On transformations groups of N linear connections on the dual bundle of k tangent bundle

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1 Sud Unv Babeş-Boya Mah o On anfoaon goup of nea connecon on he dua bunde of k angen bunde Monca Pucau and Mea Tânoveanu bac In he peen pape we udy he anfoaon fo he coeffcen of an nea connecon on dua bunde of k angen bunde T k M by a anfoaon of a nonnea connecon on T k M We pove ha he e T of hee anfoaon ogehe wh he copoon of appng n a goup Bu we gve oe goup of anfoaon of T whch keep nvaan a pa of coponen of he oca coeffcen of an nea connecon Maheac Subec afcaon 2010: 53B05 Keywod: ua bunde of k angen bunde nonnea connecon -nea connecon anfoaon goup ubgoup 1 Inoducon The noon of Haon pace wa noduced by cad R Mon n [7] [8] The Haon pace appea a dua va Legende anfoaon of he Lagange pace The dffeena geoey of he dua bunde of k ocuao bunde wa noduced and uded by cad R Mon [13] The poance of Lagange and Haon geoee con n he fac ha he vaaona pobe fo poan Lagangan o Haonan have nueou appcaon n vaou fed a: Maheac Mecanc Theoeca Phyc Theoy of ynaca Sye Opa ono Boogy Econoy ec In he peen econ we keep he genea eng fo cad R Mon [13] and ubequeny we eca ony oe needed noon Fo oe dea ee [13] Le M be a ea n denona anfod and e T k M π k M k 2 k be he dua bunde of k ocuao bunde o k coangen bunde whee he oa pace : T k M T k 1 M T M 11

2 122 Monca Pucau and Mea Tânoveanu Le x y y k 1 p 1 n be he oca coodnae of a pon u x y y k 1 p T k M n a oca cha on T k M The change of coodnae on he anfod T k M : x x x 1 x n x de x 0 ỹ x x y 12 k 1 ỹ k 1 ỹk 2 x y k 1 ỹk 2 y k 1 y k 2 p x x p whee he foowng eaon hod: ỹ α x ỹα1 ỹk 1 y y α 0 k 2; y 0 x k 1 α 13 T k M a ea dffeena anfod of denon k 1 n Wh epec o 11 he naua ba of he veco pace T u T k M a he pon u T k M : { } x u y u y k 1 u p 14 u anfoed a foow: u x x x u x ỹ x u ỹk 1 ỹ x u ỹ u ỹk 1 u y y ỹ y ỹ k 1 u ỹk 1 u y k 1 y k 1 ỹ k 1 u p x u x p ỹ k 1 u p x p u 15 he condon 13 beng afed The nu econ 0 : M T k M of he poecon π k defned by 0 x M x 0 0 T k M We denoe T k M T k M \ {0} Le u conde he angen bunde of he dffeenabe anfod T k M T T k M dπ k T k M whee dπ k he canonca poecon and he veca dbuon { V : u T k } M V u T u T k M ocay geneaed by he veco fed: y y k 1 p a evey pon u T k M The foowng F T k M nea appng: defned by: J x y J y k 1 J J : χ T k M χ T k M y y k 1 J y 2 y k 2 0 J 0 16 p

3 On anfoaon goup of nea connecon 123 a evey pon u T k M a angen ucue on T k M We denoe wh a nonnea connecon on he anfod T k M wh he coeffcen: x y y k 1 p k 1 x y y k 1 p x y y k 1 p 1 2 n The angen pace of T k M n he pon u T k M gven by he dec u of veco pace: whee: T u T k M 0u 1u k 2u V k 1u W ku u T k M 17 oca adaped ba o he dec decopoon 17 gven by: { } x y y 1 2 n 18 k 1 p x x y y y y 2 y k 1 y k 1 p p k 1 y k 1 p k 2 y k 1 19 Unde a change of oca coodnae on T k M he veco fed of he adaped ba anfo by he ue: x x x whee: x x y x ỹ x y k 1 x ỹ x k 1 p x p 110 The dua ba of he adaped ba 18 gven by: {x y y k 1 p } 111 dx x dy y x dy k 1 y k 1 y k 2 y x k 2 k 1 dp p x Wh epec o 12 he coveco fed 111 ae anfoed by he ue: 112 x x x x ỹ x x y ỹ k 1 x x y k 1 p x x p 113

