49

Jnanabha, Vol. 37, 2007 (Dedicated to Honour Professor S.P Singh on his 701h Birthday)

ON DECO:MPOSITIONS OF PROJECTIVE CURVATURE TENSOR IN CONFORMAL FINSLER SPACE

By

C.K. Mishra and Gautam Lodhi Deapartment of Mathematics and Statistics, Dr. R.M.L. Avadh University

Faizabad-224001, Uttar Pradesh, India

E-mai[ :chayankumarmishraca.yahoo.com, lodhi_gautam(g:Tediffmail.com

1 ReceiL·ed : April 20, 2007 j

ABSTRACT M.S. Knebelman [l] has developed conformal geometry or generalised metric

spaces. The projective tensor and curvature tensors in conformal Finisler space

were discuS-ssed by Mishra 1J2][3]). The decomposition ofrecurrent curvature tensor

1n an space submetric class were discussed by M. Gamma [6]. The

decomposition

Sinha and Singh

recurrent curvature tensor in Finsler manifold was studied by

Singh and Garnto [9] have also studied the decomposition of

cun-ature ten:...;;or recurrent conformal Finsler space. The purpose of the present

paper is to decompose the Projective curvature tensor and study the identities

satisfied by projective curvature tensor in conformal Finsler space.

2000 Mathematics Subject Classification : 53B40

Keywords : Finler space, Projective curvature tens, Conformal Finsler Space.

1. Introduction. Let us considered two distinct metric functions F(x,i)

and F(x,i) be defined over an n-dimentional space F~, both of which satiesfy the

requisite conditiions a Finsler space. Th ec.orresponding two metric tensor and

gJJ(x,.i:) and g'Y resulting these functions are called conformal. If there

exists a factor of proportionality between two metric tensors, Knebelman has proved

that the factor of proportionality between them is at most a point function. Thus

we have

where

g'J (x,x) = e-2" giJ (x, x)

gJJ (x,i:) = e-zcr g'1 (x, x)

cr = cr(x)

F(x,i:) = e" F(x,i:)

... (1)

... (2)

... (3)

... (4)

50

The space equiped with quantities F(x,i) and g(x,x) etc is called a conformal

Finsler space, it is denoted by F" .

B ij ( . ) - 1 p2 ij . i . j XX-- g -XX

' 2 ' ... !5

where BiJ are homogeneous of the second degree in there directional arguements.

The following geometric entities of the conformal Finsler space are given

by [7] and (8].

(; ·(x,x)= G'(x,x)-amg"'(x,x) ... (6)

G_jk(x,i) = Gj;,(x,x)- ;Bim (x,i: }um'

-, ( . ). i ( . ' Gjklnx,x = Gjhh _x,x)- ,Bi"'(x,:i.:pm J ·.

B''( . . ). lF2 ;,- _,.; ., x, x = - g - - x· x· . 2 '

... (8)

where G5kh(x,i:) are the Berwald's connection coefficients. They satisfy

i ,c;,(x,x; = G)k·

and the functions BY are homogeneous of the degree in three direcitonal

arguments.

2. Identities Satisfied by the Conformally Changed Projective

Curvature Tensor. The tensor l1T/ and w~.'. transform under the conformal n :r.:.~

change as follow [2].

W;=W;- 112Bim_(.;-,Bim\ .,. __ l_51!0BP"'_l~BP"'') -r~_ xr h h cr"'L h on hrlx n-1 hf' (pl \Gp lrix J n2-l

-~ ·J 1\1:.. B pm ) ( ' 1\(:.. B pm ~~n- NP (h)- n,- J\Ch . ·?(. ?)Br"'GP (' ?)''h "Bpm

""7""'....., .n -- rph - ,n- ....... x {Pl h

( 1 ') ' . · r J ~ Bini ,...i .; B prn n - -.. · t.; ;,,, Born ~ +a (·)x 1oh ---ohc · --n--xc'o · ·-

m i l . n -1 - P nL. -1 n P ?Bw' 2n-1 ....,; - -··.,,·····-

n"'-1

{

. 2 . 2 • · i-': pm i pm X :.. nm . L sm :.. x cPB +am(p) --6hB ---chB· ~a,,,art2B c

n-l n-l ~Bir _ (t" ,Bsm b Bi" .. . le f s

') 1

- 118~ ~Bsma_isBpr -l}"Bsm f ,Bpr i+ ~i:sl {(n+ l)(8lpBsm ~hl(JsBpr n- - n -

...;._ c E,H"r ;], ... ( 11)

= lV£1z - - nx~l ~ [? i =-: rm 'pp ) , 1 -i I +\c,kB 1 , ~-r--2-o1-~i(ri+l) - ' 1,rp J n -1 ,, -

:cpBpr: 9B'"'Gv "i 2 + ,... idrp jj+ CT nd c-Bim n

hi - -'} n- -1

1EDBpm_ XS /31;_-fnBpm ' - n+l - ,_,

2 + ---- CT _; ( -n2 -1 m(p)6fk·tiPchJ(JrBrm -

+ 2G ::-:-Cir ,"'- -Bs"' :.. __ :.Bir -~a· :.. :.. BP,- l _l_;;;i f(c-- Bsm _ _ c .. c. h'c O. ? v 1-\ hi " "' , - 1 ' P , I 1 " l, n+ ,n-

2 a BPT p ·s -_;;. BP''t: Bs"' +~Bsm:.. __ :.. °: Bpr tc' - --° - n -1 c "'c Pc-' ... (12)

RB. Mishra have introduced to obtain the conformal transformation or

projective curvature tensor 1-17) 1d, by differentiating ( 12) with respect to

:i::' ;__ ----r

n+l -.l ~Q PB pm

-c \C_;,B'-"

i;Bp· t;:~p·

Bpm p

6' n~l-Eh (EPBP"'

