [13]. 1

2

92


3. GENERIC RIEMANNIAN SUBMERSIONS In this section, we define and study generic Riemannian submersions from an almost product Riemannian manifold onto a Riemannian manifold, investigate the integrability of distributions and obtain a necessary and sufficient condition for such submersions to be totally geodesic map. Definition 3.1 Let ( M , g , F ) be an almost product Riemannian manifold and ( N , g ' ) a Riemannian '

manifold. A Riemannian submersion  : ( M , g , F )  ( N , g ) is called a generic Riemannian submersion if there is a distribution D1  ker  such that

ker  = D1  D2 , F ( D1 ) = D1 , where D2 is the orthogonal complement of D1 in (ker * ), and is purely real distribution on the fibres of the submersion  . Now, we will give an example in order to guarantee the existence of generic Riemannian submersions in locally product Riemannian manifolds and demonstrate that the method presented in this paper is effective. Note that given an Euclidean space R 8 with coordinates ( x1 ,..., x8 ) , we can canonically choose an almost product structure F on R 8 as follows:

F (a1 ,..., a8 ) =

1 (a2  a8 ,a1  a7 ,a4  a6 ,a3  a5 , a6  a4 , a5  a3 , a8  a2 , a7  a1 ), 2

where a1 ,..., a8 R. Example 3.1 Let

 be a submersion defined by :



R8 ( x1 ,..., x8 )

R2 x x x x ( 1 3 , 1 3 ). 2 2

Then it follows that

ker * = span{Z1 = x2 , Z 2 = x4 , Z3 = x5 , Z4 = x6 , Z5 = x7 , Z6 = x8} and

(ker * )  = span{H1 =

1 1 1 1 x1  x3 , H 2 = x1  x3}. 2 2 2 2

Hence we have

FZ1 = 

1 1 1 1 1 1 Z 5  X 1  X 2 , FZ 2 = Z3  X1  X 2 , 2 2 2 2 2 2

FZ 3 = FZ 5 = 

1 1 1 1 1 Z2  Z 4 , FZ 4 = Z3  X1  X 2 , 2 2 2 2 2 1 1 1 1 1 Z1  Z 6 and FZ 6 = Z5  X1  X 2 . 2 2 2 2 2

Thus it follows that D1 = span{Z3 , Z5} and D2 = span{Z1 , Z 2 , Z 4 , Z6 } . Also by direct computations, we get

g1 ( H1 , H1 ) = g 2 ( * H1 ,  * H1 ) and g1 ( H 2 , H 2 ) = g 2 ( * H 2 ,  * H 2 ),


where g1 and g 2 denote the standard metrics (inner products) of R 8 and R 2 . This show that Riemannian submersion. Thus  is a generic Riemannian submersion.

93

 is a

Let  be a generic Riemannian submersion from an almost product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then for V  (ker  * ) , we write

FV = V  V

(11)

where V  ( D1 ) and V  ((ker  * ) ) . We denote the complementary distribution to D2 in

(ker  * ) by  . Then we have (ker  * ) = D2  

(12)

and  is invariant under F . Thus, for any X  ((ker  * ) ) , we have

FX = BX  CX

(13)

where BX  ( D2 ) and CX  ( ) . From (11), (12) and (13) we have

D1 = D1,D1 = 0,D2  D2 ,B((ker  * ) ) = D2 ,

 2  B = id , C2  B = id ,  C = 0,BC  B = 0. Then by using (5), (6), (11) and (13) we get M

(U  )V = BTU V  TU V M

(U  )V = CTU V  TU V

(14) (15)

for U ,V  (ker  * ) , where M ˆ V   ˆ V (U  )V =  U U

and M ˆ V. (U  )V = AV U   U

Next, we easily have the following lemma: Lemma 3.1 Let ( M , g , F ) be a locally product Riemannian manifold and ( N , g ' ) a Riemannian manifold. Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion. Then we have 1.

M

M

AX BY  H X CY = CH X Y  AX Y M

M

V X BY  AX CY = BH X Y  AX Y , 2.

ˆ V TU V  AV U = CTU V   U

ˆ V  T V = BT V   ˆ V,  U U U U 3.

