A note on projective foliations

Pub . Ma . UAB Vol . 27 N8 2 Juny 1983 A NOTEON PROJECTIVE FOLIATIONS Izu Vaisman In [12], Nishikawaand Sa o s udied con o mal and p ojec i e olia ions de ined as olia ions whose second o de ans e sal bundle is endowed wi h ei he a con o mal o a p ojec i ep ojec able s uc u e .  (See, o ins ance [7] o such s uc u eson mani olds .) Namely, hey p o ed he exis ence o co espondingp ojec able no mal Ca an connec ions om which hey deduce ha he same s ong Bo anishing phenomenon like o Riemannian olia ions holds . Then, Nishikawa s udied cha ac e is ic classes o p ojec i e olia ions in [13] . Independen ly, I discussed con o mal olia ions in [16] using he classical de ini ion o con o mal s uc u es by means o Riemannian me ics, and Mon esinos [11] p o ed he s ong Bo anishing heo em o con o mal olia ions, by using his classical app oach . The aim o his No e is o p esen p ojec i e olia ionsby using he  , al e na e app oach o p ojec i e s uc u es known as geome yo pa hs [6], and by cons uc ing he no mal connec ion wi h a ec o bundle e sion o he o iginal Ca an echnique [4] . This app oach will p o ide us no only wi h he Bo - Nishikawa-Sa o anishing heo em o [12, 13] bu also wi h p ojec i ely in a ian ep esen a i e o ms o he eal Pon jagin classes o mani oldsand o ans e se bundles o olia ions . Fu he mo e,we shall ob ain a cohomological obs uc ion o he exis ence o a ans e sal p ojec i ep ojec able (I am using he na e olia e ins ead) s uc u e . Beyond all his, since wo app oaches o p ojec i es uc u es on mani olds a e a ailable, i seems na u al o use hem bo h in s udying p ojec i e olia ions as well . 109 1 . P ojec i eS uc u es on e o mula ion o hosegi en W his pape , we a e always in he endowed wi h a o sionless linea Two c-cha s o U l U' 0 ¢, and o e U l U' one has whe e X,Y a e local ec o ields and ~is a well-de ined 1- o m . A amily F={(Ua,V a )} o c-cha s is a e p ojec i ely ela ed, and {U a } is a las is a p o ec i e s uc u e unique p ojec i e s uc u e . s uc u e on i is a p ojec i e a las . Hence a p ojec i e s uc u e can ion, bu no in a canonical manne . Mani olds . The de ini ions o his sec ion a e a in [6J . Le V n be a di e en iable mani old (in C -ca ego y), and U an open subse o V connec ion V . Then we call (U,V) a c-cha . a e called p ojéc i ely ela ed i ei he U l U' = a p ojec i e a las on a co e ing on V . O cou se, A pai consis ing o mani old . Following a e some well-known ac s conce ningp ojec i e V i any wo o . i s cha s o V . A maximal p ojec i e any p ojec i e a las yields a a mani old V and a p ojec i e mani olds [6] : a Riemannian i) A global o sionless linea connec ion on V, and in pa icula connec ionp o ide a p ojec i e s uc u e . c- .cha s bya pa i ion o uni y, we ge a global c-c ia o he alwaysbe ep esen ed by a global connec- Con e sely, i we glue up local same maximal ü) The unpa ame ized . geodesics o ie connec ions Va o a p ojec i e s uc u e a e he same, and hey yield ie sys em o pa is o ie s uc u e which, in ac , is he cha ac e is ic objec o he p ojec i e mani old . ü i)  A p ojec i e s uc u e can always be de ined by a Ricci symme ic a las (i .e ., one whose connec ions Da ha e a symme ic Ricci cu a u e enso ) .  (Then, ip o (1 .1) is a closed o o .) Mo eo e , we can e en ep esen he s uc u e by a single o sionless Ricci-symme ic linea connec ion . i ) Le n = dim V > 1, and se (1 .2)  W(X,Y)Z = R a CX,Y)Z + n+1 [B a (X,Y)  - B a, CY,X)]Z + + n  {[nB a CZ,Y) + B a (Y,Z)]X - [nB a CZ,X) + Ba(X,Z)]Y} whe e Ra is he cu a u e o Va , and B a is he co espondingRicci enso . Then W does no depend on he index a, and i de ines he Weyl p ojec i e cu a u e enso .  W =0  o n = 2, and o n > 3, W = 0  i V is a p ojec i elyEuclidean mani old , i .e ., one which has a p ojec i e a las consis ing o la connec ions .  (E .g ., he Riemannian mani olds o cons an cu a u ebelong o his class .) )  [5] The Che n-Pon jagin o ms o a Riemannianmani old a e in a ian by p ojec i e ans o ma ions be ween Le i-Ci i a connec ions . This la e ac can be ex ended as ollows . Using he usual local componen s o he enso (1 .2), le us de ine he local 2- o ms (1 .3)  wj = 2 Wljkk dx kn dx L and he global di e en ial o ms (1 .4)  P . C )  =  1 .  n 6 k, . . . k29  W h1  n  . . .  n  Wh2j (27 ) 2j (2j)! h,k=1 F, . . .h 2i  k, The o ms (1 .4) can be compu ed by using a global Ricci-symme ic o sionless linea connec ion as a p ojec i e a las o V (see iii) abo e) . In his case he compu a ion o A . A ez .[l .] Co iginally done o con o mal s uc u es) applies, and P j (V) a e seen o be equal o he usual Che n-Pon jagin o ms o he chosen connec ion . Hence, he Pon jagi n classes o a gene al p ojec i e mani old V can be ep esen ed by p ojec i ely in a ian o ms, and hese o ms a e gi en by (1 .4) . Pa icula ly, e e y p ojec i elyEuclidean mani old has anishing Pon jagin classes . 2 . P ojec i eFolia ions . Now, we shall apply he schema o Sec ion 1 o he ans e se bundle o a olia ion . Le M n be a mani old, and F a olia ion o codimension Q on M (see, o ins ance [2] o gene ali ies on olia ions) . Le E be he angen bundle o F, and Q = T F = TM/E be i s ans e se bundle . Then, we ha e he na u al p ojec ion u : 'DI + Q, and we shall deno e T CX) = X, and  X  any elemen o  The dual bundle  0*  is a subbundle o T*M . Ou connec ion will be o a ach he label olia e o e e y hingwhich is cons an on he lea es o F, and he label basic o e e y hingwhich depends only on he "di e en ials in 0* " . Pa icula ly, a basic connec ion V on 0, is cha ac e ized by [2] (2 .1)  V X Z = [X,Z] . Such a connec ion has ¡he o sion (2 .2)  T(X,Y) -- VXY - VYX - [X,Y] (which does no depend on he choice o X,Y), and i is o sionless i T = 0 . Mo eo e , 0is a olia e bundle, and he basic connec ion V is olia e i o e e y olia e sec ions Z,X, he Sec ion V X Z is also olia e . I is known ha Q has always basic connec ions bu may ha e no olia e connec ions [9J . Finally, wo o sionlessbasic connec ions on Q will be called ans e sally p ojec i ely ela ed i o any ec o ield X on M, and sec ion Z o 0 one has (2.3)  1 XZ = V X Z + a(x)z + a(z)x , o some basic 1- o m a (i .e ., aE Q*), which implies ha a(Z) depends on Z alone) . Now, we shall e e o basic connec ions on Q, de ine like in Sec ion 1 ans e sal c-cha s and a lases , and ge he eby he no ion o a ans e sal p ojec i e s uc u e o he olia ion F . Fu he mo e,i all he connec ions D a o a ans e sal p ojec i e a las a e olia e connec ions we shall say ha his a las de ines a olia e ans e sal p ojec i e s uc u e . A olia ion F endowed wi h a olia e ans e sal p ojec i e s uc u e is called a p ojec i e olia ion . (In his case, he 1- o ms a o (2 .3) a e olia e o ms .) I is ob ious ha he ans e sal p ojec i e s uc u e o a p ojec i e olia ion F o codimension q is locally he pulí-back o a p ojec i e s uc u e o Rq by he local sub- me sionswhich de ine F [2], and he la e a e ela ed by p ojec i e di eomo phisms .  This p o es ha ou de ini ion o a p ojec i e olia ion is equi alen o ha o [12] . Mo eo e , one can ge ans e sal pa hs which a e he pull-backs o he pa hs o he p ojec i e s uc u e o Rq men ioned abo e . Like in Sec ion 1, we see ha a global o sionless basic Q-connec ion de ines a ans e sal p ojec i e s uc u e o F, and e e y such s uc u e has a lases consis ing o a single global cha . Pa icula ly, he ans e sal pa o he second connec ion o a Riemannian me ic o M wi h espec o F [14,15] o e s a ans e salp ojec i e s uc u e o F, which p o es he exis ence o such s uc u es