-:-~>~ n-;-l'

,;;.. ~ch,Bpnt

5s J

n+l

. /· pm) - nc j \C PB •(h)J

-c\i{c:pBP"' ~J+ 2o-m1k hJ\gm) 2 _, noikf (-;, '"'Bpm-) (-a.'"' BPm)}

n2-ll°.m(h)JC/)p -O"m(p) johl _

-"" 2o-ma,[2J 1~ -t,B"' - n-: 1 {c-);!k (a PBsm ~ h/3,Bpr +al!, (a PB""~ j (ah 1E,B1" )

+a_ .Bsm')S . .? '-- Bpr +(\-Bsrn)3 ~,,("':.. Bpr)l- Sj {"' (;, Bsm\;., .tJ Bpr( " . r:t~Pc, - ctr. - /·n;.cpos _J n+l op_clh Pki s j

52

!i' + :z-~\ {(a )31i1Bsm ~ pasBpr -(3 ):PBsm ~hl3sB_vr-'- (3;,:B'·n \i_,C;pasBpr

_, ( ~ sm \~ ~ ; pr l ' oik - cPB f)JOhJcsB )-i--2--

n -1

. (" Bsm \;.;., :., ~ _Bpr {;., B-'il'' + c _1 Y h !f pc ~ - I,( p

.I~ Bsm b :., Bpr - "- I" B,"' '1::: , .. C, 1--r;Cs C 1: \C_,, L SB pr

r :C\Bpr <8lli k J lf;,ftBsm f p<",Bpi­n" -1

- " '> (" B'"' 1" " Bpr C ;C;, ! CD P1Cs + !•;., B"" \'- . ;., .:, Bpr _ ;., J '; Bsm \:., _:., .:, Bpr Cl ;:'JC_,,c, C;,1\C p O_,C1C5

.:, ("' Bsm _b (" ;., Bpr )- ;., (:., Bsm +cJ c1 l-'hJ cpcs _ c_;pp

_ ('"' Bsm \; .:, ;., .:, B pr op PJOhJOzOs

X U!k f :., '; '; pm 2 -1,,i (

+-9--'(0m(i)( j (hi( PB n- -1

26/c \"'·

n 2 -1

CTm(p{/';,

I"':., BP (.:,BS!" 13 .c!cs P: .~ c/·sBpr

c,_.!(; PBP'" (a ,Bpm )j

Multiply (13) by g1"" we get

-w:ikhgi11 = e1(T giu wjkh + 2e 2'7 umgiu [alk B'r D.~' - ·m

1 ~i -3 . (c. · B' )(h 1: - n-'- 1 ' ./., !!?. '

151 r- (" ) r - '\ J ':"\ -.. pm -. vr m +--v;P o1kB 1111 -\c1kB· )S\,_r n+ 1 \- , I • ' ' "!-'

61 :;;

n2 -1

+a, ,la Bpm 1, \ J -3h1k Bpm

- p ,_ r ~ i: -c ~c

f -i - (• ·. ') x ;., 1:. ...... ...., !,n ____ r·l(

+2e2ag;u(crmlk(hicJB · n-rlcr PBp-rr:: S'.:, c:;)t

n-.l

+ n(~,h I . ' . u (~ -n 2 - 1 I m(h II c c BP!H . J J1 a 12 .? BP"'

ndp)\ J.._,·hi ') ·Jc- f.:, ~e- -g,,,um CY,- le.;

·I .

BPf7' ~;:;

(13)

X ):., :., (°' Bsm\:., "Bpr. '.:') /:., Bsm'i:., ;., Bpr '- {:. Bsm ---1 )('J·clk cp FhJc, +ctk\,cp r- i ics - c; ,CrJ; n+ \- . . - . - ,c,pf;sBpr

+ (1"'1hBsm }"/1i1k/"sBpr )}- nbl l -~ p((\hB;r" };. _,BP,. n;s;.

~R --)-

n- -1

-(3/\,Bsm ~h 1c\Bpr +(8hiB"" ~)':j3,Bpr -v~PB'm ~/:1z/3,Bpr

;. Bsm \.:., ;. BP' _;Ch] r pc s

·I

~2;!k ~hl(tJ.Bsm ~pf;sBpr -CJh1(E

0B'112 0 .3

5Bpr -'-((; ,Bsm E~,c .. BP'-

n -1 \- · · · · J · -·

-(3Pgmi ~[ha jasBpr }+ x~iS[kl [a )ahiaiBsm f /sBpr -313h1(8 PBsm ~J:'5BP,. n -

where

B

I.

~ ~;: G_,::lr ~L

f

P)a B~,. -c, le PB-'" f,

BP' 12 B"' ~

:;C; l.~.B-'"'" f-u

(:, .:. BP'" c~,c ., · X G "' ~ ,,

w =g,,,Hr'

We have the follov.ing indentities.

;., Bor . ;., (-;., Bsm le,,' -:--C 1 C1

!:, :, B",.) (\(_'p{ .-..: ,- .

