A note on projective foliations

Pub . Mat . UAB Vol . 27 N8 2 Juny 1983 A NOTEON PROJECTIVE FOLIATIONS Izu Vaisman In [12], Nishikawaand Sato studied conformal and projective foliations defined as foliations whose second order transversal bundle is endowed with either a conformal or a projectiveprojectable structure .  (See, for instance [7] for such structureson manifolds .) Namely, they proved the existence of correspondingprojectable normal Cartan connections from which they deduce that the same strong Bott vanishing phenomenon like for Riemannian foliations holds . Then, Nishikawa studied characteristic classes of projectivefoliations in [13] . Independently, I discussed conformal foliations in [16] using the classical definition of conformal structures by means of Riemannian metrics, and Montesinos [11] proved the strong Bott vanishing theorem for conformal foliations, by using this classical approach . The aim of this Note is to present projective foliationsby using the  , alternate approach to projective structures known as geometryof paths [6], and by constructing the normal connection with a vector bundle version of the original Cartan technique [4] . This approach will provide us not only with the BottNishikawa-Sato vanishing theorem of [12, 13] but also with projectively invariant representative forms of the real Pontrjagin classes of manifoldsand of transverse bundles of foliations . Furthermore,we shall obtain a cohomological obstruction to the existence of a transversal projectiveprojectable (I am using the narre foliate instead) structure . Beyond all this, since two approaches to projectivestructures on manifolds are available, it seems natural to use them both in studying projective foliations as well . 109 1 . ProjectiveStructures on reformulation of thosegiven W this paper, we are always in the endowed with a torsionless linear Two c-charts or U fl U' 0 ¢, and over U fl U' one has where X,Y are local vector fields and ~is a well-defined 1-form . A family F={(Ua,V a )} of c-charts is are projectively related, and {U a } is atlas is a pro ective structure unique projective structure . structure on it is a projective atlas . Hence a projective structure can tion, but not in a canonical manner . Manifolds . The definitions of this section are a in [6J . Let V n be a differentiable manifold (in C -category), and U an open subset of V connection V . Then we call (U,V) a c-chart . are called projéctively related if either U fl U' = a p rojective atlas on a covering on V . Of course, A pair consisting of manifold . Following are some well-known facts concerningprojective V if any two of . its charts of V . A maximal projective any projective atlas yields a a manifold V and a projective manifolds [6] : a Riemannian i) A global torsionless linear connection on V, and in particular connectionprovide a projective structure . c- .charts bya partition of unity, we get a global c-cfiart of the alwaysbe represented by a global connecConversely, if we glue up local same maximal ü) The unparametrized . geodesics of tfie connections Va of a projective structure are the same, and they yield tfie system of patfis of tfie structure which, in fact, is the characteristic object of the projective manifold . ü i)  A projective structure can always be defined by a Ricci symmetric atlas (i .e ., one whose connections Da have a symmetric Ricci curvature tensor) .  (Then, ip of (1 .1) is a closed forro .) Moreover, we can even representthe structure by a single torsionless Ricci-symmetric linear connection . iv) Let n = dim V > 1, and set (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} where Ra is the curvature of Va , and B a is the correspondingRicci tensor . Then W does not depend on the index a, and it defines the Weyl projective curvature tensor .  W =0  for n = 2, and for n > 3, W = 0  iff V is a projectivelyEuclidean manifold , i .e ., one which has a projective atlas consisting of flat connections .  (E .g ., the Riemannian manifolds of constant curvaturebelong to this class .) v)  [5] The Chern-Pontrjagin forms of a Riemannianmanifold are invariant by projective transformations between Levi-Civita connections . This latter fact can be extended as follows . Using the usual local components of the tensor (1 .2), let us define the local 2-forms (1 .3)  wj = 2 Wljkk dx kn dx L and the global differential forms (1 .4)  P . Cv)  =  1 .  