On homogenity and transitivity of fields of geometric objects

Pub . Ma . UAB ol . 30 nó 1 Maig1986 ON HOMOGENITY AND TRANSITIVITY OF FIELDS OF GEOMETRIC OBJECTS Je zy Konde ak ABSTRACT . I a is a ield o geome ic objec son amani old M hen we can associa e wi h i a p incipalsubbundle o H (M) . We show ha (in ini esimal) homogeni y and (iñ ini esi mal) ansi i i y o his subbundle a e equi alen o some in _e g alcondi ions o he Lie eaua ions gene a ed by a . I . INTRODUCTION l . Le M be a di e en iable mani old . In he p esen pape , mani olds, ec o ields and so on always mean di e en- iablemani olds, di e en iable ec o ields and so on . Di e en iabili y always means hedi e en iabili y o class I U,V a e open subse so M hen a di eome phism U - V is called a lccal di eómo phism o M . These (M) o all local di eomo phisms o M is a pseudog oup . By TPA we shalldeno e a se o ec o ields de ined on opensub- se s o M . We deno e by H (M) he se o all -je s a 0 o di- di eomo phisms o openneighbou hoods o  0  in 1R n  on o open subse s o  M .  Le  i : H (M) -+ M  be he a ge p o- jec ion . Then H (M)  is ap incipal ib ebundlewi h he s uc u e g oup L o all -je s wi h he sou ce and wi h he a ge a 0 o local di eomo phisms o IR n . 2 . Le F deno e a na u al bundle om he ca ego y o n-dimensional mani olds ( o he de ini ion o he na u al bund- le see [71 o [81) . Di e en iable sec ions o he bundle F(M) --> M a e called ields o geome icobjec s . Fde assu- me ha F is o o de . I means ha i U,V a e n-dimen- _sional mani olds and 9,  : U --> V a e di eomo phisms such ha  jx  = jx 0 o a ce ain  x E U  hen  F (-p) I -l 17 (x) = F(O)  , ha is F(9) and F(O) a e equal on he ibe o 7 : F(U) -> U abo e he poin x . Le F 0 = 7 (0), whe e  7 :F (IR n )  _  IR n ,  be called a s anda d ib e o he bundle F . I X E TM hen he low o X induces a low on F(dom(X)) . The ec o ield on F(dom (X)) de ined by his low is called a comple li o X and is usually deno edby F (X)  (c . [9]) . A unc o H which a aches o each n-dimensional mani old M he p incipal ib e bundle H (M) is an example o a na u albundle .  I W : N i -> N 2 is an embedding o n-dimen sional mani olds  hen  H (,) (j ) = j0(~p . )  whe e  j EH (Nl ) . The bundle H is o cou se o o de . A comple li o X E TM o he bundle H (M) willbe deno ed by H (X) . The concep o 'na u albundle' was in oducedby A . Ni- jenhuis ([7]) as a mode n app oach o he classical heo y o geome ic objec s (c . [l]) . Le a be a iel o geome ic objec s on M . Then a induces a mapping a : H (14) - F 0 such ha a  0 )  =  F( -1 )a ( (0)) . Le  1 0 E im a .  We assume ha hese  E =  ( U ) -1 (1 0 )  is a p incipal subbundle o H (M) . I is easy o see ha his is equi alen o he ollowing ac : o each x,y E M he e exis s  -p EI'(M)  such ha  F(sp)a (x) = a(y) .  In he o he wo ds onecansay ha a is 0-de o mable (c . [10]) . Conce - ning ields o geome ic objec s one usuallyassumes ha hey a e 0-de o mable . The bundle E is no uniquely .de e mi ed by a  and i also depends on he choice o an elemen o  1 E F 0 . Al houghwe ix he bundle E, all he esul s a e ue o all subbundles o H (M) induced by a . II . LIE EQUATIONS ASSOCIATED WITH THE FIELDOF GEOMETRIC OBJECTS . We shall ecall some ac s om he hoe yo Lie equa- ions . These ac s will be applied in he hi dpa o his pa- pe . The basic de ini ions o he hoe y o Lie equa ions one can ind in [21, [31, [4] . . 1 . The e a e gi en he na u albundle F o o de and he ield o geome icobjec s a on M (c . I .3) . Le Ii (M)  =  {j ,  :  ~p E (M)  and  x E dom  p} . Then n (M) is a Lie g oupoid (c .