:, "Bor (" Bsm\;.,:.,:., ;C pcs · - op P/'1z:C B pr l]

' I

,((:.BPm'j1. . J I

! ;:_BP"' l'l] .,l ! i

53

(14)

(15)

Theorem 1. V.7hen F., and F.. are in conformal correspondence, we have

J' -20 :..~\'iBzni -( .(c:, Bim

cJ' k

. x' r. , ..,..----:-nl 'r'i1c"1 !(" BP"',)

T \ L ''· p 'Ud]

.i· ' k'.Bprr J t'' }ij}

0 --.-_---- (

-_i

:, Bpm ). -(" (/: BP"' . ihl] P · h I

-\t Bpr bm ' h I hipr

!:, Bmi: .) :, ('-· pm). :, (:, ""' i/1c,\c1,, · ,, .-ne;,,c1,B 11 +c;, 1c 1

,B· j ' l I '~ n l f j < ] J • f

-i" . .:. (" B"'m X C;JCI i _c 1 '

·.i -1 1 (" Bpm 1 -x c 1jc 111 op

b:1i n 2 -1

,la -Bpm _r,' fi.

-n + tJt, Bi"" +( 12 Bl"'' 1 ... \ p /;. Bpm

-1 r).

- w!: 1

-20. ~c ~<" B' --( ;,e: n

2 .; c\.(8 Bpm) ---1 "'' p ' n+l

5; ,;_f ; , [/., I .

-n-.,-'­n- -1

-2G -'- .:. .:. (.:. B pm ) J ' ;., I'- B pm ) --( ' ( ----( (

1 J -~ p ' 2 1 kl p -n_;_i n -

- :. Ir" J ,Bu" 11T1 Jc

.--! -i . • '). i · (. · ) Of/ · (· 'j 'j_ __ J_' _ "-. - B pnt _ __, _ -. . -: B pnt . ·) • ,_ 1 r n ·) 1 ,1 1 v n--1" } · n--1 _,,. · ·

n 51 J, · I· + /na . __ ,,_. - -: i-: Bpm ~ m\p) ·) C li!lC h

8{1, . ~Ci· n- -1 ._1

)l [ ( \ ' pn1 ; ; sni .=... .; tr

1B .12ama,. cu 011B .k31i1c8 B -

+

'n- -1 · ·

;(' {, . . . '. '' - . - (', }3 . -- snr -: .-., pr -._ --... snl -.,. -.. -.. nr ..... sn? -.. ..... ""' pr k cPB c;, 1c,B -c:h\.cpB ~'jici.c,B' +81j ,ohB . h(Jpo,B

n+l

.B'.111 ~h }

((: gor '.\ p

;s: ;J !;..,,

-.,- ·c n- -l ! _:;

,B51" }3._a Bpr -8 (a Bsr" \~- a Bpr l

, • 1. s p k Pnl s J

54

,, - ~t1z

1 {n3.J 1akBsm ~i.,Bpr -ni)1\?PB"m}?

n -Bpr-"- r " "BP,. · ,:( n( s

- n(/3 J!E1:B·"" fJ PE-'Bpr + c\ (E ;;B0''' f,i,Bv -3 )C:PB''" f j/~,.BP"

+tJJIBsm(Jk(J/3sBpr -(EPBsm ~kC:j:EsBpr -i1Ej . . ! :., :., (:., B-"m Y:., :., B pr -r-xcjiokcp .'1PiC.,. :/ 2)3,H'"' ~ .

"" \ ' .t ) i

.[:.. (" Bsm\:.. :. :... Bpr -+ x ('j] .cz r 1<1 ·pc, (/ ~Bsn

!" :/(:..

. (."' B-',,, \; :, :, "' BP,. -X GP Y--'_;iC1:C1C, ie B'm •, p

(': B"m ,c; c Bpr) p s

B pr i- ·) ·\ ("' Bsm . .X C .. CP

.;.-'i_:...Bsm \:. ~... 1.{ :' r -pt'sBpr

t-_Bv )- n\i f, ;,Bsm ~ ,/3PC: _Bpr . . ! •• _; ~ s

- "; (':., B-'171 \; '! Bpr _I:.. B'm Clh CP Y::.11Cs \CiJ ,C:JBF--iE B f

' F ~ "BP"

x1811z(apB"" \:0i"a1asBP,. -x1a, )3,,,la:B""' }3, c Bp· -.1:. (: 1c: fJ JJ l-< ~n" . ,

I o~ .:,_ • . ---0 I

T 2cr ml u) I n 2 -1 h.

l

BP"' p

81 [ /; .

n2 .

1ck1,a_ B-°"'

- p

·) 5;, -'-~~CT•:, n- -1 ""Pich

; . ;, ~ . JP~ :j1 k::: tr c B· l'J ,.. • ... ! s ,J

BP"' ji

8' ·2·/ I/· i - - B''" '1 - ,_ -B"°" 11 16 - x -_,-' - /CT,,,,,-,c ,[ ; (I) !- Ci, I c \C ,,.C' rl ..J' "' l ' \ ' J < , f ,._; n -Proof. Interchange the indices k and h in (13) a..n.d subtracting the equation thus

obtained from (13) and using the symmetric property of the

the result (16).

, we get

Theorem 2. When F,, and F,, are in conformal correspondence, we have

W -W = e20 (W -W . )+ 9e20 c; a, :. -B'" V::.,. - :. · B'" ;ukh .1hlw .1uklz .Jkiw · ~ mb1[:L_ r. IJ.~;_-.r ::.

-a (a B,-m J. k

' :.., (.; . ,c. c ,B'm J \ h1

j:' I·

,- n-l (Jck!(;PBPm c J

(Ci Bpm .P

?/-+ J L {· n2 -1 'PP \CkBpm ~ ("' -c/J .ch1Bpm ~ Bpr \nm , .; BP' 'Y,.m

ck .J .. JhJpr7"Ch] ,1-Ikpr

+~l.;/· n2 -l ('10 j\DpBpm l] - n\a (;.,BPI" J f2j

- c. , I(; -Bpm 1'1J' ·_;

+a"'a Bpm ni\ p

.[;.., {;,, ("' Bpm \ -l; .; ("' Bpm -x oi\uh] Ot ~p)+x ojohl .op

-z 0 hJ f .:, --ino· n 2 -l · .1

; (; Bpm\ '.:, {;,, Bpm') :. ('- Bpm - no j\o P l(k) 'o k \u i 1(p l] - ck ,c P

T·

+x'c

. / "' ;., (a·. Bpm) l+ 9 2cr r . l" ( . .; Bim +x ojok P (l)f, ,,_,e gi[ulC>m(h)(hJ c; x :..

