A study on Ricci soliton in S -manifolds.

Transcription

1 IO Joual of Mathmatc IO-JM -IN: p-in: olum Iu I Ja - Fb 07 PP - wwwojoualo K dyavath ad Bawad Dpatmt of Mathmatc Kuvmpu vtyhaaahatta hmoa Kaataa Ida Abtact: I th pap w tudy m ymmtc ad pudo ymmtc codto -mafold tho a ad wh th occula cuvatu to ad a th mooth fucto o M futh w dcu about cc olto Kywod: -mafold -Et mafold Et mafold cc olto I Itoducto Th oto of f -tuctu o a -dmoal mafold M a to fld of typ atfy f f 0 wa ftly toducd 96 by K ao 8] a a alzato o M of a of both almot cotact fo ad almot complx tuctu fo 0 Du th ubqut ya th oto ha b futhly dvlopd by val autho ] ] ] ] 5] 6] 7] Amo thm H Naaawa 6] ad 7] toducd th oto of famd f -mafold lat dvlopd ad tudd by I Goldb ad K ao ] ] ad oth wth th domato of lobally famd f -mafold A maa mafold M calld locally ymmtc f t cuvatu to paalll 0 wh dot th v-vta cocto A a alzato of locally ymmtc mafold th oto of mymmtc mafold wa dfd by 0 TM ad tudd by may autho 8] 9] 6] 0] I zabo 5] av a full tc clafcato of th pac Dzcz 8 9] wad th oto of mymmty ad toducd th oto of pudoymmtc mafold by ] wh mooth fucto o M ad a domophm dfd by Dfto A cc olto a atual alzato of a Et mtc ad dfd o a maa mafold M A cc olto a tpl wth a maa mtc a vcto fld ad a al cala uch that 0 wh a cc to of M ad dot th dvatv opato alo th vcto fld Th cc olto ad to b h tady ad xpad accod a atv zo ad potv pctvly Th autho hama ] ad MMTpath 7] tatd th tudy of cc olto cotact mafold But ad amaau 6] Bawad ad Ialahall ] Dbath ad a l ABattachaya 7] hav tudd th xtc ad alo obtad ult o cc olto f -motu mafold -aaa mafold otza -aaa mafold Ta-aaa mafold u PEhat poblm 0] But Bawad Ialahall ad Aho Bawad ad Kdyavath hav tudd cc olto Kmotu mafold almot mafold u m-ymmtc ad pudoymmtc codto ] I th pt pap w tudy cc olto -mafold atfy m ymmtc ad pudo ymmtc codto tho a ad wh th DOI: 09790/ wwwojoualo Pa

2 occula cuvatu to ad a th mooth fucto o M II Plma t M b a -dmoal mafold wth a f -tuctu of a If th xt lobal vcto fld o M uch that; f I f 0 f 0 f f wh a th dual -fom of mafold th xt a maa mtc uch that f f w ay that th f -tuctu ha complmtd fam Fo uch a fo ay vcto fld ad o M A f -tuctu f omal f t ha complmtd fam ad f f ] d 0 wh f f ] Njhu too of f t F b th fudamtal -fom dfd by F f T M A omal f -tuctu fo whch th fudamtal fom F clod 0 fo ay ad d d d F calld to b a -tuctu A mooth mafold dowd wth a -tuctu wll b calld a -mafold Th mafold toducd by Bla ] hav to ma that f w ta -mafold a atual alzato of aaa mafold I th ca om tt xampl a v ] ] If M a -mafold th th follow lato hold tu ]; f T M 5 f f f f } T M 6 wh th maa cocto of t b th dtbuto dtmd by th pojcto to- f ad lt N b th complmty dtbuto whch dtmd by f I ad pad by It cla that f th 0 fo ay ad f N th f 0 A pla cto o M calld a vaat f -cto f t dtmd by a vcto x xm uch that f} a othoomal pa pa th cto Th ctoal cuvatu of calld th f -ctoal cuvatu If M a -mafold of cotat f -ctoal cuvatu th t cuvatu to ha th fom f f f f f f f f } f f f f f f f f} F F F F F F } 7 wh T M uch a mafold N K wll b calld a -pac fom Th Euclda pac E ad th hpbolc pac H a xampl of -pac fom DOI: 09790/ wwwojoualo Pa

