The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria. Foliated CR Manifolds

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1 ESI The Erwin Schrödinger International oltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Foliated CR Manifolds Sorin Dragomir Seiki Nishikawa Vienna, Preprint ESI 1290 (2003) March 12, 2003 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL:

2 Foliated CR Manifolds Sorin Dragomir Seiki Nishikawa Abstract We study foliations 1 on CR manifolds and show the following 2. 1) For a strictly pseudoconvex CR manifold M, the relationship between a foliation F on M and its pullback π F on the total space C(M) of the canonical circle bundle of M is given, with emphasis on their interrelation with the Webster metric on M and the Fefferman metric on C(M), respectively. 2) With a tangentially CR foliation F on a nondegenerate CR manifold M, we associate the basic Kohn-Rossi cohomology of (M, F) and prove that it gives the basis of the E 2-term of the spectral sequence naturally associated to F. 3) For a strictly pseudoconvex domain Ω in a complex Euclidean space and a foliation F defined by the level sets of the defining function of Ω on a neighborhood U of Ω, we give a new axiomatic description of the Graham-Lee connection, a linear connection on U which induces the Tanaka-Webster connection on each leaf of F. 4) For a foliation F on a nondegenerate CR manifold M, we build a pseudohermitian analogue to the theory of the second fundamental form of a foliation on a Riemannian manifold, and apply it to the flows obtained by integrating infinitesimal pseudohermitian transformations on M. Contents 1 Introduction 2 2 Foliations and the Fefferman metric CR and pseudohermitian geometry The normal bundle The Fefferman metric Foliated Lorentz manifolds Tangentially CR foliations The basic tangentially Cauchy-Riemann complex The filtration {F r Ω 0, } r The Graham-Lee connection Partly supported by the Grant-in-Aid for Exploratory Research of the Japan Society for the Promotion of Science, No Mathematics Subject Classification: 53C12, 53C50, 32V05, 32V40 2 Key Words: tangentially CR foliation, Fefferman metric, Tanaka-Webster connection, Graham-Lee connection, basic Kohn-Rossi cohomology, infinitesimal pseudohermitian transformation 1

3 4 Foliations and the Tanaka-Webster connection The second fundamental form The characteristic form Flows Introduction Foliations on CR manifolds appear naturally in several contexts. For instance, if a CR manifold (M, T 1,0 (M)) is Levi flat, then the maximally complex distribution H(M) of M is completely integrable so that M carries a foliation (the Levi foliation) by complex manifolds (cf. [16], [38]). Cf. Section 2 for notation and conventions. To see another example of this sort, let Ω = {ϕ < 0} C n+1 be a strictly pseudoconvex domain with real analytic boundary M = Ω. Let O(Ω) be the algebra of functions on Ω which admit a holomorphic extension to some neighborhood of Ω. Let Σ M be a real analytic submanifold which is not C-tangent at any of its points. y a result in [8], if Σ is locally a maximum modulus set for O(Ω) (cf., e.g., [14] for definitions), then L = T (Σ) H(M) is completely integrable and gives rise to on Σ a C-tangent foliation F of codimension one. On the other hand, by a result in [5], if Σ is tangent to the characteristic direction T of a pseudohermitian structure θ on M, then Σ is a contact CR submanifold (in the sense of [43], and thus a CR manifold), F is a Riemannian foliation and the metric g Σ induced on Σ by the Webster metric of (M, θ), where θ = (i/2)( )ϕ, is bundle-like (also, Σ is Levi flat and F is its Levi foliation). Opposite to the Levi flat case, if (M, T 1,0 (M)) is a nondegenerate CR manifold of hypersurface type whose pseudohermitian structure θ is a contact form on M, then the characteristic direction T of (M, θ) defines a flow on M (the contact flow, cf., e.g., [17]). Also, foliations by Riemann spheres appear (cf. [27]) on twistor spaces (nondegenerate 5-dimensional CR manifolds) of 3-dimensional conformal manifolds (a generalization of the example to n dimensions is due to [39]). A converse of this situation is known as well, namely if M is a nondegenerate CR manifold of CR dimension n = 2m carrying a foliation by compact complex manifolds of complex dimension m, then m = 1, the leaves are CP 1 s, and M arises from a twistor construction (cf. [28]). Furthermore, it should be noted that in [10] one considers foliations F on a CR manifold M such that, for any defining local submersion f : U U (i.e., the leaves of F U are the fibres of f), the local quotient manifold U is a CR manifold, f is a CR map, and df : H(U) H(U ) is surjective. Such F has a transverse CR structure (in the sense of [6]) and also a tangential CR structure (so that each leaf of F becomes a CR submanifold of M). While foliations with a transverse CR structure have been investigated (cf. [3] and [1]), a systematic treatment of foliations with tangential CR structure is still missing in the mathematical literature. The purpose of the present paper is to study basic properties of foliations on CR manifolds, in particular, tangentially CR foliations on nondegenerate CR manifolds, and prove the following as the first step. If M is a strictly pseudoconvex CR manifold with a fixed contact form θ whose corresponding Levi form G θ is positive definite, and F is a foliation on M which is tangent to the characteristic direction T of θ, then the pullback foliation 2