--(;

n+l ..

tBpm

(a BPm) . p i

55

8'_ - J :.. i-

n2_1 ci,;lc~ ;;BP"' : :. (·:... Bom )l J:... I;" Bim .)

n-_:2 lcj cp . 1-CTm(h)ff"h cj n -

x' - . - n-'-l 8i"k(a BP"' , .. ·. ,D

5' __ J_

·? n- -1

(:... BP"'-)-n~ :.(°i Bpm cP . 2

1c 1 ,c P

n -8/1 n -­? n- -1

G

-aj

'

-c'

~ .:..._ 1 -. !.' \CI- IBpln

I.:. r:. 1ino'", ·) [· (-. \; . -

!~ +2e-"gilu0mCT,. Ci t,.Bs.r:1 PhJcsB1r n~ -1 · · · ' ' ·

B ir x' };'.: ~ 1'.:.: Bsm b ;,, Bpr ;_, :... /:...- Bsm \o, ;,, BPT s - n+l ~ jck cp Fhlt.s -cjchl\cp f.Jkcs

{8 Bsm .p ) \ ' )" ( ( ;_, pr ;_, :. sm :. . :. :... pr :. · sm .:, · · pr

1csB -ch1(cPB ~)c11osB +cj 8 11 B ~hlapasB

Bsm'1:..:.:. Bor _ {:. Bsm\:... ;_, ;_, /.:.. BPr) (;_, Bsm\:.:.:... (:... Bpr) l ch:cpc, · +;ck P_iC/1]cp\cs 1 - ,ch] pjchCp\Cs ,

+ n5~ l -~~P I.:._ ;: Bpr :. (';_, Bsm);,, ;_, Bpr l+ 8/, f ;, {:... Bsm \.:. :... Bpr C.1z0s -cp,ch Fhlcs 5 ~l lnOj\OhJ ppos

n -

-nc Jc _Bsm vr- . ;.,, ; szn ; .; vr ; sm · ;.,, .:... pr ( \ (" \

0 B· +nch]cpB pjosB· -ncPB ~1chJcsB k'- " B :;,- .:.. 1·.:.. B"'m b cpc.s . -ch] .cp r': sBpr +(8JB"m ~hlhpasBpr

,Ev +x1C':ih 118zBsm f pc\Bpr -i/8 f:. Bsm b,-': Bpr ,C p f- 10 5

. -}.:.. II:. Bsm\.:,:; ;_, Bpr . .z;_, [;_, Bsmb.:,;,, Bpr, -£;,I"' Bsmk:; :. :... Bpr -r-..l.chJ\\OJ P/pCs -xohJ,Op f-!jOzu 8 ,-xcjpl FhJ°pOs

(: )c~ PBsm ,lt:h18/3sBpr + x1(azBsm )3 /3h]a pasBpr - xi (epBsm ~ )h/3/3sBpr f

oh] . .

r.:... Bsm ~ :: Bpr - ;_, I.:.. Bsm f J en pc s nc J ,c p B pr-'- ~ fc:... B"m 'i::,. l Bpr

s . no k.. P f-J Jc s 2 1 n -j_

- j:l C:

,,BS!" P

i(C:JB'9n

,Bpr-'- 12 ,Bsm )~ ,.,f,BlJr -dk(aPBsm ·_\:...Ji70Bpr

- .J . j-1 •. , \ p oJ

BV _/[: ::tBsm ;., BP' , ·; ;., (:. Bsm \; ;_, Bpr Jes -r-XG jok Pl p pas

(.:, Bsm b .:.. Bvr \cp ,.c)zCs . - h Bsm \;., :... .:. Bpr ._1;., (;., Bsm \;_, ;_, ; Bpr

1 Pjopos -x Ok _op pjo1c8

· l .:... .:.... sm :.. :.... .:... .Dr .:... . .:... :c;m ..:... .:.,, :... pr · l · sm ~ .:... .:.... ; pr j ' .. ( ~ + x c JczB ~he pc,B· - c)c PB ,0hciosB + x a1B ' joke pcsB

-xl(aPBsm ·13 )3hf318cBPr ·, f .! ·~ ~ v

9~

2e-v gi[u 8k - . ~nahJ(a Bpm -1 p

8/ii :... . l 2--ch(a Bpm)l n -1 . P J

+ -i ~i 'l f 8i . ; bk ~ .:.. pm b h l :. ;., pm , -I k :... :... ;_, pm

- 2 -c1:1(c jB )--2 -ck(ojB ), --r-X CTm(l) - 2 -ojohl(oPB ) n -1 n -l i ! n -l

) \.

_ bh] B-Bk(BPBpm)~-X1Gm(p) 2 1 J ' . n -

8i ;., 3'i ('; Bpm)' 8~1 :... ; la- BPm)j(l7) -2-CJUh] Gt +-2-ojok~ l . n -l n -1

'i6

Proof. bterchange the indices u and h in 14) and subtracting the equation thus obtained to (14), we obtain the identity (17 .