[6]) . Le now n (a) = {ix~P E n (M) : F(Sp)a(x) = a(~0(x))} . Wi h he ield a we associa ed also he bundle E . Lemma II .l . The ollowingequali y holds II (a)  =  {p , -P - 1  :  P .P'  E E} whe e (j g) . (j ) -1 = j (g . -1 ) o j ,j g E H (M) and x = (0) . P oo . Le jx, E'II (a) and y = W(x) . Le j E Ex hen  a (j (W° ))  = F(('° ) -1 ) a (Y)  = F( -1)F(~p-1)a(Y)  _ = F( -1 )a(x) _  so j (cp° ) EE . Hence we ae ha , , = ~  -1 jx~  7 0 ( .P_ ) . (jp )  and bo h je s on he igh side o his equa ionbelong o E . Since hen j W E {p'-p _1 : p', p E E} . Le now j = j (h°g -1 ) whe e j h,j g E E . Then F(O)a(x)  = F(h°g _1 )a(x)  = F(h)F(g-1)a(g(0))  = F(h)T a (j g)  _ = F(h) 0 = F(h)T a (j h)  = F (h)  F(h-1)a(h(0))  = a(h(0)) . Hence we ge ha F(O)a(x) = a(O(x)) . because i is associa edwi h he bundle E . A local di eomo phism 9 is called a local solu ion o he non-linea Lie equa ion  II (a)  i o each  x E dom9 'G jx E II (a) ; II (a)  is calledcomple elyin eg able i o each i7EII (a) he eexis s a localsolu ion 9 such ha j W =  o a ce ain xE dom~p . x 2 . Le now R (a) _ {ixX : X E TM, xE dom X and (LX0) x =0} . A . Zaj z p o ed ha R (a) is a linea Lie equa ion and he The e o e by  [6)  we ge ha  1I (a)  is a Lie g oupoid canonical p ojec ion p : R (a) --> TM is su jec i e whe e p (j X) = Xx . He e Lx a deno es a Lie de i a i e o a ield o geome icobjec s (c . [9], [101) . A ec o ield X E TM is called a local solu ion o R (a) i o each x E dom(X) we ha e j XX E R (a) . The li- nea Lie equa ion R (a) is called comple elyin eg able i o each  E R (a) he e exis s a local solu ion X such ha jXX = n o a ce ain x E dom(X) . Lemma 11 .2 . The ollowing condi ions a e eáui alen : i) .  (L x a) x= 0 ; ii) . H (X) z ET z E whe e z EEx and H (X) is a com- ple e li o he ec o ield X o he bundle H (M) . P oo . We ha e he ollowing canonical mapping H (M) - F(M), whe e 4>(j ) = F( )l 0 o j E H (M) . I was shown ha i is a di e en iable ib e mapping co e- ing he iden i y mapping on M .  (c . [10]) . I is easy o see ha (D 1 (a(M)) = E . Hence TE = (d4)-1(Ta(M)) . Le us also ema k ha i Z is a ec o ield on M hen H (Z) is p ojec able on F(Z) ia Le i s assume ha  (L x a) x = 0 . I means ha dx a(X x ) = F(X)a(x) and F(X)a(x) E TG(x)a(M) . We ha e also ha  d z (D(H (X) z )  = F(X) a(x) . Hence  H (X) E T z E  and . he impli ca ion i) . - ii) . is p o ed . Le now assume ha H (X) z E T Z E . The e o e d z ID(H (X) z )  E T a (x) a(M)  and mo eo e  F (X) a (x)  ET a (x) a(M) . Le us no ice ha he canonical p ojec ion da(x)I . Ta (x) G (M) -  Tx M  is an isomo phism and  Xx = dU(x) (F (X) a(x) ) d a (X) T (d x a (X x ) ) .  Hence  F (X) U (X)  = dx a(X x )  and his ends he p oo o he implica ion ii) . - i) . III . HOMOGENITY AND TRANSITIVITY OF FIELDS OF GEOMETRIC OBJECTS . In hissec ionwe sugges no ions o homogeni y and ansi i i y o ields o geome ic objec s . We also do his in he in ini esimal case . Then we ela e his no ions o he simi- la p ope ies o E . 1 . We shall use he ollowing no a ion : A (a)  =  { , p  E  P (M)  :  x E dom9 - F(W)a(x)  = a(~p(x))} (a)  =  {X E TM  :  SP E A (a)  whe e ~p is he low o  X} . De ini ion III .1 . We shallcall a homogenous i A(a) ac s ansi i ely on M and  in ini esimally homogenous i o e e y x E M and E T x M he e exis s XE A(a)  such ha X = . x De ini ion 111 .2 . We shallcall a ansi i e i o each . n E II (a  he e exis s W E A(a)  such ha j w x whe e x is-asou ceo 77 . We shallcall a in ini esimally ansi i e i o each 1 E R (a) he e exis s X E A(a) such ha j X =  whe e y is a sou ce o The se s A(a) and A(a) can be desc ibed in he ollo wig . way : Lemma 111 .3 . I a is a 0-de o mable ieldo geome- ic objec s hen a) . A(a) is a se o solu ions o II (a) ; b) . A(a) is a se o solu ions o R (a) . P oo : I 9 E F(M) hen T is a localsolu ion o II (a) i o each xE dom~p j , E II (a) . This is equi alen o he ac ha o each xE domsp F(W)a(x) = a«p(x)) . Hence ;p  is a local solu ion o  II (a)  i  p EA (a)  So  a) .  