4 π F of F to the total space of the canonical circle bundle π : C(M) M of M is nondegenerate with respect to the Fefferman metric F θ on C(M). Furthermore, F θ is bundle-like for π F if and only if the Webster metric g θ of (M, θ) is bundlelike for F. For a transversally oriented codimension q foliation F on M, we show that if 1) F is tangent to T, 2) the transverse volume element of F in (M, g θ ) is holonomy invariant, and 3) the mean curvature form κ of F in (M, g θ ) is d - exact, then the q-dimensional basic cohomology H q (F) of F is nonvanishing. Thus we generalize a result in [23] (cf. also Corollary 9.22 in [41], p. 125) to the case of foliations on CR manifolds. With any tangentially CR foliation F on M we associate a cohomology algebra H 0, (F), the basic Kohn-Rossi cohomology of (M, F), which has the property that H 0,0 (F) = CR (M) [the space of CR functions on M] and that H 0,1 (F) injects into the ordinary Kohn-Rossi cohomology group H 0,1 (M) of M on (0, 1)-forms. We build a decreasing filtration {F r Ω 0, } r 0 of Ω 0, (M) by M -differential ideals, and show that if {E r,s i } i 0 is the corresponding spectral sequence, then E r,0 2 H 0,r (F). Given a smoothly bounded strictly pseudoconvex domain Ω = {ϕ < 0} C n+1 and a foliation F defined by the level sets of ϕ on a neighborhood U of Ω, we give a new axiomatic description of the Graham-Lee connection, a linear connection on U which induces the Tanaka-Webster connection on each leaf of F, and then compute Faran s invariants h α β and kα (cf. [15]) in terms of the pseudohermitian torsion of the Graham-Lee connection and transverse curvature of ϕ, respectively. Also, for a foliation F on a nondegenerate CR manifold M we use the adapted connection determined by the ott connection of F and the Tanaka-Webster connection (associated to a choice of contact form θ on M) to produce a pseudohermitian analogue to the theory of the second fundamental form of a foliation on a Riemannian manifold (cf. [41], p. 62). The theory is applied to foliations which are tangent to the characteristic direction of θ and orthogonal to a semi-levi foliation, and to flows obtained by integrating infinitesimal pseudohermitian transformations on a nondegenerate CR manifold. Acknowledgments. During the completion of this paper the first named author was a guest of the Erwin Schrödinger International Institute for Mathematical Physics (November 2002) and wishes to express his gratitude to the organizers of the program Aspects of Foliation Theory in Geometry, Topology and Physics (J. Glazenbrook, F. Kamber and K. Richardson) for their kind invitation and support. Also, he had useful discussions on the matters in this paper with K. Richardson. 2 Foliations and the Fefferman metric Given a foliation F on a strictly pseudoconvex CR manifold M with a contact form θ, whose corresponding real Levi form G θ being positive definite, our main technique in this section is to consider the pullback foliation π F of F on the total space C(M) of the canonical circle bundle over M. This pullback foliation π F enjoys many of the properties of the original foliation F. For instance, π F is tangentially oriented F is tangentially oriented, the (canonical) transverse volume element of π F is holonomy invariant the transverse volume element of F is holonomy invariant, π F is har- 3