Theorem 3. When F,, and F11

are in conformal correspondence, we have

W . - W = e20 (w + W , , ;11/w u1k/1 111/;h !{lilt::

O' . x :_ " I'- B 4- b

1:!.~-- c t ,t '1 ;, \{ !' n + 1 ' ,.. ·

'),.-, 4e-·'c;n,

g 0. -~ il ~· (p

B ir hm -1-I1i: -(

B p1" I --• 1Ui.fj

r:B""

\nm PhJpr-

i) l ! . gi1u lk ;nc J·

- n2 -1 . ;Bpm I l-nt

1,i6MBP"'

' ·~P - , f-'

(e; }Bpm \ f

((::BP"' •P

'-, LC .B '- ~B CT [-g J:_ /:_

1c :f-- 1'-- ! \(

.1:{ . ,1"- BPlll I l'/ i. \C' Bpm -c' n+l JI

l p ; ·)_._· (~ n- -1

iZ0B n ·)

n- -1 " p

- 0 m( iV~ )'J ,BP"' p . }· h, 2G ~ 4e CJmurgi!u ci.i

" ,

. " BET - x' "' Jes . 1 a

n-1

(·,'- B'',,,k",," BP!' -(\.I:- B",,,k" 17: '- j) ' I} i( -" if,· \t p ~' ',i..

(" B"111 \:._ " - " B'n- 5·,. -+cik t'j1Ch(pt.,' - 2. ,t.J

n -J.

:., :,_ Bpr -'1 ("i B"m \:_ :., Bpr . £P£" -ncj1Cp 'rids ..,- B''"!2 f.

C- Bw (

c

BP' - PBsm ~ iC Bpr I . s I

. :,_ (." B"n' \:._ " Bpr :,_ f,:, B"'" \'- ,c: - ( h ! ( J i r ;} s - i h ; ".' p r .' l. B .••r

. - . H,,., ~hie B;;r s

( :., B"m \:, :., :., Bur .f:_ '- I'- B"m \:., '- Bpr .:, '- (.'- B''m ~cp ,_.h!rj_;(_., · +xcJ!Ch\_c1 F'pCs -c_;Jch.],cp _,BP''

, ~ ~ sn1 .:.., ~ ~ pr ~ ~ sni ~ .:... ~ pr , .:..., .:... sm .,~ . ~ ( ' ( ' . . .

.,-(fzi.l/B fj,cprsB -chJ,cpB ~jiC/vsB -:-Cj,(clB ,th;/!- _,Bpr

-c:,_j;kpB""12 }\1c"'1c\Bpr + (l'1B"111 ~ hjC B pr (::.· s . - c ~, .:, :_ pr )ij1

lclcsB ,J

+ 4e20 g(u011, [J n -1 Pm(j

(C; Bpm , p crm(p)ch;

Bpm jl f ;J

.;.. ~I I :,_ . ;, 'l sm '_ . j::o . .:, , . :., pm 1 18) . ' 'l] . x Pm(/lc.1J. 1z1c. PB. J CTmlp),,.. J,(onJo!B Ji,.

Proof. Interchanging the indices in each pair (j,u) and ik,hJ in equation 14J and

adding the equations thus obtained from (14), we obtain the 18).

Theorem 4. When F11 and Fn are in conformal correspondence, we have

wiuhh + wjlzlw + wldzju + VV/wjh = e20

(wjhlw + wjukh + whizih + 4 20 g e ()m iw

57

~ -B'rhm ( (~ µhi -c B1m ·/ x

_ -'-(:, ;(C:h1Bim )(kJ] + n + l )) - l j

- (· pm) i' k ! C' PB (h))

i' -- n-"-l~C[J(:h,((: Bpm . L ... \ P

8, - f - ( • ) - ( - - \ -'- n ~ 1 ·p P Ck1Bpm (h))-cli ,~il!Bim l(h))

-Z ,It. Bim _-(' . -i r -

(-'- Bprn) X ;._ ;, (" Bpm \ ' ,- 11 ' 11- n .,..1 ·•c: hiF p -(h)) - n + 1 (':/ hl cP l(k)}

6: - _·-' ~

n ~ ,_. ~ . Bpm

. -1 --

;:.. (0-: Bpm ') i; BP,. ·-\r.111 , (" Bpr \r.m 1 1)-c.p,Lh• -(k)j-\Ckl f.-Ih1pr' c,,) Pldpr)

s; , 1 -1-. ) - (- ) - {- ) tR ;_ -: :::: pnz : ...., pin , ....., ~ pn1

-~1 fCJJchJB (p)-ncJJ oPB (h)).,-oh 1 cJ 1B (p) n -

(,? ~Bpm !-'

(iBPm \ ;-·,

~B";;;m r

n) --. -. pm ~ ; pm 8'- i· - (• ' - . - ) n 2 _1 fC[j cklB )(p) - norJ{c PB (k)]

1 _ t:-:r JC: Bprn

' ;,'• \, p

' :C n 2 - l au(i.i1Bpm

- i x' -

. I -x' 8' lh ;, f- (. n 2 -1 C'Jlfh1 ftBpm \ }(p)

-C:rklaPBpm ~ 51

4 ·)" [ 1:.. (; Bim -:- e- gi111 Cim!k'(C°h!\Cj] -a (; -

n + 1 j I 0 h /!' B pm p J ----c'-

n Tl lli B pm )'.

p )

- Gndh}}

"}

n'" -1

- - - i -i _ (·-': --Bim )-'-_X_; .(-=: ,-1 Bpm \, ~ -':, !.? Bpm 1- ( , 1 • C[ I ( k ,( n f+ CI• I,, n " ' .1 ' · n + 1 - · · ' ,, -· n -:- l ·· · ic

r.,

n.1Bp.u

B pr. p

n 2 -l

n 2 -1 p

BP"')'

