WEYL MANIFOLDS WITH SEMI-SYMMETRIC CONNECTION. Füsun Ünal 1 and Aynur Uysal 2. Turkey.

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1 Mateatca and Coputatona Appcaton, Vo. 0, No. 3, pp , 005. Aocaton for centfc eearc WEYL MANIFOLD WITH EMI-YMMETIC CONNECTION Füun Üna and Aynur Uya Marara Unverty, Facuty of cence and Letter, Departent of Mateatc, Turey. Itanbu Tecnca Unverty, Facuty of cence and Letter, Departent of Mateatc, Turey. Abtract- We defne a e-yetrc connecton on a Wey anfod and tudy proectve curvature tenor and confora curvature tenor after vn oe properte of te curvature tenor wt repect to e-yetrc connecton. Keyword- Wey anfod, e-yetrc connecton, curvature tenor, roup anfod.. INTODUCTION Hayden [] ntroduced e-yetrc etrc connecton on a eannan anfod and t defnton wa deveoped by Yano [] and Ia [3-4]. In t paper, we defne a e-yetrc connecton on a Wey anfod and defne te curvature tenor wt repect to e-yetrc connecton. We ve oe teore by ean of a reaton between curvature tenor wt repect to eyetrc connecton and yetrc connecton. After defnn proectve curvature tenor and confora curvature tenor wt repect to e-yetrc connecton,we obtan oe teore by un properte of tee tenor. In te at ecton of te paper, wt te ep of [5], we exane Wey roup anfod.. THE CUVATUE TENO WITH EPECT TO EMI-YMMETIC CONNECTION An n-denona anfod wc a a yetrc connecton and a confora etrc tenor ad to be Wey anfod, f te copatbe condton n te for of T= 0. In t cae, Wey anfod denoted by W n (, T ). T, wc a covarant vector, caed copeentary vector of te anfod. If T = 0 or T radent, ten eannan anfod obtaned. T cane by Tˆ = T + ( n λ ), under te tranforaton of te etrc tenor n te for of ĝ = λ wt λ a pont functon. Accordn to t tranforaton, f te quantty A cane by  = λ A, ten te quantty A caed a and te quantty &, wc defned by & A= atete of wt te wet of { } A T A, caed enerazed covarant dervatve of A.In t defnton, A denote ordnary covarant dervatve of A. A enerazed connecton on a Wey anfod ven by [6] Γ = İ Γ +a, were a = Ω A + Ω + Ω (.) If Ω coen by Ω = a a n (.), ten a e-yetrc connecton on a Wey anfod defned by

2 35 Füun Üna and Aynur Uya Γ Te toron tenor T =Γ +, were =a (.) wt repect to e-yetrc connecton defned by T = (.3) Anaoou to te defnton of te curvature tenor wt repect to yetrc connecton, we defne te curvature tenor wt repect to e-yetrc connecton by = Γ Γ +Γ Γ Γ Γ (.4) eebern te defnton of te curvature tenor wt repect to yetrc connecton and un (.) and (.4), we ave: = + + (.5) Te tenor n (.5) defned by were = + (.6) denote covarant dervatve wt repect to yetrc connecton. Mutpyn (.5) by, = + + (.7) Mutpyn (.7) by, = +(n-) +, were = (.8) By un te defnton of and, = + (n-) (.9) Lea.: Te tenor yetrc f and ony f radent. Proof: It own eay by un (.6). Teore.: Te curvature tenor wt repect to e-yetrc connecton a te foown properte: () + = 0. () + = (T, T, ) () (v) = + = [ ] + = ( [ ] + [ ] + [ ] Proof: () It obtaned by addn (.7) and te equaton by cann te at two ndce n (.7). () By un (.7) and reebern te ae reaton wt repect to yetrc connecton, te requred reut obtaned.

3 Wey Manfod wt e-yetrc Connecton 353 () It eay een by an contracton wt repect to ndce and I n (.5). (v) It own by cann ndce, and cyccay n (.5). Coroary.: If radent on a Wey anfod wt e-yetrc connecton, ten te foown od: () + + = 0. () T + T + T = 0. Teore.: If te curvature tenor wt repect to e-yetrc and yetrc connecton concde, ten radent. Proof: Let =. Fro (.7), + = 0 Mutpyn bot de of (.0) by, (n-) + =0 (.) If (.) utped by, = 0 (.) By un (.) n (.), t found tat = 0. T ow tat radent. Defnton.: A e-yetrc connecton ad to be oca-fat, f te curvature tenor wt repect to e-yetrc connecton. Teore.3: If te e-yetrc connecton defned on a Wey anfod ocafat, ten T radent. Proof: Fro (.7) and Defnton., = + (.3) Mutpyn (.3) by, T, T, ) = 0 (.4) (.4) ow tat T radent. L 3. THE POJECTIVE CUVATUE TENO WITH EPECT TO EMI- YMMETIC CONNECTION A enerazed connecton on a non-eannan anfod denoted by = Γ ~ + Ω ~,were Γ ~ yetrc part and ~ Ω curvature tenor wt repect to t connecton ven by L = B B B ant-yetrc part, and te + Ω, were denote te ter wt repect to yetrc part of te enerazed connecton. ven wt te foown defnton: ~ B = Γ ~ Γ + Γ ~ ~ Γ Γ ~ ~ Γ

