ON A STRUCTURE SATISFYING FK--(--)

ON A STRUCTURE SATISFYING FK--(--)

^K*F=O LOVEJOY S. DAS

Departmentof Mathematics Kent StateUniversity

Tuscarawas Campus New Philadelphia,OH 44663

(Received October 23, 1993 and in revised form July 6, 1994)

ABSTRACT. In this paper we shallobtain certain results onthe structure definedby

F(K,

)K+t)

and satisfying F^K

)K+XF

0, where

F

is a nonnull tensor fieldof the type(1,1) Such a structure on an n-dimensional differentiable manifold

M"

has been called

F(K, (-)K+I)

structureofrank

"r",

^{where the}rankofFis constant on

M

andisequalto"r" Inthiscase

M

iscalled an

F(K, )K+I)

^{manifold ThecasewhenKisoddhas}been considered in thispaper

KEY WORDS AND PHRASES.

f-structure,

Integrability Conditions, Conformal Diffeomorphism, Nijenhuis Tensor.

AMS SUBJECT CLASSIFICATION CODE. 53C15.

1. INTRODUCTION.

LetFbe a non zerotensorfieldofthetype(1,1)andof classC onM suchthat[2]

F

^K

)K+IF

^{0 and F}

)’+XF

0 (1 1)

for 1

<

^w

< K,

whereKisafixedpositiveinteger greater than² Thedegreeof the manifold being

K, (K _> 3).

Letus defineoperatorson

M

by:

i de__f(_ )K+I FK-1, r de=f/_ (_)K+IFK-1

(1.2) where

I

denotes theidentityoperatoron

M’.

Thus from

(1.1)

^and

(1.2)

^thefollowing^resultsare obvious

f+=I, i2= i, 2 r.

For

F

satisfying(1.1),^thereexistscomplementarydistributions

-

^and

^)(,/,

correspondingtotheprojection operators

i

and rhrespectively. Nowwestatethe following theorems

[2].

THEOREM (1.1). Wehave

F. F F

and Frh rF 0 (1.3)

THEOREM

(1.2).

^{Letthetensorfield}

F( :/: 0)

^{satisfy (1}

1)

^andlet the operators andthdefined by

(1.2).

^{Then itadmits analmost}productstructure on

,

and null operator

on/17/.

That is

F

k-li i

and

FK-lrh

rhF^K-1 0

(1.4)

ThenF

=1

actson as analmostproductstructure and on

AS/as

anulloperator.

2(--)K+FK-!

Then

Also,

Thus,

tI

^if

^F

^u-

-I

)2K+2F2K-2 )K+I Fn-a

=4(-

^+I-4(-

4F

KF

^K-2

+ ^I 4( ) F

4F^N-

+ I

4F^N-

1,

from

(1.1)

--y.

- ^I

^if

^F

^-1

^{- I,}

and

0.’2

-I

Hence isanalmost productstructure.

2. METRICFOR

F(K, )K+I)

STRUCTURE.

THEOREM (2.1). Let

M

be an

F(K,-(-)z"+)

manifold of degree

K

defined by

F

^K

)K+IF

0and

F )+IF :/:

0for1

<

^w

<

K,and

K

is a fixed positive integer greater than 2, then:

thereexistsa positive definite Riemannian metricgwithrespecttowhich

and/r

areorthogonal andsuchthat:

HH:gts Wfntgt, gs,

where

and therank ofFisodd.

H F-

K-1 and

H

PROOF. Letusconsiderlocal coordinate system in the manifold

M

andlet usdenote the local componentsofthe tensor intheset

{F, i,, H}

^by

.

^{Hereweconsider}r-mutally orthogonalunit vectors

u(a,

b,c, 1,2, 3,

r)

^{in and}

(n r)

mutually orthogonalunitvectors

UA( ^A,B,G,...

^r

+

^1,r

+

²

^,n)

ⁱⁿ

^/r’

(w,, &)

denotesthe inverse matrixof

(u’, u).