B pm ') ("' ;, Bpm )l p ,-CTm(p)OlJCk] j

2rr r;, {;, sm '}:.. ;, ir ;, (;, sm \; - ir ~4e c;m0.rgiiulcu\cklB Ph!csB -G[j\OhiB PklosB

-1 x

n +1 i'1)C PBsm }:-. BP' _ ~ _;, . (-=: Bsm \; · ::i Bpr

•0 s clJchicp Phlcs

.:.... .:.., ..:.., sni .:.., .:..., pr . .:.. ; sm · .:.. ;, pr · ~ sin , , }3 {. ) (' ) -cijch1\opB FkJcsB +c!k(oPB JJ PhJosB -c1i 1 cPB ~ 'i ..'.... pr i ..:... .:.. sm .; · · pr ; · sni :... .:._ :.. pr . ) (. ' ( p -cLi(,ck!csB -r-c[j PklB phJoposB -cfJ akJB "hJaposB

{" Bsm \; :.. (" ;, Bpr .) (" Bsm \; 0- (:., " BP,. )1 - erk p j]Ohi 0 pas +Ph) PrJ kJ\O pos f

. .J J:.. (" Bsm \;: ;, Bpr :.. (" Bsm \;, 2- Bpr l -:- n+l f'p ch, fk1°s -cp cki F'h1 s J

'58

. c)l ii J /.:., '; sm \; '; pr 1.:., .:., .-;m ~ , pr

.-.,--in\cilch!B PpcsB -n(cd-:JJB r111 csB n~-1 · · - ·

. (~ Bsm \;, .:., .:., Bpr (.:., Bsm \::. . .:., .:., BP,. , n c 111 r: j]c pas - n\C 0 Fpc hies . . . . :.. ~ sni .:., pr .; ~ srn ,;.., ;., pr , ;.,, sm ~ .:., ..:.._ pr f (.· .f ! •

-r-(1iJ 11 B pcsB -chicpB . ncsB -,.-(cJ 1B ~h!cpc~B

(.:., Bs111 \.:., .:., .:., Bpr - cl' K-'1i 1Cilcs ')

n- -1 ·Bsm f .:., Bpr ! . pcs

f.:., ':i Bom 1-1 .:., B pr , B"m t :... ;, B pr (" B:;m \ -npuc,; Ph,cs T h . Jlcpc". -n.,cp I

: .... ".'.'l pr ""\ ......, srn --; -. pr , ~ -:- srn ....., .... or -: snz -... """' -. pr • • • • (. • • • • (" ! • • i. • . . .

c[jch]csB +clh cj 1B ~pcsB -rC[k.CpB ,BJlcsB- +\cuB ~-k)cpcsB

-(.:., Bsm \.:., .:., "?, Bpr cv Prhr,,~,

. ! < x·o,;, !.:., __ ,. tc 2 1 .. n -

.atBsm f :')c,BP' - c Bsni i":i Bpr 0 ,_ s

-r th. ;(("1-B'm f ;,/:;Pa _Bpr -

'' , J j .'}

;C ,BJ)' ..;.. c ,-B 0m fh,fc ;/,B

"""' -. Si1'1 ·-. t .-.. ..... ;Jr . l.. '- I - . -cj]cpB Ph!lclcsB· f c B .!:c i:' B·'"' i:'::. ;:; ' ' B s -,cp ,Ll]L-h..JCJC.s

ih5'· J.:., . - \.:, . - --~ru(c;d171Bsm ppcsBpr -cr;3k1\80Bsm

n- -1 · · -J • •

. ~ ;; - L --. B pr . ~ I--. Bsm 15 cs ...,. c[h ,_c i .. Pc,Bpr

..... -. snz ......, .... -. vr .-.. sm -.. ....., -.. or . (. ·~ - - . ~'- . . . -c1kcpB ,111 o 1c 8 B· +oij B , .l.?JlcpcsB-

.:., (''.:., Bsm \; (.:., .:., Bor ·J. C[j _Gp Pl?J.OfCs . I

B sm b .:., .:., .:, Bpr (" Bsm b .:., .:., .:, Bp·· Fuc ldr Po s - c P 1-·uo k;c ic.-; .7,

4e~0 g, _ n2' - I

-Gm(p)ah) (a j]Bpm )+ x1 CT m(l)a j)(ahl(; PB pm

51 , ( . (· ) h1 l "· .., om - -"-- )<J ml 1·)C h CJ /JB · - CJ mi_p n~ -1 · ·· · '

' 18 -Bpm \ .. n

, . . 1· )1 x18!, - I_ .. · J -, .., --, pm . i. f, --. J-, - ~ DT'1

-x CT ( 101 ch' c1B 1--.--c,(·h:lc1B' mp, .J ! .. - .· n2 -1 .J ... , , -

(7 . ~ .m~ C'

~ '._/,_.,

-en

IE:BP·T..

_B 7 '' l

((!)Bpm) tl - l(!j iJ

. 2rr .:., (:., im \ x' .:., +2e g;,Gmlk ch] ojB )---.. cj

n + .i

'j 51, - .. B pm - __ .! - --. • • I --- B pm p , n -i- l C n1 .C p

2n8fk 2 i (. . ·) 1-. . ., ) • --' - () . < • pm - . , ..... ""'' - pnl r ' . :..c r-

T 2 le g1u'(fm(h)]OJ8PB . c;m\pi\C;CnJB !1'2e giuGmur n -

•l . , . . .