4 124 Monca Pucau and Mea Tânoveanu Le be an nea connecon on T k M wh he oca coeffcen n he adaped ba 18 : Γ H h h h α 1 k α n nea connecon unquey epeened n he adaped ba n he foowng fo: x x x H p H p x y α y α p p p α x α x x x x p p y α y α H α 1 k 1 y α y β α y β p α β 1 k 1 p y α α 1 k 1 y α The e of he anfoaon of nea connecon Le be anohe nonnea connecon on T k M wh he oca coeffcen x y y k 1 p x y y k 1 p x y y k 1 p k n Then hee ex he unquey deened eno fed τ 1 1 T k M α 1 k 1 α and τ2 0 T k M uch ha: { α α 1 2 k 1 α α 1 2 n 21 onveey f α and α α 1 2 k 1 epecvey and ae gven hen α α 1 2 k 1 epecvey gven by 21 ae he coeffcen of a nonnea connecon Theoe 21 Le and be wo nonnea connecon on T k M k 2 k wh oca coeffcen: x y y k 1 p x y y k 1 p 1 2 n epecvey k 1 k 1 x y y k 1 p x y y k 1 p x y y k 1 p x y y k 1 p

5 On anfoaon goup of nea connecon 125 If an nea connecon on T k M wh oca coeffcen Γ H h h h α 1 k 1 α hen he anfoaon: gven by 21 of nonnea connecon pe fo he coeffcen Γ H h h h α 1 k 1 α of he nea connecon he eaon 22 ha he anfoaon: Γ Γ gven by: H H [ 2 k 2 k 1 3 k 3 k k 2 k 2 k 1 k 2 k 3 k 1 k 1 k 1 k 3 k 1 [ 2 3 k 1 k 1 k 3 k 1 k 3 k 2 k 3 k 2 k 1 k 1 h 0 k 2 k 1 h 0 h 1 2 n [ k 1 k 1 ] whee denoe he h covaan devave wh epec o Γ k 2 k 1 22

6 126 Monca Pucau and Mea Tânoveanu Poof I foow f of a ha he anfoaon 21 peeve he coeffcen h h k 1 Ung he eaon and 21 we oban: x x y y k 1 k 1 y y y 2 y k 1 y k 1 p p Ung and 19 we ge: x y k 1 H y k 1 k 2 y k 1 H y k 1 p 23 x y k 1 x y 2 y 2 k 1 y k 1 y k 1 p H k 2 k 1 y k 1 y k k 1 k 1 y k 1 y 2 2 y 3 y k y 3 k 3 y k 1 y 4 k 4 y k 1 y k 1 y k 1 y k 1 y k 1 y 3 y 4 k 4 y k 1 k 3 k 1 H 2 2 y k 1 k 1 k 1 2 y k y k 1 y k 1 y k 1 2 y 3 y 4 y k 1 y k 1 2 y k 1 y k 1 k 2 k 1 k 3 k 1 k 4 k 1

7 On anfoaon goup of nea connecon 127 y k y k 1 2 y 4 So we have obaned 21 1 y y y k 1 y k 1 2 y k 1 y k 1 y k 3 k 1 k 4 k 1 y k 1 y 2 2 y 3 k 2 y k 1 y 3 y k 1 2 y 4 y k 1 2 So we have obaned 22 2 y k 2 y k 2 y k 1 k 2 y k 1 k 2 k 2 k 1 y k 1 y k 1 y k 1 y k 1 k 3 k 1 y k 1 k 4 k 1 y k 1 k 2 y k 1 k 1 y k 1 y k 1 y k 1 p y k 1 y k 1 ; y k 1 p So we have obaned 22 k 1 p y k 2 p y k 1 p y k 2 y k 2 y k 2 y k 2 y k 1 ; y k 1 y k 2 y k 1 y k 1 p y k 2 p y k 1 y k 1 So we have: 24

