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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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