:., .:., Bir X )°' .:, {.:., Bsm \:., .:., Bpr , -'-. ('.:, Bsm b {;, .:., Bpr ') Oh]Cs - n+l 'f jO[h\cp PhJOs TC[k _op f-'j\Oh]Cs

( . . ~ ( , · ~ sn1. · ~ ~ pr ; sm ; ..:.., ; .:.... pr -r-8jPlkB f1zJcposB -(clkB jchJ,CposB

8' . l I· -- y: !"' ~~

n 1 ,~ v .C·. B·""

T ' ~ · _ f1

(8,,.Bsm) ' ,._f~ i

Bpr}

· 8 lh ~nfa -8 Bsm \:., a Bpr - n(ao:., Bsm b a Bpr ~ nla, .Bsm T 2 1 ( ~ J h] pp s . J p Fhl s , \ f1•

n -cpcsBpr

59

_B-~m \:.. ~u ,t'

. '-. _Bpr _ ):.. _ .f '- -Bsm \;., ;., Bpr _ ', _ _{:... Bsm \;.,.:.. Bpr _,_ ics '/Cn;·Cj 1t'pos ohj\cp P/'s '

./.,;

Fe Bsm \;., :.. ::-: Bpr (:.. Bsm \;., ;., :.. Bpr I, X b[k L, (" ~ Bsm \;., ;., Bpr -,c P;,icpcs -,cp Prhcjos 1+--z-ltc'J ch1cz Ppc,

(.:.. Bsrn '1:, :.. B pr j C p LfC, . -

" 1·:.. Bsm \;., I :.. ::. B or - c Ct l'."'hl ,_cpc, .

-l8_Bsm \;., , p F

+ _-i;i r. V"-"J

;C:c,BP-'

~Bpm ,...

n -I'- Bsmh;., ;., Bpr ;., (;., Bsmk;.,;., Bpr

•iCJ CjOpC,_. -Oh] Cp 1CjCtCs

:.. I'- Bsm \::_ (;.,;., Bpr) (;., Bsm\::i::. ;., ;., Bpr c·J ,cp ,ch]\c1c_, + c 1 t- Jehl° pcs

gm [k :.. ;., pm ;., ;., pm , 2e2G . bi rr .. ')

fJm(JPhJ(oPB )-crm(p)Oh](ojB )J

"' "' (ia. Bpm)l] crmip)C jch], I f

. .. (19)

Proof. The proof follows in consequence of (14) anad (17). 3. Decomposition of Conformal Projective Curvature Tensor. We

considered the decomposition of conformal projective curvature tensor in the form

=.)Co (20)

\vhere ~ is a homogeneous conformal decomposition tensor and Xi lS a non

zoro •;ector such

X''Vi = l (21)

Similar manner the decomosition of projective curvature tensor WJkh in

the form

l-V)1c1i =Xi qi Jkh

where the decomposition Yecto: ,yi also satisfies the condition

X'\~ = 1

Transvecting (22) by :e, >ve get

where

=X'¢!?h

.t.. . i == -'+' 'hbx­J ...

The decomposition tensor $ ikh satisfies the identity

=

(22)

(23)

(24)

(25}

(26)

We notice that the decomposition vector xi and the recurrence vector v;

60

are trans£'ormed conformally as under :

J(i =e-0X1 (27

and

V =e"V l I

{28l

respectively.

Using equation ( 20 and 22 l in equation ( 13), the obtained equation

transvercted by ~ and using the equation (21), 23), {27) and (28), •ve get

:+: - G ~ ' v 2 f;, Blr 'l'jkh -e 'Vjkh T ; amL01k

;, \i - --o, a .. B'"' _; -.A ---~~ n -'-l ·

1. BDm \ap · i1;,i]

81- f

_} i;, i' -f.J(31k(3111Bpm ti n + 1 '(-'p\_O\kBP"' )} - pr G"~ B .h;pr n 2 -1

f;, 1.ohlBpm

;.,_ (" Bpm) ;., (; Bpm -no.ioP (h)]+chlloj -ah la BP"'

t ., p - i'5[k ? -

9 '-' i n- -1 , (81Bpm

- (} h l (/3 p B pm )(!) }] + V; 20 ml k (: h i (C: j B im y'

n .. 1 a I (c\ .(1 BP112')

-,- • , .. I p

' l '

8j ;, (" pm )I Vin6f 1c ---c,·O B r+---n + 1 · ii P · J n 2 -1

,C DBpn1 f-G J •

' v I;, ( • Bsm \:,, :.._ Bir '• x' _\;., - Bsn: L'.: - B [J.!' -r i20m0,.10;8r1c P111Cs -.---lie C"<; ;,• L-C. , L . n- .

;., ;,, (';, Bsm );, ;, (;, Bsm \;:; ;, (.;, Bsm ~ ;:; B_:;r ;., B·""' \;:._ [" ;., Bpr j'

-cju[k,Op pjo[k\Op f"jo[k,Op 1-'n]Ls -c:k _1--'j\cpcs ,

; (·a· Bsm \; I;, 0· Bpr) :.._ (;, Bsrn \;, :.. :.. Bpr :.. (,,:.. Bsm \;, /;, ;, BP,. -O[k ·p pj~uh] s -C.iPU: :Chlcpcs -Oj,Op PLk\O;zJCs

(·,_ Bsm\;:., ('"' :.,Bpr\(':,, Bsm~:., {;, ;,Bpr)l 8) ):... {;, Bsm\:., :...Bpr -o1h pjchJopos J-CP t"'jc[k\Oh]os J-n+l!Gp\p[h PkJos

n8' 1·(. . ~ . , _ _ ,_ . _ _ ._ lk \t"1 - Bsm - ; Bpr !- - Bsm13 - BP'. I- B.-sm\;:: +-2--- tcjohl pos -pjcp . hies -,ch: ,._

n - .1

Bpr

-(aPBsn ,0/ihlasBpr}+ ;rh ~ ~h 1 (8jBsm "pp(\BP,. -c\1(3PBsm}3.ic\BP,. n -

(. ~ _ _ (· ,_ . . '] Vih;! [,· ...... ::;rn '"' ~ ""'\ pr -i snz ""'l ; pr , r lk ~ - cjB _ h)c:pcsB - cPB ~ihojosB f.,. n 2 -l c j

