A NOTE ON PROJECTIVE FOLIATIONS

A NOTE ON PROJECTIVE FOLIATIONS Izu Vaisman In [12], Nishikawa and Sato studied conformal and projective foliations defined as foliations whose second order transversal bundle is endowed with either a conformal or a projective projectable structure . (See, for instance [7] for such structures on manifolds .) Namely, they proved the existence of corresponding projectable 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 projective foliations 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 foliations by using the , alternate approach to projective structures known as geometry of 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 manifolds and of transverse bundles of foliations . Furthermore, we shall obtain a cohomological obstruction to the existence of a transversal projective projectable (I am using the narre foliate instead) structure . Beyond all this, since two approaches to projective structures on manifolds are available, it seems natural to use them both in studying projective foliations as well .

Following are some well-known facts concerning projective 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 connection provide a projective structure .Hence, the Pontrjagin classes of a general projective manifold V can be represented by projectively invariant forms, and these forms are given by (1 .4) .
2 .Projective Foliations .Now, we shall apply the schema of Section 1 to the transverse bundle of a foliation .Let Mn 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 attach the label foliate to everything which is constant on the leaves of F, and the label basic to everything which depends only on the "differentials in 0* " .Particularly, a basic connection V on 0, is characterized by [2] (2 .1)VXZ = [X,Z] .
Such a connection has ¡he torsion (which does not depend on the choice of X,Y), and it is torsionless if T = 0 .
Moreover, 0 is a foliate bundle, and the basic connection V is foliate if for every foliate sections Z,X, the Section V XZ is also foliate .It is known that Q has always basic connections but may have no foliate connections [9J .Finally, two torsionless basic connections on Q will be called transversally projectively related if for any vector field X on M, and section Z of 0 one has for some basic 1-form a (i .e., a E 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 submersions which 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 transversal projective 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 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 All the following tensor components are with respect to (2 .5),(2 .6) .
A basic connection 0 has the following important associated 2-form 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 The idea of the above proof 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) .Corollary 2 .4 .If F is a projective foliation of codimension q > 2, one has necessarily PontkQ = 0 for k > q .
3 . .The Cartan Normal Connection .Let (M,F) be a foliate manifold with transversal projective structure defined by a global basic torsionless connection 0 of Q .(We shall use again the notations of Section 2) .Inspired by Cartan [4], let us consider the canonical line bundle K(F) = AqQ* of Q, then define which is a foliate vector bundle of rank q+1, with the local bases Then we get the following new connection forms and it is differentially but not necessarily foliately cohomologous with (3 .3) .
Hence (3 .3)and (3 .12)define different foliate vector bundles .Hereafter, we shall always refer to T(F) as the fóliate bundle (3 .12),and to the connections and it has an invariant meaning to ask T to be skew-symmetric, which gives (for q % 2) 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 conditions (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.F is 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 language of 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 PT of the projective frames of the fibres of 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 vector proportional 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) define the same element of P(q,R) iff and we can always take aa -daa0 as the representative of the corresponding element of P(q,R) .The central projective group P0 (q,R) and the corresponding Lie algebra p 0 (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 PT .Because of (3 .12), the matrices obtained from (Ko) by replacing Ka with 0 will yield a connection on BT , and this induces a connection on PT , whose P 0 (q,R)-valued local forms are represented by (Ka -d K') .As shown by (3 .11)and a (3 .22), the latter matrices do not depend on n0 any more .
Finally, let us also note another important property of PT .We start by introducing in the manifold BT the local coordinates (xa ,xu , where ~a are the components of the vectors of a frame of B T with respect to the bases (3 .10) .
Then ~a are "homogeneous coordinates" in PT , and, in view of (3 .12), the local equations x a = const ., quotients of la = const .define on P0 a foliation F0 whose leaves cover the leaves of F (like in the Riemannian case [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 PT is given by (3 .25)..Rj o w 0 = n o dEx -6 0n 0 dj a + ( n ' EY -0n01Y)(Ka -5xK0) `a a0 a a aa 0 aa CL a0 y y0 ' which are q 2 +2q linearly independent 1-forms on P T .Furthermore, the restriction of the forms (3 .25),E a 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 JOdxb , 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 coframes of 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 these coframes 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 transverse projective 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 parallelism is independent of g, and is foliate as well .
Therefore, for projective foliations we 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's original method [3] 1 .Projective Structures on reformulation of those given 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 = {(U a ,Va )} of c-charts is are projectively related, and {Ua } 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 Vn 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 .
c-.charts by a partition of unity, we get a global c-cfiart of the always be represented by a global connec-Conversely, 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 D a have a symmetric Ricci curvature tensor) .(Then, ip of (1 .1) is a closed forro .)Moreover, we can even represent the 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 [Ba (X,Y) -Ba, CY,X)]Z + + n {[nBa CZ,Y) + B a (Y,Z)]X -[nBa CZ,X) + Ba(X,Z)]Y}where R a is the curvature of Va , and B a is the corresponding Ricci 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 projectively Euclidean manifold , i .e ., one which has a projective atlas consisting of flat connections .(E .g., the Riemannian manifolds of constant curvature belong to this class .)v) [5] The Chern-Pontrjagin forms of a Riemannian manifold 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. .k29 Wh1 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 .
local coordinates (xa ,xu ) (a,b, . . .= 1, . ..,q ; u,v, . . .= q+l, . ..,n), such that xa = const .define the leaves of F, and the changes of the coordinates are locally of the form (2 .4)la = ¡a (xb), Xu = ¡u (xb ,xv) .7) O x = Y ab X c ' vX Xa = 0 b a .u and it has no torsion iff Y ab = Yba ' A projective transformation (2 .3)takes the form (2 .8)where X = X adxa 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(Xa' Xu)X c = Re cauX e ' R(Xu' Xv )X c = 0, where Now, a basic connection on Q has local equations Y ab -Yab + a a"b + abra (2 .10)Re cab = XaY cb bYca + Y cb Y ha -Ycayhb ' Recau -X uYca ' (2 .11)Recab + Re cac + Recaa = 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 dxa , and r forms 0u in their local expression), and a decomposition d = d' + d" + 8 of the exterior differentiation d into components of the