3 DOI: 09790/ wwwojoualo Pa Dfto -mafold f M ad to b -Et f th cc to of M of th fom b a wh b a a cotat o M Now cotact quato 7 w t 8 ] Fom 7 w hav } 0 } } III cc olto I m-ymmtc -Mafold A -mafold ad to b m-ymmtc f w t } 0 By ta a poduct wth th w t } 0 By u 0 w hav 0 5 Ta 5 ad umm ov w t 6 Thu w tat th follow; Thom m ymmtc -mafold a Et mafold If co-la wth th cc olto alo v by Dfto t f th cotact -fam mafold f th la pa combato of th c c c ad th cc olto a

4 DOI: 09790/ wwwojoualo 5 Pa tpl wth a maa mtc a vcto fld ad a al cala uch that 0 c 7 Fom 7 w hav 0 c c w t 0 f c f c 9 Fom 6 ad 9 w hav 0 0 Ta 0 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom cc olto m-ymmtc -mafold h oollay cc olto m ymmtc -mafold tady f 0 Kahl mafold ad h f aaa mafold I cc olto -mafold atfy 0 Th occula cuvatu to v by } 0 ad w t } } } t u aum that th codto 0 hold o M th w t } 0 6 By ta a poduct wth th w t } 0 7 By u 7 w hav

5 } Ta 8 ad umm ov 9 Thu w tat th follow; 8 ad u w t Thom -mafold atfy th codto 0 a Et mafold Fom 9 ad 9 w hav 0 0 Ta 0 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom cc olto -mafold atfy th codto 0 h oollay cc olto -mafold atfy 0 tady f 0 h f aaa mafold cc olto -mafold atfy 0 hold o M th t u aum that th codto 0 Kahl mafold ad w t } 0 5 By ta a poduct wth th w t } 0 5 By u 5 w hav } 5 Ta 5 ad umm ov w t 55 Thu w tat th follow; Thom 5 -mafold atfy th codto 0 a Et mafold Fom 55 ad 9 w hav 0 56 Ta 56 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom 6 cc olto -mafold atfy th codto 0 h DOI: 09790/ wwwojoualo 6 Pa

6 oollay cc olto -mafold atfy 0 tady f 0 h f aaa mafold cc olto -mafold atfy 0 t u aum that th codto 0 hold o M th Kahl mafold ad w t } 0 6 By ta a poduct wth th w t } 0 6 By u 6 w hav Ta } 8 ad umm ov 65 Thu w tat th follow; 6 ad u w t Thom 7 -mafold atfy th codto 0 a Et mafold Fom 65 ad 9 w hav 0 66 Ta 66 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom 8 cc olto -mafold atfy th codto 0 h oollay cc olto -mafold atfy 0 tady f 0 h f aaa mafold I cc olto Pudo-ymmtc -mafold A -mafold ad to b Pudo-ymmtc f ] 7 Kahl mafold ad ] 7 H of?? DOI: 09790/ wwwojoualo 7 Pa

7 DOI: 09790/ wwwojoualo 8 Pa } 7 By ta a poduct wth th w t } 7 By u 0?? w hav } 75 Aa u H of?? w t }] 76 By ta a poduct wth th w t }] 77 By u 0 77 w hav ] 78 Fom 75 ad 78 w t ] 0 ] 79 Thfo th o } 70 Ta 70 ad umm ov w t 7 Thu w tat th follow; Thom 9 Pudo ymmtc -mafold a Et mafold povdd Fom 7 ad 9 w hav 0 7 Ta 7 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom 0 cc olto pudo ymmtc -mafold h oollay 5 cc olto pudo ymmtc -mafold tady f 0 Kahl mafold ad h f aaa mafold II cc olto -mafold atfy Q t u aum that th codto ] hold o M th ]

8 DOI: 09790/ wwwojoualo 9 Pa 8 H of?? } 8 By ta a poduct wth th w t } 8 By u?? w hav ] 8 Aa u H of 8 } 85 By ta a poduct wth th w t } 86 By u 86 w hav ] 87 Fom 8 ad 87 w t 0 ] ] 88 Thfo th o ] 89 Ta 89 ad umm ov w t 80 Thu w tat th follow; Thom -mafold atfy th codto Q a Et mafold povdd Fom 80 ad 9 w hav 0 8 Ta 8 ad umm ov w t th valu of < 0 Thu w tat th follow;