Then

w,

^{and are}both components oflinearly independentcovafiant vectors. Let

Ifq5

{a,

m,

9}

thenweput

4

^X / X X

Nowwecanshow that

W, 03

rhP.u. u

^and

^{a( A,}

^{uo) 0 (2 2)}

From Frh=0 we have

Fu’=O

^{and hence,}

Hu

^{=0 As}

^(UA,u,)=O

^{by (21), we get}

g(ua,u,) 0 Thisgivesusthat and/Qareorthogonalwith respecttoganda FromFfft thF=0 wehave

Fm

^O,

Ftwt

^{A O,}

H[wt

^{A O,} (2 3)

By

^virtueof(1 2),wehave

HJ ^{H[ 5}

^m

From(2 4), (2 5)and

F3m

^{t- --0.}

0, 0, weget

(25)

HjH

^gt

+ s,

9j, weobtain (26)

HjH +m 53

Let

Hgst nzt,

then we get

From

(2

6)and

(2 7)

weget

or

H(H,,

^H,)

^o

whichshows that

H

issymmetric.

3. CONFORMALDIFFEOMORPHISM OF

F(K, )K+)

^MANIFOLD.

Let

M"

beaC differentiablemanifold

" ^(M

⁾be the ring ofrealvalueddifferentiable function

on

M"

and

(M ’)

^{bethemoduliof}derivatives of

(M

Then

5E(M"

^{is a Lie}algebraoverthereal numbers and theelements of

Y.(Mn)

arecalledvectorfields

Let

(M ’, 9)

and

(/,

9

)

^betwoRiemannian manifolds and

M" /1 r

bediffeomorphism Let

X 3(M’), X 3(.’)

^bethe vector fields on

M and/1/

respectively

X

correspondsto the

X

inducedby Thendiffeomorphism iscalledconformal diffeomorphism provided thereexists

g(grad

a,X)

X(a)

Inaddition to(3.1)and(3 2)if

Mn .g/

forall X

(M

’) (32)

)K+I

preserves

F(K,

^{structure e}

FX= (FX)

(33)

where

F

andF are(1,1)tensor fields withrespect^to

M

and If

g

be theRiemannian metricin

21/n,

its metric satisfiesthefollowing

9

(F X , FY )

9

(X, Y

), (3 4)

for all

X ,

^{yO in thatis}

^gO

^{restrictedto/, is an}almost product structure withrespectto

F .

^The

Nijenhuis^tensor

N(X, Y)

ofFinM isexpressedasfollows,forall

X,

YE

2((M’)

N(X, Y) [FX, FY] F[FX, Y] FIX, FY] + F[X, Y] (3.5)

Wehave

[3]

[X o, yO] {[X, Y]} (3.6)

By

meansof

(3.3), (3.6)

weget

N(X,y ) {N(X,Y)}

^{for all}

^X,Y (M’), (3.7)

whereN isthe Nijenhuistensorcorrespondingto

F

in

Since isalsoan

F(K, )K+I)

structure manifoldthereforewe candefinecomplementary distribution correspondingtothe projection operators]’andrh. Let

]’

_and

_rh

be the projection operators

in/fz/’

correspondingtothestructure

F(K, )K+x)

which isdefinedasfollows:

def

--((-- )K+I FK-1 ), o ^de=f (I )K+IFK-1)

or,

def

(- )K+I F

^(r-i)

o ^{de.=f [_} (_)K+IF(K-I)

where

I

is the identity operator in

//’.

Now from (1 2),

(3.3)

^and

(3

8), ^it follows that in

F(K, (-)K+l)

^structuremanifold,wehave:

iX (1)K+IF(K-I’x ((-)K+IFK-Ix)-

(ix)

Similarly,

X X (-)K+IFK-I)x

(X- (-)K+IFK-Ix)

(,x)

(3.9)

whichshows that

,

dl preservesthe structure

THEOREM (3.1). If]. _andA’// be the dstnbutionscorrespondingtotheprojectionoperators and dl m

1/"

thenwehave

N (X ^{,Y )-}

{N(X, Y! + N(X,

^,hYI

+

N(dX,

YI +

N(dX, (3 0)

N

(X ,Y

)=

{N(iX, Y)+ N(X,

tY)+

N(tbX, tY)+,tN(X, + N(dX, YI + dnN(dnX, rhY)}

PROOF. Wehave mconsequenceof(3 10)

N(X, Y) [FX, FtY] F[FtX, iY] F[iX, FiY] + F"[X, ir]

(3 ll)

(3 12)

N(X,

^#tY)