is p o ed . Le XE A(a) and x E domX and le sp deno e he low o  X  hen  F (sp ) a (x)  = a 0p (x» .  Hence  F (X) a (x) dx a(X x ) . This means ha (LX u) x = 0 because he Lie de i a i e o he ield o geome ic objec s can be exp essed in he ollowing way  (L X a) x = dx a (Xx )  -  F (X) a (x)  (c .  [91,[101) . The e o eX is a solu ion o R (a) . Le now Y be a solu ion o R (a) hen i gene a es he ield F(Y) on F(M) . Since o e e y xE domY F(Y)a(x) = d x a(Y x )  hen  F(Y) la(domY)  is a ec o ieldon  a(domY) . I p is a low o Y hen F(0 )a(x) E a(domY) . Hence F(Ví )a(x)  = a(41 (x))  and  Y E A(a) . .  This ends he p oo o b) . F omlemma 111 .3 we ge immedia ly he ollowing co o- 1 lla y Co olla y 111 .4 . i) . The ield a is homogenous i o each x,y E M he eexis s  ~p  a local solu ion o  I (a)  such ha  ~p (x) =y ; ii) .  a is in ini esimally homogenous i o each x E M and E TxM he eexis s X a local solu ion o R (a) such ha Xx = ; iii) . a is ansi i e i 11 (u) is comple ely in e- g able ; i ) .  a is in ini esimally ansi i e i R (a) is comple ely in eg able . 2 . Wi h he p incipal ib ebundle E we can associa e he ollowing se s : A (E)  _  {Sp  E  P  H ( ~0 ) E Idomp  C  E } A(E)  _  {X E TM  :  p EA(E)  whe e `p is he low o X} The bundle  E  is calledhomogenous i o each  x,y EM he e exis s  E A(E)  such ha  q (x)  = y ;  E  is called an si i e i o each zl,z 2 E E he e exis s 0 E A(E)  such ha H (,, , )z l = z2 . The bundle E is called in ini esimally homogenous i o each xE M and each ETx M he e exis s XE A(E) such ha Xx = ; E is called in ini esimally an si i e i o . each  z E E  and  each  X E T Z E  he e exis s X E A(E)  such ha H (X) z = X . Suchmeanings o (in ini esi mal) homogeni y and (in ini esimal) ansi i i y a e used o ins ance by P . Molino [51 . Lemma 111 .5 . I E is a p incipal ib ebundle associa ed wi h he 0-de o mable ield o geome ic objec s a hen A(a) = A(E) and A(a) = A(E) . P oo . I is enough o p o e ha A(a) = A(E) . Le W E A(a)  hen i is easy o no ice ha a .H OP) _ - Ta1H (dom-p)'  I  z E EldomW  hen  T .(H (SP)z) =' , a (z) = lo - Hence  H «p) El dom(p C E  and  -p E A (E) . On he o he hand le 0 E A(E)  and le j É Ex whe- e xE domo . I implies ha a (H (0)j' ) = a (j ) = 0 . F om he de ini ion o  a we ge ha  F( -1 ., y-1 )a0(x)  = = F( -1)a(x) .  Hence  F  (Vi - 1 )aO(x)  = a(x)  and  ,~ E A(a) .  Tha ends he p oo . In he ollowing p oposi ion we compa e (in ini esimal) homogeni y and (in ini esimal) ansi i i y o he bundle E and he ield o geome icobjec s a . P oposi ion 111 .6 . I a is a 0-de o mable ield o geome ic objec s and E is a p incipal ib ebundle gene a ed by a hen i)  a is homogenous i E is homogenous ; ii)  a is in ini esimally homogenous i E is in ini esimally homogenous ; iii) ais ansi i e i E is ansi i e ; i )  a is in ini esimally ansi i e i E is in ini esimally ansi i e . P oo . The i s condi ion is a simple consec ;uence o Lemma III .5 . ; ii) . one can easy ge omlemma 11 .2 . Simila y i ) . easly ollows om lemma 11 .2 . and lemma 111 .5 . To p o e  iii) . i is enough o no ice ha i  z 1 ,z 2 EE and  WE I' (M)  hen  H (SP) zl = z2 i  jx~p =  z2 . zi1  whe e  x