5 monic (with respect to the Fefferman metric F θ on C(M)) F is harmonic (with respect to the Webster metric g θ on M), etc. Furthermore, π F lives in the presence of a Lorentz metric (the Fefferman metric F θ ). The resulting philosophy then is that one might get a better understanding of the geometry of a foliated (strictly pseudoconvex) CR manifold by establishing general theorems about foliated Lorentz manifolds. 2.1 CR and pseudohermitian geometry We start by recalling a few notions of CR and pseudohermitian geometry, which are needed throughout the paper. Let (M, T 1,0 (M)) be a CR manifold of type (n, k), where M is a real (2n + k)-dimensional C manifold and T 1,0 (M) is its CR structure, that is, a complex rank n subbundle of the complexified tangent bundle T (M) C of M such that i.e., T 1,0 (M) is totally complex, and T 1,0 (M) T 0,1 (M) = {0}, [Γ (T 1,0 (M)), Γ (T 1,0 (M))] Γ (T 1,0 (M)), i.e., T 1,0 (M) is involutive, or (formally) Frobenius integrable, where T 0,1 (M) stands for the complex conjugate of T 1,0 (M). The integers n and k are called the CR dimension and CR codimension of (M, T 1,0 (M)), respectively. Clearly, if k = 0, then (M, T 1,0 (M)) is a complex manifold of complex dimension n. We shall be mainly interested in CR manifolds of type (n, 1), which are commonly referred to as CR manifolds of hypersurface type. Let (M, T 1,0 (M)) be a CR manifold of arbitrary but fixed type. Let H(M) = Re{T 1,0 (M) T 0,1 (M)} be the Levi, or maximally complex, distribution of M. It carries the complex structure J : H(M) H(M) given by J(Z + Z) = i(z Z), Z T 1,0 (M), i = 1. The Levi form L is then defined by L(Z, W ) = i π([z, W ]), Z, W T 1,0 (M), where π : T (M) C {T (M) C}/{H(M) C} is the natural bundle map. A CR manifold with L = 0 is called Levi flat. Note that if k = 0, then L = 0 (i.e., a complex manifold is Levi flat). A CR manifold is called nondegenerate if L is nondegenerate. Assume now that k = 1 and M is orientable. Let θ be a pseudohermitian structure on M, that is, a global nowhere zero C section of H(M) T (M), the conormal bundle of H(M) defined by H(M) x = {ω Tx (M) Ker(ω) H(M) x }, x M. Consider G θ (X, Y ) = dθ(x, JY ), X, Y H(M), (the real Levi form). It is also customary to consider the complex bilinear form L θ (Z, W ) = i dθ(z, W ), Z, W T 1,0 (M). 4

6 Then L θ and the complex linear extension of G θ to H(M) C coincide on T 1,0 (M) T 0,1 (M). Also, L θ and L coincide up to a bundle isomorphism H(M) T (M)/H(M). A CR manifold (M, T 1,0 (M)), of hypersurface type, is strictly pseudoconvex if G θ is positive definite for some pseudohermitian structure θ on M. When (M, T 1,0 (M)) is nondegenerate (of hypersurface type), any pseudohermitian structure θ is a contact form on M so that θ (dθ) n is a volume form on M. If this is the case, let T be the characteristic direction of (M, θ), that is, the unique tangent vector field on M, transverse to H(M), determined by θ(t ) = 1 and T dθ = 0. As usual, we extend G θ to a degenerate metric G θ = π H G θ on M given by G θ (X, Y ) = G θ (π H (X), π H (Y )) for any X, Y T (M), where π H : T (M) H(M) is the canonical projection associated to the direct sum decomposition T (M) = H(M) R T [in particular, G θ (T, T ) = 0]. The Webster metric of (M, θ) is then defined by g θ = G θ + θ θ. If (2r, 2s) is the signature of G θ (r + s = n), then g θ is a semi-riemannian metric on M of signature (2r + 1, 2s) [and if M is strictly pseudoconvex with G θ positive definite, then g θ is a Riemannian metric on M]. For instance, let H n = C n R be the Heisenberg group with the multiplication law (z, t) (w, s) = (z+w, t+s+2 Im z, w ), where z, w = δ ij z i w j, and consider the Lewy operators Then L α = T 1,0 (H n ) x = z α izα t, 1 α n. n C L α,x, x H n, α=1 where L α = L α, is a CR structure on H n making H n into a strictly pseudoconvex CR manifold of CR dimension n (and actually into a CR Lie group, that is, a Lie group and a CR manifold whose CR structure is left invariant). A smooth map of CR manifolds f : M M is a CR map if df x (T 1,0 (M) x ) T 1,0 (M ) f(x) for any x M. A CR isomorphism, or CR equivalence, is a CR map which is a C diffeomorphism. Any smooth real hypersurface M in C n+1 is a CR manifold of CR dimension n, with the CR structure T 1,0 (M) = {T (M) C} T 1,0 (C n+1 ), where T 1,0 (C n+1 ) is the span of { / z j 1 j n + 1}. In particular, the boundary Ω n+1 of the Siegel domain Ω n+1 = { (z, w) C n C Im(w) > n } z α 2 is a CR manifold which is CR isomorphic to the Heisenberg group (the CR isomorphism is given by f(z, t) = (z, t + i z 2 ), (z, t) H n ). For any nondegenerate CR manifold M of hypersurface type, on which a contact form θ has been fixed, there is a unique linear connection (the Tanaka- Webster connection of (M, θ), cf., e.g., [13]) such that 1) H(M) is -parallel, 2) g θ = 0 and J = 0, 3) the torsion T of is pure, that is, T (Z, W ) = 0 and T (Z, W ) = il θ (Z, W )T for any Z, W T 1,0 (M), and τ J + J τ = 0, where α=1 5