o e e y F . Bu , gene ally, only local olia e ans e sal p ojec i e s uc u es exis . Following [14,15], we shall co e M, by la coo dina e neighbou hoods, wi h local coo dina es (x a ,x u ) (a,b, . . . = 1, . . .,q ; u, , . . . = q+l, . . .,n), such ha x a = cons . de ine he lea es o F, and he changes o he coo dina es a e locally o he o m (2 .4)  la = ¡ a (xb), Xu = ¡ u (xb ,x ) Then, we choose once and o e e an auxilia y Riemann me ic g, we iden i y Q wi h he co esponding no mal bundle o F, and ake he local bases and cobases (2 .5)  Xa = áa - u a u E Q(lE),  Xu = a uE E , ax ax  ax (2 .6)  dx a , 6 u = dx u+ u dx a . All he ollowing enso componen s a e wi h espec o (2 .5), (2 .6) . (2.7)  O  x  =Yab Xc  '  X X a = 0 ba  . u and i has no o sion i Y ab = Y ba ' A p ojec i e ans o ma ion (2 .3) akes he o m (2 .8) whe e X = Xa dx a is he 1- o m o (2 .3) . The cu a u e R o 0 is gi en by (2 .9)  R(Xa'Xb)X, = Re cab X e ' R(X a' X u )X c = Re cau Xe ' R(X u' X )Xc = 0, whe e Now, a basic connec ion on Q has local equa ions Y ab - Y ab + a a"b + ab a (2 .10)  Re cab = Xa Ycb  b Y ca +YcbY ha - Y ca y hb ' Re cau  -Xu Y ca ' (2 .11)  Re cab + Re cac + Re caa = 0' Re cau = Re cau . Le us also ecall ha he decomposi ion TM = E ® Q (Q 1 E) induces a decomposi ion o di e en ial o ms in o componen s o ype (p, ) (which con ain p o ms dx a , and o ms 0 u in hei local exp ession), and a decomposi ion d = d' + d" + 8 o he ex e io di e en ia ion d in o componen s o he espec i e ype (1,0), (0,1), (2,-1) [14,15] . A basic connec ion 0 has he ollowing impo an associa ed 2- o m q  _ q (2 .12)  S(X,Y) = 1 dx c (R(X,Y)X C ) = Y. dx c (R(X,Y)X C ) , c=1  c=1 which, ob iously, does no depend on he choice o g . One has P oposi ion 2 .1 . The o m R is an exac 2- o m . P oo . F om (2 .12) and (2 .10), we ge (2 .13)  S = d(Y' a dx a ) , bu we a e no ye done since ~ú = YC  dx a is only a local 1- o m . Bu deno ing h = g/Q , using he compu a ions o [18], and applying (2 .13) we shall ind ha he S- o mo he connec ion bc induced on Q by he second connec ion o g Cal eady men ioned ea lie ) is (2 .14) S = d(Fc a dx a ) = di(X c In de  dx c l = dd' In  e  = = d(d-d") In  e  = -dd" In  e He e, in iewo o mulas (2 .4), we can see ha  d" In VáeI - is a well de ined global 1- o m . Fu he mo e, c = y c - c is a " enso ", whence c dx a is a ab ab ab  ca global 1- o m . This yields (2 .15)  S = d( c a dx a - d" In  e ) , and p o es he p oposi ion . The idea o he abo ep oo is he one used in [6] o ge a p ojec i ely ela ed connec ion wi h symme ic Ricci cu a u e on a p ojec i e mani old (see iii) o Sec ion 1) .  Indeed, on a mani old, he symme y o he Ricci enso is equi alen o S = 0 . Howe e ,in ou case we canno ge B = 0 (globally) bu , i we apply he same p oo as in [6, p.88], we can ob ain a p ojec i ely ela ed connec ion on Q such ha S = da wi h a o ype (0,1) . us de ine Now, le us conside again a ans e sal p ojec i e s uc u e o he olia ion F, de ined by ans e sal c-a las wi h basic connec ions D a , and le (2 .16)  W(x,Y)2 = R a (x,Y)2 - q+1 ¡s a (x,Y)z + s a (z,Y)xJ , whe e he ield Y is angen o F . A s aigh o wa dcheckingshows ha W does no depend on he choice o Z co esponding o Z, and i is in a ian by (2 .3) . (The condi ion Y E E is essen ial .) This checking is easy by using he local componen s e  P  1 - _e_( %) P_( .l, (2 .i7)  Nl cau  () cau  q+1 ( o c~au  + da~cu~j' whe e Sau) = Rc  , and he o mulas (2 .10), (2 .8) . (a) cau The ope a o W yields a well-de ined2- o m o ype (1 .1) on M, wi h alues in he olia e ec o bundle Hom(Q,Q), which has he local componen s (2 .17) . IVe shall deno e his o m by w, and call i he auxilia y Weyl o m . I p o ides us wi h a cohomological bbs uc ion o he exis ence o a olia e ans e se p ojec