n 6 k, . . . k29  W h1  n  . . .  n  Wh2j (27r) 2j (2j)! h,k=1 F, . . .h 2i  k, The forms (1 .4) can be computed by using a global Ricci-symmetric torsionless linear connection as a projective atlas of V (see iii) above) . In this case the computation of A . Avez .[l .] Coriginally done for conformal structures) applies, and P j (V) are seen to be equal to the usual Chern-Pontrjagin forms of the chosen connection . Hence, the Pontrjagi n classes of a general projective manifold V can be represented by projectively invariant forms, and these forms are given by (1 .4) . Particularly, every projectivelyEuclidean manifold has vanishing Pontrjagin classes . 2 . ProjectiveFoliations . Now, we shall apply the schema of Section 1 to the transverse bundle of a foliation . Let M n be a manifold, and F a foliation of codimension Q on M (see, for instance [2] for generalities on foliations) . Let E be the tangent bundle of F, and Q = Tr F = TM/E be its transverse bundle . Then, we have the natural projection u : 'DI + Q, and we shall denote Tr CX) = X, and  X  any element of  The dual bundle  0*  is a subbundle of T*M . Our connection will be to attachthe label foliate to everythingwhich is constanton the leaves of F, and the label basic to everythingwhich depends only on the "differentials in 0* " . Particularly, a basic connection V on 0, is characterized by [2] (2 .1)  V X Z = [X,Z] . Such a connection has ¡he torsion (2 .2)  T(X,Y) -- VXY - VYX - [X,Y] (which does not depend on the choice of X,Y), and it is torsionless if T = 0 . Moreover, 0is a foliate bundle, and the basic connection V is foliate if for every foliate sections Z,X, the Section V X Z is also foliate . It is known that Q has always basic connections but may have no foliate connections [9J . Finally, two torsionlessbasic connections on Q will be called transversally projectively related if for any vector field X on M, and section Z of 0 one has (2.3)  1 vXZ = V X Z + a(x)z + a(z)x , for some basic 1-form a (i .e ., aE Q*), which implies that a(Z) depends on Z alone) . Now, we shall refer to basic connections on Q, define like in Section 1 transversal c-charts and atlases , and get thereby the notion of a transversal projective structure of the foliation F . Furthermore,if all the connections D a of a transversal projective atlas are foliate connections we shall say that this atlas defines a foliate transversal projective structure . A foliation F endowed with a foliate transversal projective structure is called a projective foliation . (In this case, the 1-forms a of (2 .3) are foliate forms .) It is obvious that the transversal projective structure of a projective foliation F of codimension q is locally the pulí-back of a projective structure of Rq by the local submersionswhich define F [2], and the latter are related by projective diffeomorphisms .  This proves that our definition of a projective foliation is equivalent to that of [12] . Moreover, one can get transversal paths which are the pull-backs of the paths of the projective structure of Rq mentioned above . Like in Section 1, we see that a global torsionless basic Q-connection defines a transversal projective structure of F, and every such structure has atlases consisting of a single global chart . Particularly, the transversal part of the second connection of a Riemannian metric of M with respect to F [14,15] offers a transversalprojective structure of F, which proves the existence of such structures for every F . But, generally, only local foliate transversal projective structures exist . Following [14,15], we shall cover M, by flat coordinate neighbourhoods, with local coordinates (x a ,x u ) (a,b, . . . = 1, . . .,q ; u,v, . . . = q+l, . . .,n), such that x a = const . definethe leaves of F, and the changes of the coordinates are locally of the form (2 .4)  la = ¡ a (xb), Xu = ¡ u (xb ,x v ) Then, we choose once and for ever an auxiliary Riemann metric g, we identify Q with the corresponding normal bundle of F, and take the local bases and cobases (2 .5)  Xa = áa - t u a u E Q(lE),  Xu = a uE E , ax ax  ax (2 .6)  dx a , 6 u = dx u+ tu dx a . All the following tensor components are with respect