8 128 Monca Pucau and Mea Tânoveanu naogou f we cacuae p 25 y y k 2 k 1 n wo anne we oban: k 1 k 1 26 We have: k 1 y k 1 27 k 1 α k α x k H k α α H k α 1 2 k 1 28 Ung k n he eaon obaned anaogou h fo x y we oban: 0 In he ae anne we ge h 0 Theoe 22 Le and be wo nonnea connecon on T k M k 2 k wh oca coeffcen x y y k 1 p x y y k 1 p 1 2 n epecvey If k 1 k 1 Γ x y y k 1 p x y y k 1 p x y y k 1 p x y y k 1 p H h h h α and Γ H h h h α α 1 k 1 ae he oca coeffcen of wo epecvey -nea connecon epecvey on he dffeenabe anfod T k M k 2 k hen hee ex ony one ye of eno fed B h h h h k 1 k 1

9 On anfoaon goup of nea connecon 129 uch ha: α 1 k 1 α α α H H [ 2 k 3 k k 2 wh: k 2 k 2 k 1 k 2 k 3 k 1 k 1 k 3 k 1 [ 2 3 k 1 k 3 k 1 [ k 3 k 2 k 2 k 2 k 2 k 1 k 2 k 1 k 1 k 1 { h 0 k 1 3 k 1 k 1 B k 1 k 3 h 0 h 1 2 n k 1 k 1 ] whee denoe he h covaan devave wh epec o Γ Poof The f equay 29 deene unquey he eno fed: α 1 k 1 The econd equay 29 deene unquey he eno fed α Snce h α 1 k 1 and h ae d eno fed he hd equaon α 29 deene unquey he eno fed B h Say he fouh and he a equaon 29 deene he eno fed h epecvey We have edaey:

10 130 Monca Pucau and Mea Tânoveanu Theoe 23 If Γ H h h h α 1 k 1 ae he oca coeffcen of an nea connecon on T k M and α B h h h h k 1 k 1 a ye of eno fed on T k M hen Γ H h h h α α 1 k 1 gven by ae he oca coeffcen of an nea connecon on T k M k 2 k Foowng he defnon gven by M Mauoo [4 5] n he cae of Fne pace we have: efnon 21 The ye of eno fed: B h h h h k 2 k k 1 k 1 caed he dffeence eno fed of Γ o Γ The appng: Γ Γ gven by caed a anfoaon of nea connecon o nea connecon on T k M and noed by: B h h h h k 1 k 1 Theoe 24 The e T of he anfoaon of nea connecon o nea connecon on T k M k 2 k ogehe wh he copoon of appng n a goup Poof Le and Ā B h h h h k 1 k 1 Ā k 1 Ā B h h k 1 h h be wo anfoaon fo T gven by Fo 29 we have: Ā α 1 k 1 α α α We oban fo exape: h k 2 h k 2 α h Ā h k 1 k 2 : Γ Γ : Γ Γ h h k 2 Ā h k 1 So k 2 h han he fo 29 I foow ha he copoon of wo anfoaon fo T n a anfoaon fo T ha T ogehe wh he copoon of appng n a goup