M

M

AX U  HU V = CAX U  V X U M

M

V X U  AX U = BAX U  V X U ,

94


for X , Y  ((ker  * ) ) and U ,V  (ker  * ) . Theorem 3.1 Let

 be a generic Riemannian submersion from a locally product Riemannian manifold '

(M , g , F ) onto a Riemannian manifold ( N , g ). Then the distribution D1 is integrable if and only if we have

ˆ V  ˆ U ) = C(T U  T V ) ( U1 1 V1 1 V1 1 U1 1 for U1 ,V1  ( D1 ). Proof. For U1 ,V1  ( D1 ) and Z  ((ker  ) ), since [U1 ,V1 ]  (ker  ), from (5), (11) and (13) we get M M ˆ V  FT U  F ˆ U , Z) g ( F[U1 ,V1 ], Z ) = g ( FU V1  FV U1 , Z ) = g ( FTU V1  F U 1 V 1 V 1 1

1

1

1

1

1

ˆ V   ˆ V = g (BTU V1  CTU V1   U 1 U 1 1

1

1

1

ˆ U  CT U   ˆ U , Z)  BTV U1   V 1 V 1 V 1 1

1

1

1

ˆ V  CT U   ˆ U , Z ). = g (CTU V1   U 1 V 1 V 1 1

1

1

1

Hence, we have the result. Theorem 3.2 Let  be a generic Riemannian submersion from a locally product Riemannian manifold '

(M , g , F ) onto a Riemannian manifold ( N , g ). Then the purely real distribution D2 is integrable if and only if we have

ˆ V   ˆ U  T V  T U  ( D )  U 2 V 2 U 2 V 2 2 2

2

2

2

for U 2 ,V2  ( D2 ) .

D2 integrable if and only if g ([U 2 ,V2 ], FZ ) = g1 ([U 2 ,V2 ], W ) = 0 for U 2 ,V2  ( D2 ),Z  ( D1 ) and W  ((ker  * )  ) . Since ker  * is integrable g ([U 2 ,V2 ], W ) = 0 . Thus D2 integrable if and only if g ([U 2 ,V2 ], FZ ) = 0.

Proof.

We

note

that

the

purely

real

distribution

Moreover, by using (1), (2) and (13) we have M

M

M

M

2

2

2

2

g ([U 2 ,V2 ], FZ ) = g ( F (U V2  U V2 ), FZ )  g ( F (V U 2  V U 2 ), FZ ). Using (5), (6) and (9), we get

g ([U 2 ,V2 ], FZ ) = g (B(TU V2  TV U 2  AV U 2  AU V2 ) 2

2

2

2

ˆ V   ˆ U  T V  T U ), FZ ).   ( U 2 V 2 V 2 V 2 2

2

2

2

Since B(TU V2  TV U 2  AV U 2  AU V2 )  ( D2 ) , 2

2

2

2

ˆ V   ˆ U  T V  T U ), FZ ), g ([U 2 ,V2 ], FZ ) = g( ( U 2 V 2 U 2 V 2 2

2

2

2

which proves assertion. Now, we investigate the geometry of the leaves of the distributitons D1 and D2 .

95


Theorem 3.3 Let


(M , g , F ) onto a Riemannian manifold ( N , g ). Then D1 defines a totally geodesic foliation on M 1 if and only if

( * )(U1 , FV1 )  (  ) and

ˆ FV , BX ) = g' (( )(U , FV ), CX ) g ( U 1 * 1 1 * 1

for U1 ,V1  ( D1 ), U 2  ( D2 ) and X  ((ker * )  ) . Proof. From the definition of a generic Riemannian submersion, it follows that the distribution D1 defines a totally geodesic foliation on M

M

M

1

1

if and only if g (U V1 ,U 2 ) = 0 and g (U V1 , X ) = 0 for

U1 ,V1  ( D1 ),U 2  ( D2 ) and X  ((ker * ) ) . Since  is a generic Riemannian submersion, from 

(2), we get M

g (U V1 ,U 2 ) =  g' (( * )(U1 , FV1 ), *U 2 ).

(16)

1

On the other hand, by using (13) we have M

M

M

1

1

1

g (U V1 , X ) = g1 (U FV1 , BX )  g1 (U FV1 , CX ). Since

 is a generic Riemannian submersion, using (10) and (5) we get ˆ FV , BX )  g' (( )(U , FV ), CX ). (17) g (U V1 , X ) = g ( U 1 * 1 1 * M

1

1

Thus proof follows from (16) and (17). For the leaves of D2 we have the following result. Theorem 3.4 Let


(M , g , F ) onto a Riemannian manifold ( N , g ). Then D2 defines a totally geodesic foliation on M if and only if

( * )(U 2 , FV1 )  (  ) and

ˆ BX , V )  g (T BX , V )  g (T CX , V ) g' (( * )(U 2 , CX ), *V2 ) = g ( U 2 U 2 U 2 2