to (2 .5), (2 .6) . (2.7)  O  x  =Yab Xc  '  v X X a = 0 ba  . u and it has no torsion iff Y ab = Y ba ' A projective transformation (2 .3) takes the form (2 .8) where X = Xa dx a is the 1-form of (2 .3) . The curvature R of 0 is given 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 v )Xc = 0, where Now, a basic connection on Q has local equations Y ab - Y ab + a a"b + abra (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 . Let us also recall that the decomposition TM = E ® Q (Q 1 E) induces a decomposition of differential forms into components of type (p,r) (which contain p forms dx a , and r forms 0 u in their local expression), and a decomposition d = d' + d" + 8 of the exterior differentiation d into components of the respective type (1,0), (0,1), (2,-1) [14,15] . A basic connection 0 has the following important associated 2-form 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, obviously, does not depend on the choice of g . One has Proposition 2 .1 . The form R is an exact 2-form . Proof . From (2 .12) and (2 .10), we get (2 .13)  S = d(Y' a dx a ) , but we are not yet done since ~ú = YC  dx a is only a local 1-form . But denoting h = g/Q , using the computations of [18], and applying (2 .13) we shall find that the S-formof the connection rbc induced on Q by the second connection of g Calready mentioned earlier) is (2 .14) S = d(Fc a dx a ) = di(X c In det  dx c l = dd' In  et  = = d(d-d") In  et  = -dd" In  et Here, in viewof formulas (2 .4), we can see that  d" In VáeI - rt is a well defined global 1-form . Furthermore, tc = y c - fc is a "tensor", whence tc dx a is a ab ab ab  ca global 1-form . This yields (2 .15)  S = d(tc a dx a - d" In  et ) , and proves the proposition . The idea of the aboveproof is the one used in [6] to get a projectively related connection with symmetric Ricci curvature on a projective manifold (see iii) of Section 1) .  Indeed, on a manifold, the symmetry of the Ricci tensor is equivalent to S = 0 . However,in our case we cannot get B = 0 (globally) but, if we apply the same proof as in [6, p.88], we can obtain a projectively related connection on Q such that S = da with a of type (0,1) . us define Now, let us consider again a transversal projective structure of the foliation F, defined by transversal c-atlas with basic connections D a , and let (2 .16)  W(x,Y)2 = R a (x,Y)2 - q+1 ¡s a (x,Y)z + s a (z,Y)xJ , where the field Y is tangent to F . A straightforwardcheckingshows that W does not depend on the choice of Z corresponding to Z, and it is invariant by (2 .3) . (The condition Y E E is essential .) This checking is easy by using the local components e  P  1 - f_e_(r%) P_(rv .l, (2 .i7)  Nl cau  () cau  q+1 ( o c~au  + da~cu~j' where Sau) = Rc  , and the formulas (2 .10), (2 .8) . (a) cau The operator W yields a well-defined2-form of type (1 .1) on M, with values in the foliate vector bundle Hom(Q,Q), which has the local components (2 .17) . IVe shall denote this form by w, and call it the auxiliary Weyl form . It provides us with a cohomological bbstruction to the existence of a foliate transverse projective structure since we have Theorem 2 .2 . The auxiliary Weyl form w is d"-clos ed, and it defines a d"-cohomology class which is independent on the transverse projective structure of F . The fóliation F ádmits a fóliate fraris versé'projéctive structuré iff w is also d"-exact . Proof . Let us start with a transverseprojective structure of F, and the corresponding form w . Let Da be one of the local connections of this structure, and R a be the corresponding form (2 .12) . Then, if we consider a transformation (2 .3) (or (2 .8)) where a is a basic form, such that S+ (q+l)dX = 0, we get another connection of the same projective structure whose S-form is zero . By (2 .13) such a form X exists locally . Hence, we can always choose a projectively equivalentatlaswhose connections  0  have vanishing forms  S a .  (But, generally, (x this new atlas has more than one chart .) Now, since W is projectively invariant, we can express it with these connections D a , and (2 .16) yields (2 .18)  W(x,Y)z = Ra(X,Y)f It follows that w is precisely the (1,1)-typepart of the curvature of a basic connection, and it is known from [9] that the latter is d"-closed . Now, let us note that the d"-exactness of w means that some "tensor" of local components te a exists such that (2 .19)  W e  e  . cau - Xut ca But then, it follows from (2 .10) and (2 .18) that the d"-cohomology class of w is well defined, and it does not depend on the transverseprojective structure used for F . It is known [15] that this class representsan element [w] E H 1 1 11,1 1 (Hom(Q,Q» )  , ' where the second argument denotes the sheaf of germs of foliate(1,0)-forms with values in Hom(Q,Q) . We shall say that [w] is the projective Molino-Atiyah class of F [9] . Particularly, if a foliate transverse projective structure exists, then (2 .10) and (2 .18) yield that its auxiliary Weyl form is w=0, whence [w]=0, and w of any other transverseprojective structure is d"-exact . (3 .21)  T ac  `"acb - E) acb - (q-1)K ac ' and it has an invariant meaning to ask T to be skew-symmetric, which gives (for q % 2) (3 .22) _ 1 (b b Kac  2(q-1) ~acb + ~cab This means that we are able to determine a canonical connection K, if Ir00 is chosen, and we proved Theorem 3 .1 . Let F be a foliation of M of codimension q % 2, endowed with a transverse projective structure . Let us choose the auxiliary Riemannian metric g, and a basic connection  n o0  on  K(F) .  Then, there is a unique connection on T(F), which satisfies the co nditions (3 .11) and (3 .22) . 124 The connection of Theorem 3 .1 will be called the normal Cartan connection (compare with [4] and [16]) . If,the . projective structure of . Fis foliate,we may use in the above computations of Ká local foliate connections o, and we see that the normal connection of T(F) is "equal up to the choice of mo0 11 to the lifts of the normal connections of the "local bases" of F . Now, in order to escape from the arbitrary connection form n00 we have to go over to the projectivization of T(F), and it is nice to do this in the languageof principal bundles . Let us consider the principal bundle B T of the bases of T(F), factorize it by the relation of proportionality, and get the bundle P T of the projective frames of the fibresof T(F), whose structure group is the q-dimensional projective group . Then, let us take the principal subbundle B0 of B T consisting of bases with the first vectorproportional to e of (3 .2), and perforen the same factorization to get a subbundle P0 of P T for which the structure group is the central-projective group (i .e ., the group of the projective transformations with a iven fixed oint) . Followin  4 g  p  g [],  it is  P 0 T  which plays the main role ;  we consider it as a foliate principal bundle with the transition cocycle (3 .12) . It is known that the general projective group P(q,R) is GQ(q+1,R)/centre, whence the corresponding Lie algebra p(q,R) is gk(q+1,R)%{pI} (I is the unit matrix) . Hence, (a'a) and (aa~ of g£(q+1,R)  definethe same element of P(q,R) iff (3 .23)  a 'a - saa'o =aa - daa0 and we can always take aa - daa 0as the representative of the corresponding element of P(q,R) . The central projective group P0 (q,R) and the corresponding Lie algebra p0 (q,R) are defined similarly but using only matrices  (aa) with Now, the normal connection of T(F) yields a connection on B T with the g£(q+1,R)-valued local connection forms (Ko), and this induces a connection on P T . Because of (3 .12), the matricesobtained from (Ko) by replacing Ka with 0 will yield a connection on BT , and this induces a connectionon PT , whose P0 (q,R)-valued local forms are represented by (Ka - d K') . As shown by (3 .11) and a (3 .22), the latter matricesdo not depend on n0 any more . Finally, let us also note another important property of PT . We start by introducing in the manifold B T the local coordinates (xa ,x u ,  where ~a are the components of the vectors of a frame of B T with respect to the bases (3 .10) . Then ~a are "homogeneouscoordinates" in PT , and, in view of (3 .12), the local equations xa = const ., quotients of la = const . define on P0 a foliation F0 whose leaves cover the leaves of F (like in the Riemanniancase [10]) . The mentioned property (which is a reason of refering to P0 ) is that F0 ádmits a transverse parallelization . (This is known for q=n [7] .) Indeed, let (n o ) be the inverse matrix of (Ca) .Then, the global gk(q+1,R)- valued connection form of the normal connection on B T is the matrix [8] (3 .24)  Ea = nodIX + noEaKY , and the induced p(q,R)-form on P Tis given 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 are q 2 +2q linearlyindependent 1-forms on PT . Furthermore, the restriction of the forms (3 .25), Ea excepted, to P0 define the normal connection on P0 , whence they provide q 2 +q independent 1-forms on PT . But, it is easy to see that  ~0/p T = Ti JO dxb , and if we add them we obtain in all  q?