11 On anfoaon goup of nea connecon 131 Reak 21 If we conde 0 α 1 k 1 and 0 n 210 we α oban he e T of anfoaon of nea connecon coepondng o he ae nonnea connecon : T 0 0 B h h h h T k 1 k We have: Theoe 25 The e T of he anfoaon of nea connecon o nea connecon on T k M k 2 k ogehe wh he copoon of appng a goup Th goup ac effecvey and anvey on he e of nea connecon Poof Le 0 0 B h h h h : Γ Γ be a anfoaon fo T gven by 211 : k 1 k α 1 k 1 α α H h H h B h h h h α 1 k 1 α α α h h h h 1 2 n 211 The copoon of wo anfoaon fo T a anfoaon fo T gven by: 0 0 B h h k h h k B h B h h h k 0 0 B h h h h k 1 k h k 1 k 1 h h h The nvee of a anfoaon fo T he foowng anfoaon fo T : B h h h h : Γ Γ k 1 The anfoaon 211 peeve a nea connecon f: B h h h h 0 h 1 2 n k 1 Theefoe T ac effecvey on he e of nea connecon Fo he Theoe 22 eu ha T ac anvey on h e

12 132 Monca Pucau and Mea Tânoveanu Le u conde: T H h 0 0 h h T k 1 k1 T 0 0 B h h 0 h h T 2 k 1 k T k B h h h h 0 T k 2 k T 0 0 B h h h 0 T k 1 T k 1 k 0 0 B h 0 0 T k k k 2 k Popoon 21 The e:t H T T T T ae bean ubgoup of T k 1 k 1 Popoon 22 The goup T peeve he nonnea connecon T H peeve he nonnea connecon and he coponen H h of he oca coeffcen Γ ; T peeve he nonnea connecon and he coponen h of he oca coeffcen Γ T coponen k 1 peeve he nonnea connecon and he h of he oca coeffcen Γ T peeve he nonnea con- k 1 necon and he coponen h of he oca coeffcen Γ and T k 1 peeve he nonnea connecon and he coponen h h h of k 1 he oca coeffcen Γ Refeence [1] anau Gh Tânoveanu M ew pec n he ffeena Geoey of he econd ode oangen Bunde Unv de Ve dn Tşoaa [2] anau Gh The nvaan expeon of Haon geoey Teno S Japona [3] Ianuş Ş On dffeena geoey of he dua of a veco bunde The Poc of he Ffh aona Se of Fne and Lagange Space Unv Başov

13 On anfoaon goup of nea connecon 133 [4] Mauoo M The Theoy of Fne onnecon Pub of he Sudy Goup of Geoey 5 ep Mah Okayaa Unv 1970 XV 220 pp [5] Mauoo M Foundaon of Fne Geoey and Speca Fne Space Kaeha Pe Ou 1986 [6] Mon R Haon Geoey Senau de Mecancă Unv Tşoaa [7] Mon R Su a géoée de epace Haon R cad Sc Pa Se II no [8] Mon R Haon Geoey naee Ş Unv Iaş S-I Ma [9] Mon R Ianuş S naae M The Geoey of he dua of a Veco Bunde Pub In Mah [10] Mon R On he Geoeca Theoy of Hghe-Ode Haon Space Sep n ffeena Geoey Poceedng of he ooquu on ffeena Geoey Juy 2000 ebecen Hungay [11] Mon R Haon pace of ode k gae han o equa o 1 In Jouna of Theoeca Phy [12] Mon R Huc Shada H Sabău VS The geoey of Lagange Space Kuwe cadec Pubhe FTPH [13] Mon R The Geoey of Hghe-Ode Haon Space ppcaon o Haonan Mechanc Kuwe cad Pub FTPH 2003 [14] Saunde J The Geoey of Je Bunde abde Unv Pe 1989 [15] Udşe Şandu O ua onnea onnecon Poc of 22 nd onfeence ffeena Geoey and Topoogy Poyechnc Inue of Buchae Roana 1991 [16] Yano K Ihhaa S Tangen and oangen Bunde ffeena Geoey M ekke Inc ew-yok 1973 Monca Pucau epaen of Maheac and Infoac Tanvana Unvey of Başov Roâna Mea Tânoveanu epaen of Maheac and Infoac Tanvana Unvey of Başov Roâna