i e s uc u e since we ha e Theo em 2 .2 . The auxilia y Weyl o m w is d"-clos ed, and i de ines a d"-cohomology class which is independen on he ans e se p ojec i e s uc u e o F . The ólia ion F ádmi s a ólia e a is e sé'p ojéc i e s uc u é i w is also d"-exac . P oo . Le us s a wi h a ans e sep ojec i e s uc u e o F, and he co esponding o m w . Le Da be one o he local connec ions o his s uc u e, and R a be he co esponding o m (2 .12) . Then, i we conside a ans o ma ion (2 .3) (o (2 .8)) whe e a is a basic o m, such ha S+ (q+l)dX = 0, we ge ano he connec ion o he same p ojec i e s uc u e whose S- o m is ze o . By (2 .13) such a o m X exis s locally . Hence, we can always choose a p ojec i ely equi alen a laswhose connec ions  0  ha e anishing o ms  S a .  (Bu , gene ally, (x his new a las has mo e han one cha .) Now, since W is p ojec i ely in a ian , we can exp ess i wi h hese connec ions D a , and (2 .16) yields (2 .18)  W(x,Y)z = Ra(X,Y) I ollows ha w is p ecisely he (1,1)- ypepa o he cu a u e o a basic connec ion, and i is known om [9] ha he la e is d"-closed . Now, le us no e ha he d"-exac ness o w means ha some " enso " o local componen s e a exis s such ha (2 .19)  W e  e  . cau - Xu ca Bu hen, i ollows om (2 .10) and (2 .18) ha he d"-cohomology class o w is well de ined, and i does no depend on he ans e sep ojec i e s uc u e used o F . I is known [15] ha his class ep esen san elemen [w] E H 1 1 11,1 1 (Hom(Q,Q» )  , ' whe e he second a gumen deno es he shea o ge ms o olia e(1,0)- o ms wi h alues in Hom(Q,Q) . We shall say ha [w] is he p ojec i e Molino-A iyah class o F [9] . Pa icula ly, i a olia e ans e se p ojec i e s uc u e exis s, hen (2 .10) and (2 .18) yield ha i s auxilia y Weyl o m is w=0, whence [w]=0, and w o any o he ans e sep ojec i e s uc u e is d"-exac . (3 .21)  T ac  `"acb - E) acb - (q-1)K ac ' and i has an in a ian meaning o ask T o be skew-symme ic, which gi es ( o q % 2) (3 .22) _ 1 (b b Kac  2(q-1) ~acb + ~cab This means ha we a e able o de e mine a canonical connec ion K, i I 00 is chosen, and we p o ed Theo em 3 .1 . Le F be a olia ion o M o codimension q % 2, endowed wi h a ans e se p ojec i e s uc u e . Le us choose he auxilia y Riemannian me ic g, and a basic connec ion  n o0  on  K(F) .  Then, he e is a unique connec ion on T(F), which sa is ies he co ndi ions (3 .11) and (3 .22) . 124 The connec ion o Theo em 3 .1 will be called he no mal Ca an connec ion (compa e wi h [4] and [16]) . I , he . p ojec i e s uc u e o . Fis olia e,we may use in he abo e compu a ions o Ká local olia e connec ions o, and we see ha he no mal connec ion o T(F) is "equal up o he choice o mo0 11 o he li s o he no mal connec ions o he "local bases" o F . Now, in o de o escape om he a bi a y connec ion o m n00 we ha e o go o e o he p ojec i iza ion o T(F), and i is nice o do his in he languageo p incipal bundles . Le us conside he p incipal bundle B T o he bases o T(F), ac o ize i by he ela ion o p opo ionali y, and ge he bundle P T o he p ojec i e ames o he ib eso T(F), whose s uc u e g oup is he q-dimensional p ojec i e g oup . Then, le us ake he p incipal subbundle B0 o B T consis ing o bases wi h he i s ec o p opo ional o e o (3 .2), and pe o en he same ac o iza ion o ge a subbundle P0 o P T o which he s uc u e g oup is he cen al-p ojec i e g oup (i .e ., he g oup o he p ojec i e ans o ma ions wi h a i en ixed oin ) . Followin  4 g  p  g [],  i is  P 0 T  which plays he main ole ;  we conside i as a olia e p incipal bundle wi h he ansi ion cocycle (3 .12) . I is known ha he gene al p ojec i e g oup P(q,R) is GQ(q+1,R)/cen e, whence he co esponding Lie algeb a p(q,R) is gk(q+1,R)%{pI} (I is he uni ma ix) . Hence, (a'a) and (aa~ o g£(q+1,R)  de ine he same elemen o