9 Thom cc olto -mafold atfy th codto h oollay 6 cc olto -mafold atfy tady f 0 ad h f aaa mafold III cc olto -mafold atfy t u aum that th codto Kahl mafold ] hold o M th ] ad 77 9 w t ]} 0 9 o } 9 9 ad umm ov w t 9 Thfo th Ta Thu w tat th follow; Thom -mafold atfy th codto Fom 9 ad 9 w hav 0 95 Ta a Et mafold povdd 95 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom cc olto -mafold atfy th codto h oollay 7 cc olto -mafold atfy tady f 0 ad h f aaa mafold Kahl mafold I cc olto -mafold atfy ] hold o M th t u aum that th codto ] ad 86 0 w t ] 0 0 DOI: 09790/ wwwojoualo 0 Pa

10 Thfo th Ta o } 0 ad umm ov u w t 0 Thu w tat th follow; 0 Thom 5 -mafold atfy th codto a Et mafold povdd Fom 0 ad 9 w hav 0 05 Ta 05 ad umm ov w t th valu of < 0 Thu w tat th follow; Thom 6 cc olto -mafold atfy th codto h oollay 8 cc olto -mafold atfy tady f 0 ad h f aaa mafold Kahl mafold ocluo It how that cc olto -mafold atfy m-ymmtc ad pudo-ymmtc codto a h Hc f th aaa mafold a h whch accodac wth ] 5] ] ad f 0 th Kahl mafold a tady ] fc ] Bawad ad G Ialahall cc olto otza -aaa mafold Acta Mathmatca Acadma Padaoca Nyyhaz vol 8 o pp ] Bawad ad Kdyavath cc olto of almot mafold Bull al Math oc ] DE Bla Gomty of mafold wth tuctual oup O J Dfftal Gom ] DE Bla G udd ad K ao Dfftal omtc tuctu o pcpal toodal budl Ta Am Math oc ] hxu H ad M hu cc olto o aaa mafold Axv:0907v mathdg] 6] a l ad M amaau Fom th Ehat poblm to cc olto f-kmotu mafold Bullt of th Malaya Mathmatcal cc octy vol o pp ] Dbath ad A Battachaya cod od paalll to Ta-aaa Mafold ad cocto wth cc olto obachv Joual of Mathmatc ol No 0-6 8] Dzcz O pudoymmtc pac Bull oc Math Bl A ] Dzcz ad Gya O om cla of wapd poduct mafold Bull It Math Acad ca ] P Ehat ymmtc to of th cod od who ft covaat vat a zo Ta Am Math oc 5 o ] I Goldb ad K ao O omal lobally famd f -mafold Tohou Math J ] I Goldb ad K ao Globally famd f -mafold Illo J Math ] I Haawa Oyama ad T Ab O th p-th aaa mafold J Hoado v Ed ctii A ] G Ialahall ad Bawad cc olto -aaa mafold IN Gomty vol 0 Atcl ID 8 pa 0 5] Ihhaa Nomal tuctu f atfy f f 0 Koda Math m p ] H Naaawa f -tuctu ducd o ubmafold pac almot Hmta o Kahla Koda Math m p ] H Naaawa O famd f -mafold Koda Math m p ] K Nomzu O hypufac atfy a cta codto o th cuvatu to Tohou Math J ] A Oawa odto fo a compact Kahala pac to b locally ymmtc Natu c pot Ochaomzu v 8977 DOI: 09790/ wwwojoualo Pa

11 0] D Po otact maa mafold atfy 0 oohama math J ] M M Pava ad Bawad O almot pudo Boch ymmtc alzd complx pac fom Acta Mathmatca Acadma Padaoca Nyyhaz ] hama cod od paalll to al ad complx pac fom Itatoal J Math ad Math c ] hama cod od paalll to o cotact mafold Alba Goup ad Gomt ] hama cod od paalll to o cotact mafold II Math p Acad c aada III No ] I zabo tuctu thom o maa pac atfy 0 I Th local vo J Dfftal Gom ] Tao ocally ymmtc K-cotact maa mafold Poc Japa Acad ] M M Tpath cc olto cotact mtc mafold 8] K ao O a tuctu dfd by a to fld f of typ atfy f f 0 To N DOI: 09790/ wwwojoualo Pa