+2q independent 1-forms on the manifold PT , which constitute a global field of transverse coframesof the foliation F0 . Clearly, these coframes depend only on the transverse projective structure of F, and on the auxiliary Riemann metric g of M . Moreover, if F is a projective foliation thesecoframes do not depend on , g, and they are foliate with respect to FO . By going over to the corresponding dual frames, we see that we have obtained Theorem 3 .2 . Let  F be a foliation of codimension q ó 2  on M, and g be an auxiliary Riemannian metric . Then, for every transverseprojective structure of F, there is a uniquely defined global transverse parallelism (the "normal parallelism") of the foliation F0 on PT .  If the given projective structure is foliate,this pa rallelism is independent of g, and is foliate as well . Therefore, for projective foliationswe have a situation which is similar to the one encountered in the case of the Riemannian foliations [10], and one might try to use the methods of [10] in the study of the projective foliations . Remark . Cartan'soriginal method [3] could be used similarly in order to write down the normal connection of a conformal foliation . Namely, if F is a conformal foliation, and if {ha} is a set of foliate metrics of OJUa, (where {U .,} is a flat open covering of M), which defines the conformal structure [16], then one has some relations ha = 0ashs, where d as are positive foliate real functions on Ua fl U s , and define a 1-cocycle of the covering {U a } . This provides us with a foliate line bundle e on M having Das as its transition cocycle, and one can see that the normal Cartan connection [3] can be obtained on e® (0i®0) ®e -1 'REFERENCES 1 . A . Avez, Characteristic Classes and Weyl Tensor : Applications to General Relativity, Proc . Nat . Acad . Sci ., U .S .A ., 66 (1970), 265-268 . 2 . R . Bott, Lectureson Characteristic Classes and Foliations, Lect . Notes in Math ., 279, Springer-Verlag, Berlin, 1972, 1-94 . 3 . E . Cartan, Les espaces á connexion conforme, Ann . Soc . Polon . Math ., 2 (1923), 171-221 (Oeuvres Complbtes, III 1, Gauthier-Villars, París, 1955, 747-798 .) 4 . E . Cartan, Sur les variétés á connexion projective, Bull . Soc . Math . France, 52 (1924), 205-241 . (Oeuvres Complbtes, III 1, Gauthier-Villars, Paris, 1955, 825-862 .) 5 . S .S . Chern, Geometry of Characteristic Classes, Proc . 13th Biannual Sem . Canad . Math . Congress, Halifax 1971, 1-40 . 6 .  L .P . Eisenhart, Non-Riemannian Geometry, American Math . Soc . Colloquimm Publ . VIII, New Printing : American Math . Soc . Providence, R .I ., 1972 . 7 . S . Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, New-York, 1972 . 8 . S . Kobayashi and K . Nomizu, Foundations of Differential Geometry, I . II, John Wiley and Sons, New York, 1963, 1969 . 9 . P . Molino, Propriétés cohomologiques et propriétés topologiques des feuilletages á connexion transverse projetable,Topology 12 (1973), 317-325 . 10 . P . Molino, Géométrie globale des feuilletagesriemanniens, Proc . Koninklijke Nederlandse Akad ., Series A, 85 (1982), 45-76 . 11 . A . Montesinos,ConformalCurvature for the Normal Bundle of a Conformal Foliation, Ann . Inst . Fourier, Grenoble 32 (1982), 261-274 . 12 . S . Nishikawa and H . Sato . On Characteristic Classes of Riemannian, Conformal, and Projective Foliations, J . Math . Soc . Japan 28 (1976), 223-241 . 13 . S . Nishikawa, Residues and Characteristic Classes for Projective Foliations, Japanese J . of Math . 7 (1981), 45-108 . 14 . I . Vaisman, VariétésRiemanniennes Feuilletées,Czechosl . Math . J . 21 (1971), 46-75 . 15 . I . Vaisman, Cohomology and Differential Forms, M . Dekker, Inc . New York, 1973 . 16, 1, Vaisman, Conformal Foliations, Kodai Math . J . 2 (1979), 26-37 . Rebut ee . 3 de juny del 1983 Departmentof Mathematics University of Haifa ISRAEL