( 2 .
13) S = d(Y'a dxa) , 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-form of the connection rbc induced on Q by the second connection of g Calready mentioned earlier) is (2 .14)S = d(Fc adxa ) = di(X c In det dx c l = dd' In et = = d(d-d") In et = -dd" In et Here, in view of formulas (2 .4),we can see that d" In VáeI-rt is a well defined global 1-form .Furthermore, t c = yc -fc is a "tensor", whence t c dxa is a ab ab ab ca global 1-form .This yields (2 .15)S = d(tc adx a -d" In et ) , and proves the proposition .

3
(aia /8x b ) .The elements of the inverse matrix of (3 .3)will be denoted by qa , i .e ., págy = 6a Now, with respect to (3 .2), a connection on T(F) is given locally by a matrix (ná~of connection forms with the transition law (us note that T(F) admits connections such that Tr0 is an arbitrary basic connection on K(F), and (3 .6)TT a = dxa , Tra -(q -1)n0 = 0 c Particularly, let us look at the connections 0 (w b and V, (w ,b) of the a) a projective structure of R, with coefficients Y ac , y , b related by (2 is a chosen basic connection on K(F) .This yields the unique global connection of the projective structure of Q for which w,a + Tr0 = 0 .Then we see by (3 .5)that (3 .8)ub = w , b + 6bu0 fits into the definition of a connection of T(F) satisfying (3 .6) .From (2 .8),(3 .7),(3 .8),we also get (the coefficients o£ the projective connection of [6, p .98] .(3 .10)e = e, u a = e a + rae , Hereafter, on T(F) we are refering to the connections (3 .6),(3 .9)only .Now, let us consider the following Cartan change of the local bases (3 .2) [4]

Now, let us
denote by Ká = dicá -Ka n KB the curvature forms of our connections, whence The transition cocycle of the bases (3 projectively invariant forms .Furthermore, let us note that (3 .6),(3 .9)and (3 .11)imply (3 .15)Ka -q K0 = 0 .This suggests us to impose as a further condition (3 .16)Ka -qK0 = 0 (which has an invariant meaning by (3 .12)) .After explicitation, (3 .16)turns out to be equivalent to (3 .17)dx a A Ka = 0 , and, if we denote Ka = K ab dxb , (3 .17)becomes K be no part of type (0,2) in view of.(3 .13)),and define also similar components for Oá .Then, consider Kb -dbKO=Oá-K0Adxb a , and the corresponding coefficients (3 .20)"ace -Kace -a a KOce h ace -( K ac d e -Kaedc) These coefficients yield the "tensor" (3 .21)Tac`"acb -E)acb -(q-1)K ac ' 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 {h a} 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 das are positive foliate real functions on U a fl U s , and define a 1-cocycle of the covering {Ua } .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 But, any two of the connections Y have the same difference as the corresponding connections [9]re the field Y is tangent to F .A straightforward checking shows 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 .)Thischecking is easy by using the The operator W yields a well-defined 2-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 transverse projective structure of F, and the corresponding form w .Let D a be one of the local connections of this structure, and Ra be the corresponding form (2 .12).Then, if we consider a transformation It follows that w is precisely the (1,1)-type part 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 then (2 .10)and(2.18)yieldthatitsauxiliaryWeyl form is w=0, whence [w]=0, and w of any other transverse projective structure is d"-exact .(2.20)Conversely, if w is d"-exact, we have (2 .19),andwecan go over from everyone of the connections Ybc of the S-vanishing atlas to the connection Y bc = Y bc -t bc ' which will obviously be (local) foliate connections .Formally, the formulas (2 .8),(2.10),and(2 .11)are the same as in the case of a projective structure on a manifold (where Xa = 8/2xa ), whence we conclude that the computations of [6, p .87-89] hold good, and, if q 3 2, the "tensor" Now, with (2 .17)and(2 .20),wecan define the local differential forms Since IIh are closed forms, and S is exact, it follows that the forms IIh are cohomologous to 11h .Moreover, we get in fact (2 .21)We= Z We cabdxa t, dxb + Wecaudxa n eu .whereBu is defined in (2 .6) .Theorem 2 .3 .The real Pontrjagin classes of the transverse bundle Q of a fo liation F o £ co dimension -> 2are representable via the de Rham isomorphism