IST Iteratoal Joural of Egeerg Scece, Vol 7, No3-4, 6, Page 8-85 The Le Algera of Smooth Sectos of a T-udle Nadafhah ad H R Salm oghaddam Astract: I ths artcle, we geeralze the cocept of the Le algera of vector felds to the set of smooth sectos of a T-udle whch s y defto a caocal geeralzato of the cocept of a taget udle We defe a Le racet multplcato o ths set so that t ecomes a Le algera I the partcular case of taget udles ths Le algera cocdes wth the Le algera of vector felds Keywords: Vector udle, Le theory Itroducto We ow that f s a smooth mafold, the χ(, the set of all smooth vector felds o (or, the set of all smooth sectos of χ( forms a Le algera (See for eample [] I ths paper frst we defe the cocept of a T-udle, whch s a geeralzato of taget udle The, we defe a Le algera structure o the set of all smooth sectos of a T-udle such that the partcular case of taget udle, t cocdes wth the Le algera of vector feld We are ale to geeralze may of deftos ad theorems aout vector felds to the set of smooth sectos of T-udles T-udle Defto Let E ad e smooth mafolds wth dmesos + ad respectvely, such that = for some N Also suppose that p:e s a smooth map By a T-chart o ( E, p, we mea a ordered par (, where s the doma of a chart (, u ad s a fer respectg dffeomorphsm as the followg dagram, E p : = ( p pr where pr s the proecto o the frst compoet Two Paper frst receved Nov, 8, 4 ad revsed: ay, 4, 5 Nadafhah s wth the Departmet of athematcs, Ira versty of Scece ad Techology, m_adafhah@ustacr H R Salm oghaddam s a PhD studet at the same Departmet Ira versty of Scece ad Techology, salm_m@ustacr T-chart (, ad (, are called T- compatle : =, v R ; ( ο ( v, = (, ( v For some mappg : GL(, R y J ( L J ( a L L J ( ( ( Where J ( = [ ( ] The mappg s u called the trasto fucto etwee the two T-charts Defto A T-atlas A= (, for ( E, p, s a set of par-wse T-compatle T-charts (, such that ( I s a ope cover of Two T-atlases are called equvalet, f ther uo s aga a T-atlas 3Defto A T-udle ( E, p, cossts of two mafolds E (the total space ad (the ase ad a smooth mappg p : E (the proecto together wth a equvalece class of T-atlases 4 Lemma Let ( E, p, e a T-udle The for ay there s a uque vector space structure o fer E : = p ( whch s somorphc wth R
8 The Le algera of smooth sectos of a T-udle Proof: Let (, ad (, e two T-charts wth, the we have : E { } E a(, h ( : E { } E a( h, ( (3 Where h ad h are two vertle mappgs (These two mappgs are est ecause ad are dffeomorphsms for adv R, we have ( ( v ( h( ( v Therefore,, =,, (4 or ( v h( ( v =, ; also, we have,, =, =, (5 ( h( ( v ( ( v ( v Ad therefore h ( ( v, = v Therefore, h( ( v h( ( v, =, (6 I other words, ( h ( = h ( We ow also that GL(, R ad therefore E has a uque vector space structure 5 Corollary Let { e, K, e } e the stadard ases for the vector space R, the { ( e,, L, ( e, } ad { ( e,, L, ( e, } are two ordered ases for E Let = y e, = z e, = = (7 ( ( that y, z R the y z ( = (8 y z The followg eample proves the secod part of theorem Let E = J ( RR, e the vector udle of et's of secod order from R to R at, that p : E R, p( f = f ( s the atural proecto Cosder the followg two compatle charts for R : u : := R R, t at( u : := R (,+, t ae t (9 The, A= { u, u} s a atlas for stadard structure of R Cosder the two mappgs : p ( ( ( yt zt yz ( ad : p ( ( ( + yt+ zt a ; e ye, y+ e z ( It s clear that {, } ad{, } are two vector udle charts such that o ( ; y, z = ( ; ( ( y, z, where y e y z a e e z ( But ( [ J ] = e Therefore, E s a vector udle ut s ot a T-udle 7 Defto Let ( E, p, ad ( F, qn, e two T-udles By a T-udle homomorphsm we mea a par (, of a smooth maps ( : E F, : N such that : E F s ferwse lear ad the followg dagram s commutatve: E F P q 6 Theorem Ay T-udle admts a vector udle structure ut the verse s ot correct N Proof: The frst part of theorem s trval, for ths t s suffcet to otce that ay T-atlas s a vector udle atlas Therefore, f A s a mamal T-atlas o E the there s a mamal vector udle atlas A such that A A Therefore, for ay the map : E F ( s lear I ths case we say that covers If s vertle the we say s a somorophsmt-udles