4 354 Füun Üna and Aynur Uya In t cae connecton on a non-eannan anfod defned n [ ] W = B + were B = B. nce te proectve curvature tenor wt repect to a enerazed 7 by + ( ) n B B n + ( ) n B B (3.) ( ) n+ B B ~ Γ = Γ +( ) + and ~ Ω = ( ) for te e- yetrc connecton on a Wey anfod, we defne B wt repect to eyetrc connecton a foow: B = + [ ] + (3.) were = r + r By an contracton on te ndce and n (3.), we et: B = + [ ] ( n) (3.3) Te equaton by cann ndce and n (3.3) ubtracted fro (3.3), B B= ( ) (n-3) [ ] (3.4) By un (3.), (3.3) and (3.4) are ued n (3.), we obtan proectve curvature tenor wt repect to e-yetrc connecton on a Wey anfod a foow: W = + ( ) { n [ ]} n+ + ( ) + ( H n H ) (3.5) were H = n + + ( n) [ ]. Teore 3.: Proectve curvature tenor wt repect to e-yetrc connecton a te foown properte: () W + W = 0 () () W = 0 ( n) W = [ ] n+ (v) W + W + W = 0 Proof: () It obtaned by addn (3.5) and te reaton by cann te ndce and n (3.5). () By contracton on te ndce and n (3.5), te requred reut obtaned. () It eay een by contracton on te ndce and n (3.5). (v) I.Banc dentty for proectve curvature tenor own by cann te ndce, and cyccay and un (3.5). Coraary 3.: W = 0 f and ony f radent. Teore 3.: Proectve curvature tenor wt repect to e-yetrc connecton and yetrc connecton are reated wt te foown reaton:

5 Wey Manfod wt e-yetrc Connecton 355 W = W + [ ] + ( ) r r K K n + r r (3.6) were K = n + + ( n+ ) Proof: By coon = 0 n (3.5), te proectve curvature tenor wt repect to yetrc connecton obtaned by W = + ( ) + ( ) H H n+ n (3.7) were H = n +. By vrtue of (.5),(3.5) and (3.7), (3.6) obtaned. Defnton 3.: A e-yetrc connecton ad to be proectvey-fat, f te proectve curvature tenor wt repect to e-yetrc connecton vane. Teore 3.3: If a e-yetrc connecton oca-fat and radent, ten te connecton ao proectvey fat. Proof: By un Defnton. and [ ] = 0, te requred reut found. 4. THE CONFOMAL CUVATUE TENO WITH EPECT TO EMI- YMMETIC CONNECTION Under confora tranforaton, te etrc tenor and te e-yetrc connecton were P Γ on a Wey anfod are tranfored a foow: Γ = Γ + = (4.) P + P = T and (P Q ) (P Q ) (4.) = Q. Wt te ep of (4.) and (4.), we defne te cane of curvature tenor under confora tranforaton wt repect to e-yetrc connecton by ( ) = + [ P ] + P[ ] + W W + W W + P Q G were W = P Q + P( Q ), P = P P, Q = Q Q (4.3) Mutpyn (4.3) by = =, we ave te equaty nubered by (4.4): P + P + W W + W W + P Q G ( ) + [ ] [ ] Mutpyn (4.4) by =, ( ) = + [ P ] + P[ ] + ( n ) W + W ( n ) P Q (4.5) Conequenty fro (4.5), = + ( n)( W n P Q ) (4.6) were W = W. Fro (4.5), we ave W defned by te foown euaton by (4.7)

6 356 Füun Üna and Aynur Uya ( n)( ) + ( ) + ( n )( Q [ ] [ ] n ) + Q n W + n) P Q Fro te defnton of W, W = + n P Q ( n ) or equvaenty n P Q = W ( ) ( n ) On te oter and, f (4.4) utped by =, (4.8) = + n ( [ P ] + P[ ] ) (4.9) By un (4.9) n (4.7), we obtan: ) ( ) + ( n ) W W = n ) Puttn (4.9) and (4.0) n (4.4); we ave: were C defned by C ( ) (4.0) = C (4.) C = n + A B G n n ) ( n)( n ) were A = + and B = + (4.) Mutpyn (4.) by ; we fnd te confora curvature tenor wt repect to eyetrc connecton n te foown for: C = + n were A = and B = n A + B G (4.3) n ( n ) ( n)( n ) + Teore 4.: Te confora curvature tenor wt repect to e-yetrc connecton a te foown properte: () C + C = 0 () C + () (v) (v) C Proof: C = 0 C = 0 C = C = 0 + C + C = 0..