2

2

for U1  ( D1 ) , U 2 ,V2  ( D2 ) and X  ((ker * )  ) . M

Proof. The distribution D2 defines a totally geodesic foliation on M if and only if g (U V2 ,U1 ) = 0 and 2

M

g (U V2 , X ) = 0 for U1  ( D1 ) , U 2 ,V2  ( D2 ) and 2 M

M

2

2

X  ((ker * ) ) . Since we have 

g (U V2 ,U1 ) =  g1 (U U1 ,V2 ) , from (10) we have M

g (U V2 ,U1 ) =  g' (( * )(U 2 , FU1 ), *V2 ). 2

In a similar way, by using (13) we obtain

(18)

96


M

M

M

2

2

2

g (U V2 , X ) =  g (U BX , FV2 )  g (U CX , FV2 ). Using (5), (6), (11) and if we take into account that

 is a generic Riemannian submersion, we obtain

ˆ BX , V )  g (T BX , V )  g (T CX , V ) g (U V2 , X ) =  g ( U 2 U 2 U 2 M

2

2

2

(19)

2

 g' (( * )(U 2 , CX ), *V2 ). Thus proof follows from (18) and (19). From Theorem 3.3 and Theorem 3.4, we have the following result. Theorem 3.5 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then the fibers of  are the locally product Riemannian manifold of leaves of D1 and D2 if and only if

( * )(U1 , FV1 )  ( *  ) and

( * )(U 2 , FV1 )  ( *  ) for any U1 ,V1  ( D1 ) and U 2  ( D2 ) . For the geometry of leaves of the horizontal distribution ((ker * )  ) , we have the following theorem. Theorem 3.6 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then the distribution (ker * )  defines a totally geodesic foliation on M if and only if M

M

1

1

AX BX 2  H X CX 2  ( ), V X BX 2  AX CX 2  ( D2 ) 1

1

for any X1 , X 2  ((ker * ) ) . M

M

1

1

Proof. Since M is a locally product Riemannian manifold, from (1) and (2) we have  X X 2 = F X FX 2 for X1 , X 2  ((ker * ) ) . Using (13), (7) and (8) 

M

M

M

1

1

 X X 2 = F ( AX BX 2  V X BX 2 )  F (H X CX 2  AX CX 2 ). 1

1

1

Then by using (11) and (13) we get M

M

 X X 2 = BAX BX 2  CAX BX 2  V X BX 2 1

1

1

1

M

M

M

1

1

1

 V X BX 2  BH X CX 2  CH X CX 2  AX CX 2  AX CX 2 . 1

M

Hence, we have  X X 2  ((ker * )  ) if and only if 1

M

M

1

1

B( AX BX 2  H X CX 2 )   (V X BX 2  AX CX 2 ) = 0. 1

M

Thus  X X 2  ((ker * )  ) if and only if 1

1

1

97


M

M

1

1

B( AX BX 2  H X CX 2 ) = 0,  (V X BX 2  AX CX 2 ) = 0, 1

1

which completes proof. In the sequel we are going to investigate the geometry of leaves of the vertical distribution ker * . Theorem 3.7 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then the distribution (ker * ) defines a totally geodesic foliation on M if and only if

ˆ Z  T Z  ( D ) TZ Z 2  AZ Z1  (D2 ),  Z 2 Z 2 1 1

2

1

1

for any Z1 , Z 2  (ker * ) . Proof. For any Z1 , Z 2  (ker * ) , using (2), (5), (6) and (11) we get M M M M ˆ Z  A Z  T Z )  Z Z 2 = F Z FZ 2 = F (Z Z 2   Z Z 2 ) = F (TZ Z 2   Z 2 Z 1 Z 2 1

1

1

1

1

1

2

1

ˆ Z   ˆ Z = BTZ Z 2  CTZ Z 2   Z 2 Z 2 1

1

1

1

 BAZ Z1  CAZ Z1  TZ Z 2  TZ Z 2 . 2

2

1

1

From above equation, it follows that (ker * ) defines a totally geodesic foliation if and only if

ˆ Z  T Z ) = 0. C(TZ Z 2  AZ Z1 )   ( Z 2 Z 2 1

2

1

1

M

Thus  Z Z 2  (ker * ) if and only if 1

ˆ Z  T Z ) = 0, C(TZ Z 2  AZ Z1 ) = 0, ( Z 2 Z 2 1

2

1

1

which completes proof. From Theorem 3.5 and Theorem 3.6, we have the following result. Theorem 3.8 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then the total space M is a generic product manifold of the leaves of D1 , D2 and (ker * )  , i.e., M = M D  M D  M 1