P(q,R) i (3 .23)  a 'a - saa'o =aa - daa0 and we can always ake aa - daa 0as he ep esen a i e o he co esponding elemen o P(q,R) . The cen al p ojec i e g oup P0 (q,R) and he co esponding Lie algeb a p0 (q,R) a e de ined simila ly bu using only ma ices  (aa) wi h Now, he no mal connec ion o T(F) yields a connec ion on B T wi h he g£(q+1,R)- alued local connec ion o ms (Ko), and his induces a connec ion on P T . Because o (3 .12), he ma icesob ained om (Ko) by eplacing Ka wi h 0 will yield a connec ion on BT , and his induces a connec ionon PT , whose P0 (q,R)- alued local o ms a e ep esen ed by (Ka - d K') . As shown by (3 .11) and a (3 .22), he la e ma icesdo no depend on n0 any mo e . Finally, le us also no e ano he impo an p ope y o PT . We s a by in oducing in he mani old B T he local coo dina es (xa ,x u ,  whe e ~a a e he componen s o he ec o s o a ame o B T wi h espec o he bases (3 .10) . Then ~a a e "homogeneouscoo dina es" in PT , and, in iew o (3 .12), he local equa ions xa = cons ., quo ien s o la = cons . de ine on P0 a olia ion F0 whose lea es co e he lea es o F (like in he Riemanniancase [10]) . The men ioned p ope y (which is a eason o e e ing o P0 ) is ha F0 ádmi s a ans e se pa alleliza ion . (This is known o q=n [7] .) Indeed, le (n o ) be he in e se ma ix o (Ca) .Then, he global gk(q+1,R)- alued connec ion o m o he no mal connec ion on B T is he ma ix [8] (3 .24)  Ea = nodIX + noEaKY , and he induced p(q,R)- o m on P Tis gi en by (3 .25)  . .R -j ow 0 =no dE x - 60n0dja +  ( n ' E Y - 0n01Y)(Ka - 5xK0) `a  a0  a a  aa 0  aa  CL a0  y  y0  ' which a e q 2 +2q linea lyindependen 1- o ms on PT . Fu he mo e, he es ic ion o he o ms (3 .25), Ea excep ed, o P0 de ine he no mal connec ion on P0 , whence hey p o ide q 2 +q independen 1- o ms on PT . Bu , i is easy o see ha  ~0/p T = Ti JO dxb , and i we add hem we ob ain in all  q?+2q independen 1- o ms on he mani old PT , which cons i u e a global ield o ans e se co ameso he olia ion F0 . Clea ly, hese co ames depend only on he ans e se p ojec i e s uc u e o F, and on he auxilia y Riemann me ic g o M . Mo eo e , i F is a p ojec i e olia ion heseco ames do no depend on , g, and hey a e olia e wi h espec o FO . By going o e o he co esponding dual ames, we see ha we ha e ob ained Theo em 3 .2 . Le  F be a olia ion o codimension q ó 2  on M, and g be an auxilia y Riemannian me ic . Then, o e e y ans e sep ojec i e s uc u e o F, he e is a uniquely de ined global ans e se pa allelism ( he "no mal pa allelism") o he olia ion F0 on PT .  I he gi en p ojec i e s uc u e is olia e, his pa allelism is independen o g, and is olia e as well . The e o e, o p ojec i e olia ionswe ha e a si ua ion which is simila o he one encoun e ed in he case o he Riemannian olia ions [10], and one migh y o use he me hods o [10] in he s udy o he p ojec i e olia ions . Rema k . Ca an'so iginal me hod [3] could be used simila ly in o de o w i e down he no mal connec ion o a con o mal olia ion . Namely, i F is a con o mal olia ion, and i {ha} is a se o olia e me ics o OJUa, (whe e {U .,} is a la open co e ing o M), which de ines he con o mal s uc u e [16], hen one has some ela ions ha = 0ashs, whe e d as a e posi i e olia e eal unc ions on Ua l U s , and de ine a 1-cocycle o he co e ing {U a } . This p o ides us wi h a olia e line bundle e on M ha ing Das as i s ansi ion cocycle, and one can see ha he no mal Ca an connec ion [3] can be ob ained on e® (0i®0) ®e -1 'REFERENCES 1 . A . A ez, Cha ac e is ic Classes and Weyl Tenso : Applica ions o Gene al Rela i i y, P oc . Na . Acad . Sci ., U .S .A ., 66 (1970), 265-268 . 2 . R . 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Rebu ee . 3 de juny del 1983 Depa men o Ma hema ics Uni e si y o Hai a ISRAEL