:., :., ('"' Bsm \:., :., BPI' . :... (:., B"m '\;,, :... ;., Bpr :.. i;., Bsm b -cjc-hl op Plus +oh] u 1 /-'jopos -ch \Op c

,0;, -Bsm 'i:., ;, Bpr

j 'i r pcs

,SB pr

-,.--c

_ 1_;; k

61

1" .i-t;, 12 B pr

' cjlcpB-"" fh Bpr )_._ (~ B"m \.:. _;_,. ~ r~ Bpr s . ' .c l ;= J '- n 1( p - s

1- - - - IJ Xi - r- · ) 0[1k - -· . ; . ~ ~ ~-.. pr .' _ -r-_ ') _ _· _ -: . "':'! --: _ pnz __ ' _ ; . ~. prn

-f.e-,,,Crc,B I V,~G,.,1" CJ C[kC1, 1B + 2 c 1,,c 1 B .) n, • _, - - ... 1.~ 11 -c- l - ' '· n - l J

0 .,, ,ZiBP"' J .. --:-- "\-~5lh 2,-u,,,i, h ' n- -1 -

[i _Bpm p

v -l~i ix ork ! ..

n2k_1

2Pmu/' 1 (2hf Bpm)l - . I p f ·)

n- -1

___ (29)

represents the conformal transformation of the decomposition tensor under the change r l

Thus \Ve state

Theorem 5. Under the decomposition (20), the conformal decomposition tensor

is expressed in the form i29)_

Transve(:-ting equation \20) by _;;,),we have

= x 10:.i

Using the equation (23), (28) and (30) in the equation (12), we obtain

,,B'm xi (r_

il - - ~c _ t-,. BP"' ' n --:--1 P '"'

-'--\ D l 'Phl1pl

{30)

(31)

- l .,.. -;-,-- o~ -1 ,._ B::- ~PB 11 - l/,,, J- BPm 1'111 p ;-)] + 2B G1z111) l n- - 1 !

-'-

+e"Vi

f.::·

r n:B12: _ _ • ~r '

n -l

'J - ,, p

I B pm l

D [ . J

e"V? +~CT -

n- -1 ndp tt:P<\J,.Bm' - hJBpm uvY 2 rr.:. Bsm f.:. ;:, Bir

e i O"mO",-t\r·[k "('h(.s

- t x -- --;;-1 i'\ ,(-- c~ B pr 1 (, -r j p .s ~ -7- - sr

I 71 -1 Ufk h]Bsm » i SB pr -(2 lz]2 SB pr}.:. PBsm

2 --:- -Bsm:.. - _ n -1 'h ,1-, r Bpr , ,. .. } s

(32)

Interchange the indices k and h in the equation (32), we get

=-~hk (33)

62

by vritue or relation Wkh = -iv:i:k [ 4].

Li the veiw of the equations (23) and (24), the equation (32), becomes I

J. CT,i_ av2 i(;., Bim '+'kh =e '+'kh -e i Gm! 0[k

L i'

i]- n -c-1 pafkBprn ,Erm QP,

R 0 hJp

_._ 1 ~i J . . nz -1 oi1: in+ l~c 111Bpm i -n(3PBpm

_(.~ ;., ,

l] ,~hfpBpm ll B rmap. r + 2 k1rp- .d

+ecr~2crmi(h)i B im n s:.i ;., Bpm i' ;., ;., Bpm , ----v"•C ---oh.a j n 2 -l ''' P n+l '' P

"TT 9 e •,~ ..,.-2--

n -1

'

ii:P c"';, if Erm -, -~" r

x' ;., ;., ;., BPrl 1 _, ---C111CnC~ ,--Op,

n + 1 ' " ·· 'n -1 ·"'

' 2 Bsm '-: 0., ;., B pr ,-- C-111C µC,

n-1 · ·

.B·:J."'''-

1B"m l~P(: ,BPT - ';., BP'"'§ B"m JCS , p

J;, ;, ; f';,jc,B'r

Which gives the conformal transformation of decomposition tensor

under the conformal change. Theorem 6. Under the decomposition

tensor ¢ hh is expressed in the form 1 34).

' a.no

REFERENCES

the conformal decomposition

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15 (1929), 376-379.

! 21 R.B. Mishra, Projective tensor in conformal Finsler space, Bull. De Ia Classes des Science, Acad

Royale de Belgique. f1967J, 1275-1279.

131 R.B. Mishra, The Bianchi identitites satisfied by curvature tensors in a conformal Finsler space. Tensor NS., 18 ( 1967), 187-190.

! 41 H. Rund, The Differentiai Geometry o{Finsler space. Springer-Verlage 1.1959;,

15 i B. B. Sinha and S.P. Singh. recurrent Finsler space of second order H Rep. Indian J. Pure and

Applied Math. 4 No. 1, (1973l. 45-50.

161 M. Gamma, On the decomposition of the recurrent tensor in an Areal space of submetric class, ! 'l'T. of Hakkaido Univ. (Section) 28 (1978), 77-80.

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! 8 l A.K Kumar, The effect of conformal change over some inti ties in finsler space, Atti delta Accad.

Nax Linceli Rendiconti, (1-2), LII, H972l, 60-70.

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