2

( ker  * ) 

, if

and only if

( * )(U1 , FV1 )  ( *  ), ( * )(U 2 , FV1 )  ( *  ) and M

M

1

1


1

for any U1 ,V1  ( D1 ) , U 2  ( D2 ) and X1 , X 2  ((ker * ) ) , where M D , M D and M 1

leaves of the distributions D1 , D2 and (ker * )  , respectively. From Theorem 3.6 and Theorem 3.7, we have the following result.

2

( ker  * ) 

are

98


Theorem 3.9 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . Then the total space M is a generic product manifold of the leaves of (ker * )  and ker * , i.e., M = M

( ker  * ) 

 M ker  , if and only *

if M

M

1

1


1

and

ˆ Z  T Z  ( D ) TZ Z 2  AZ Z1  (D2 ),  Z 2 Z 2 1 1

2

1

1

for any X1 , X 2  ((ker * ) ) and Z1 , Z 2  (ker * ) , where M

( ker  * ) 

and M ker  are leaves of the *

distributions (ker * )  and ker * , respectively. Now, we give necessary and sufficient conditions for a generic Riemannian submersion to be totally geodesic. The Riemannian submersion map  is called totally geodesic map if the map  * is parallel with respect to  , i.e.,  * = 0 . A geometric interpretation of a totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths.

Theorem 3.10 Let  : (M , g , F )  ( N , g' ) be a generic Riemannian submersion from a locally product Riemannian manifold ( M , g , F ) onto a Riemannian manifold ( N , g ' ) . only if

 is a totally geodesic map if and

ˆ FV = 0, CTU FV1   U 1 1

1

ˆ W  T W ) = 0, C(TU W  AW V )   ( V V ˆ BX  T CX ) = 0, C(TV BX  ACX V )   ( V V for any U1 ,V1  ( D1 ) , W  ( D2 ) , U  (ker * ) and X  ((ker * )  ) . Proof. For X1 , X 2  ((ker  ) ), since

 is a Riemannian submersion, from (10) we obtain (  )( X 1 , X 2 ) = 0.

For U1 ,V1  ( D1 ) , using (2) and (10) we have

( * )(U1 ,V1 ) =  * ( FU FV1 ). 1

Then from (5) we arrive at

ˆ FV )). ( * )(U1 ,V1 ) =  * ( F (TU FV1   U 1 1

1

Using (11) and (13) in above equation we obtain

ˆ FV   ˆ FV ). ( * )(U1 ,V1 ) =  * (BTU FV1  CTU FV1   U 1 U 1 1

1

1

ˆ FV  (ker ) , we derive Since BTU FV1   U 1 * 1

1

ˆ FV ). ( * )(U1 ,V1 ) =  * (CTU FV1   U 1 1

1

1

99


Then, since

 is a linear isomorphism between (ker * ) and TM , ( * )(U1 ,V1 ) = 0 if and only if ˆ FV = 0. CTU FV1   U 1 1

(20)

1

For U  (ker * ), W  ( D2 ) , using (2), (10) and (11), we have M

M

M

( * )(U ,W ) = U  *W   * (U W ) =  * ( FU FW ) =  * ( FU (W  W )). Then from (6) we arrive at

ˆ W )  F ( A V  T W )). ( * )(U ,W ) =  * ( F (TU W   V W V Using (11) and (13) in above equation we obtain

ˆ W   ˆ W ) ( * )(U , W ) =  * ((BTU W  CTU W )  ( V V

 (BAWV  CAWV )  (TV W  TV W )). Thus ( * )(V ,W ) = 0 if and only if

ˆ W  T W ) = 0. C(TU W  AW V )   ( V V

(21)

On the other hand using (2), (5), (6) and (13) for any V  (ker * ) and X  ((ker * )  ) , we get M

M

M

( * )(V , X ) = V  * X   * (V X ) =  * ( FV FX ) =  * ( FV (BX  CX ))

ˆ BX   ˆ BX =  * (BTV BX  CTV BX   V V

 BACX V  CACX V  TV CX  TV CX ). Thus ( * )(V , X ) = 0 if and only if

ˆ BX  T CX ) = 0. C(TV BX  ACX V )   ( V V

(22)

The result then follows from (20), (21) and (22).

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