[F X, _{FdY] F[F} iX, IY] F[X, FrhY] + F2[X, Fnr]

(3 13)

N(FnX, Y) [FFnX, FY] F[FFnX, Y] F[,’hX, FY] + F"[FnX, Y]

(3 14) N Fn X Fn

Y

F^qtX F Fn Y F

F

FnX Fn Y F FnX F Fn Y

+

F Fn X Fn Y (3 15) Adding(3 12), (3 13),(3 14)and(3 15)we get

N(IX, Y) + N(iX,

dnY)

+ N(FnX, Y) + N(FnX,

dnY)

N(X,Y)

(3 16) Soinconsequenceof(3 7)we get

N"(X,Y ) {N(X, iY) + N(X,

#tY)

+ N(X, IY) + N(X, Y)}’ {N(X,Y)}

Thisprovesthe first

pa

ofthe theorem Theproofof the second

pa

follows from(1 2)

4. INTEGBILICONDITIONSOF

F(K, )K+

STRUCTU

Ifthe distribution in M is integrable then

N(IX, Y)

is exactly the Nijenhuis tensor of

F

THEOM(4.1). For anytwo vectorfieldsXandYwehave

(i) the distribution isintegrablein

M

iffthe distributionL isintegrablein (ii) thedistribution isintegrableinM iffthedistribution isintegrablein

PROOF. We know that the distribution

g

_{is integrable in}

_M

_iff

[ X, Y]

^{0 d the}

distribution is integrable in M iff

i[X, Y]

^0, for y ^two vector fields

X,

Y

X(M )

Hence inview of

(3.6)

^d

(3 7)

d by mes of integrability conditions of

g

_d

_[4]

_{weobtMn the}

proofofthe theorem

(4 1)

(i)and(ii).

THEOM (4.2). The distribution

g

_{and are} both integrable in

M

iff

go

_and

o

^e

integrablein

PROOF. Theproof follows directlywiththe help of(4 l) (i)d(ii)^and(3 10)

THEOM

(4.3).

If the distribution is integrableinM then the almost product stcture definedbyF*

de

oneach integrM manifold of isintegrablein

M

iff the

Mmost

productstcture defined by ^{def F} on each integralmifoldof

{

isintegrablein provided isimegrablein PROOF. Wesuppose thatthe distribution{ _{isintegrable in}

M"

then

F

induces oneach integrM manifold of almostproduct structureifF is

F(K, (-)K+)

^{structure In both thecasesthe}

DEFINITION (4.1). We say that an

F(K, -(- )I. )

structure in

M

endowed with (1,1) tensor field

F

satisfyingF^c

)K/IF

0 isp-partially integrableandthealmostproduct structure F* ^{def F}

-

^isintegrable

THEOREM (4.4). The

F(K, (-),-,1)

^structure/J-partially integrable in

M

iffit is also p-partiallyintegrablein

PROOF. Theprooffollowsin viewolDer(4 1),^Theorems(4 1)(i)and(4 3)

DEFINITION (4.2). We say that

F(K, (-)1+1)

^{structure to be}partially integrableiff it is p-partiallyintegrable and thedistributionof

&

isintegrable

THEOREM (4.5). ^{The structure}

F(K, (-)K+I)

^ISpartially integrable in

M

iff it is so in PROOF. Theproofof the theoremfollows fromDefinition(4 2)and Theorems(4.4)and(4.1) (i).

[]

[2]

[3]

[4]

[5]

REFERENCES

YANO, K. and ISHIHARA, S, Integrability conditions of a structure satisfying

f3+ f

^0,

Quarterly J. ofMaths.,¹⁵(1964),pp.217-22.

DAS, LOVEJOY S., Completeliftofa structure satisfyingF^K

)K+

F O, International Journal

of

MathematicsandMathematical Sciences, 15, 4(1992),803-808.

HICKS, N.

J.,

Notes onDifferential

Geometry,

D. VanNostrand

Company, Inc.,

Princeton, New York(! 969).

GRAY,

ALFRED,

Some examples ^{of almost} Hermitian manifolds, Illinois Jour.

of

^{Math, 10}

(1966),353-366.

FLORENCE,

GOULI-ANDREOU,

On a structure defined by a tensor field

F

of type (1.1) satisfying

f5 f

^0,Tensor N.S.,36(1982),^180-184