Nadafhah ad H R Salm oghaddam 83 together wth ther homomorphsms form a categorytvb 3 The Le Algera of Smooth Sectos of a T-udle 3 Defto Let e a pot We defe T := T L T (3 = where for ay = KT, s the taget space at, for dstcto, the de s attruted Also we defe T := T (4 = = as a mafold s dffeomorphc to Whtey-sum of copes of T (see [] I other words, = T = {( v, L, v v T,, : π( v = π( v } The proecto s defed aturally 3Theorem ( T, π, s a T-udle = (5 Proof: Let (, u e a chart o, t s suffcet to defe the map : π ( y a ( a ( a, L, = = ( a ;, L, a (6 33 Defto Let( Epe,, a T-udle wth T-atlas(, I Assume e ad(, u e a chart such that, ad (, e the T-chart correspodg to t I ths case we defe the mappg : E T y = ( y = = y := L m + m + L y ( + = where E ad = y ( e,, that we L = assume{ e, K, e } s the stadard ases for for ay f C (, R we defe f f := 34 Lemma ( s well defe (7 R, also (8 Proof: It s suffcet to show the defto s depedet of charts Let ad = y e, = z e, = = (9 ( ( Therefore, we have ( = y ( e, = = L y L y = = L y( + = = z ( L ( = = u u L L z ( = = L z( + ( = = By usg (36, we have
84 The Le algera of smooth sectos of a T-udle = z L z = u L = u ( z = ( = ( + u Therefore, s well defe 35 Theorem Let ( E, p, e a T-udle such that for ay, dm( ( p = the the mappg : E = T wth ( : = ( s a somorphsm of T-udles Proof: Let (, u e a chart o ad (, e the correspodece T-chart o E It s clear that s vertle ad y usg the proof of theorem (3 the followg dagram s commutatve p ( π ( Φ T Where ( a, = ( a, that s somorphsm for ay 36 Corollary Ay two T-udles o the same ase such that ther fers have the same dmeso are somorphc 37 Defto Suppose ( E, p, e a T-udle By a secto of the T-udle E we mea a smooth map : E such that po = Id We deote the set of all sectos of a T-udle ( E, p, y C ( E 38 Defto Let ( E, p, e a T-udle; η, C ( E ad f C (, R the we defe: ( + η := + η, ( f := f( ( 39 Defto Let ad η e C ( E ad (, u e a chart o, ths case we ca wrte local coordates f e = = (, g e = ad η = (, (3 where f, g C (, R The we defe η := ( ( η ( (4 ad g ( η ( := f L = = = = g L f L (5 L = = f p= = = g ( + ( + Therefore, we have η = f g (, e (6 p+ p+ p+ 3 Defto Let e a secto of T-udle(,, ad (, u s a chart of such that Ep, suppose that Also we assume that f = ( f, K, f C (, R (7 where f C (, R, ths case we defe : (, C R C (, R y ( f( = ( ( ( f, L, f ( f :=, L, = ( g( ( = f g ( ( ( + = = Where g (, e (8 Now we ca defe the Le racet of to sectos of a T- udle 3 Defto Let η, C ( E, the we defe the Le racet of adη y[ η, ] = η η, f we assume that = ad f (, e = the we have η = g (, e =
Nadafhah ad H R Salm oghaddam 85 [, η] = ( f f g, p= = = p + p + ( e p + g p + p + (9 3 Theorem Suppose that ( E, p, s a T-udle the C ( E wth Le racet ad the addto of fuctos ad scaler product s a Le algera Proof: The product that we defe defto (39 s assocatve (It s clear from the deftoso C ( E s a assocatve algera ad also we defe the Le racet of two sectos ad η y η η, therefore the C ( E s a Le algera 33 Corollary If ( E, p, e a T-udle wth stadard fer R such that dm( = the the Le algera C ( E s somorphc to multplcato of = / copes of Le algera χ ( (the Le algera of all vector felds o 4 Cocluso T-udle whch defed ths artcle s a vector udle such that t s a atural geeralzato of taget udle Also we defed a Le algera structure o the set of smooth sectos of a T-udle such that partcular case of taget udle t cocdes wth the Le algera of vector felds Refereces [] Esrafla E ad Nadafhah, " E -fuctor, a ew geometrc oect whch s a geeralzato of the ordary taget oect," Proceedg of The 3 st Iraa athemathcs Coferece 7-3 August, pp 56-67 [] Kolar I, chor PW, ad Slova J, "Natural operatos dfferetal geometry", Sprger-Verlag, Hedelerg 993