7 Wey Manfod wt e-yetrc Connecton 357 () Wt te ep of (4.), t wrtten te defnton of C. By addn t and (4.), te requred reut obtaned. () If te proce n () apped to te frt two ndce n (), te reut found. () It found by contracton on te ndce and n (4.3). (v) Te ae proce apped to te ndce and n (4.3). (v) By cann cyccay te ndce, and, tree equaton are wrtten. And by addn tee equaton, te requred reut obtaned. Teore 4.: Confora curvature tenor wt repect to yetrc connecton and e-yetrc connecton on a Wey anfod concde. Proof: Confora curvature tenor wt repect to yetrc connecton defned wt te ep of (4.3) by were and C = A + n n A = B = B G (4.4) n ( n ) ( n)( n ) + +. Un (.5) n (4.3) and wt te ep of (4.4), te requred reut obtaned. Defnton 4.: A e-yetrc connecton caed confora-fat, f te confora curvature tenor wt repect to e-yetrc connecton vane. Defnton 4.: A anfod caed confora-fat, f te confora curvature tenor wt repect to yetrc connecton vane. Coraary 4.: e-yetrc connecton defned on a Wey anfod conforafat f and ony f te Wey anfod confora-fat. Teore 4.3: If e-yetrc connecton defned on a Wey anfod oca-fat, ten t ao confora-fat. Proof: = 0 pe =0 and = 0. If te reut are ued n (4.3), we fnd C = 0 eann te connecton confora-fat. 5. WEYL GOUP MANIFOLD Defnton 5.: If te curvature tenor wt repect to e-yetrc connecton and te tenor van on a Wey anfod, ten tat one caed Wey roup anfod. By above defnton we ave te foown ea: Lea 5.: Wey roup anfod oca-fat. Lea 5.: radent on a Wey roup anfod. Teore 5.: Proectve curvature tenor wt repect to yetrc connecton and e-yetrc connecton concde on a Wey roup anfod. Proof: By un Defnton 5. and Lea 5. n (3.7), te requred reut obtaned. Coraary 5.:e-yetrc connecton defned on a Wey roup anfod proectvey-fat f and ony f Wey roup anfod proectvey-fat.

8 358 Füun Üna and Aynur Uya Teore 5.: A Wey roup anfod confora-fat. Proof: If te condton = 0, = 0, = 0 are ued n (4.4), te reut found. Teore 5.3: A Wey roup anfod proectvey-fat. Proof: If te ae condton n te proof of Teore 5. are ued n (3.8),te requred reut obtaned. Teore 5.4: Confora curvaure tenor and proectve curvature tenor wt repect to e-yetrc connecton concde on a Wey roup anfod. Proof: Confora curvature tenor, C, defned n ter of te proectve curvature tenor wt repect to yetrc connecton by te foown reaton: W + + ( n)( ( n)( n ) In t reaton D defned a foow: n+ ) D + [ ) G ] ( n ) [ ] [ ] [ ] [ ] [ ] ( [ ] [ ] ) + +. On te oter and by un te fact tat roup anfod, te equaton of C = W obtaned. ) n C = C (away) and W = W for Wey Coroary 5.: e-yetrc connecton defned on a Wey roup anfod confora fat f and ony f t proectvey-fat. EFEENCE. A.Hayden, ubpace of a pace wt toron, Proc.London Mat.oc. 34, 7-50, 93.. K.Yano, On e-yetrc etrc connecton, ev.ouane Mat.Pure App. 5, , T.Ia, Hyperurface of a eannan anfod wt e-yetrc etrc connecton, Tenor N.. 3, , T.Ia, Note on e-yetrc etrc connecton, Tenor N.. 4, 93-96, N..Aae and M..Cafe, A e-yetrc non-etrc connecton on a eannan anfod, Indan J.pure app.mat. 3, , V.Murecu, Epace de Wey a toron et eur repreentaton confore, Ann.c.Unv.Toara,-8, L.P.Eenart, Non-eannan Geoetry, Aercan Mat.ocety Cooqu Pubcaton 8,97.