7 τ(x) = T (T, X), X T (M). The vector valued 1-form τ on M is called the pseudohermitian torsion of and satisfies g θ (τ(x), Y ) = g θ (X, τ(y )) for any X, Y T (M), that is, τ is self-adjoint with respect to g θ. 2.2 The normal bundle Generally, given a codimension q foliation F on a C manifold N, we denote by T (F) the tangent bundle of F and by ν(f) = T (N)/T (F) its normal (or transverse) bundle, and by Π : T (N) ν(f) the natural bundle map. Let (M, T 1,0 (M)) be a strictly pseudoconvex CR manifold of CR dimension n. Let θ be a contact form on M such that G θ is positive definite. Let F be a codimension q foliation of M. Note that if 2n q, then θ is not basic. Indeed, if T (F) θ = 0 then T (F) H(M), and if T (F) dθ = 0, then for any X T (F) 0 = dθ(x, JX) = G θ (X, X) = X = 0. Hence F is the foliation by points, that is, q = 2n + 1. Let us extend G θ to the whole of T (M) as a degenerate metric G θ, by requesting that T is orthogonal to each V T (M), and let us consider T (F) 0 = {Y T (M) G θ (X, Y ) = 0 for all X T (F)}. We collect a few elementary facts in the following Proposition 1. The tangent bundle T (F) is nondegenerate in (T (M), G θ ) if and only if the characteristic direction T of (M, θ) is transverse to T (F). In general, let T (F) H(M) = π H (T (F)) be the projection of T (F) to H(M). Then we obtain (2.1) T (F) 0 = [ T (F) H(M) ] R T, where the orthogonal complement [T (F) H(M) ] of T (F) H(M) is taken in (H(M), G θ ). If T is tangent to F, then the following hold: (1) T (F) H(M) = H(M) T (F). (2) The natural bundle map σ 0 : ν(f) T (F) 0 is a bundle monomorphism and corestricts to a bundle isomorphism ν(f) [ T (F) H(M) ]. (3) H θ (r, s) = G θ (σ 0 (r), σ 0 (s)), r, s ν(f), is a Riemannian metric on the normal bundle ν(f) M. Proof. Let us prove the first statement in Proposition 1. Assume that T is transverse to T (F). Let X T (F) such that G θ (X, Y ) = 0 for any Y T (F). Then 0 = G θ (X, X) = G θ (π H (X), π H (X)) = π H (X) 2, so that π H (X) = 0. Thus T (F) X = θ(x)t, which yields θ(x) = 0, i.e., X = 0. Vice versa, assume that T (F) is nondegenerate in (T (M), G θ ). The proof is done by contradiction. If T x T (F) x for some x M, then G θ,x (v, T x ) = 0 for 6

8 any v T x (M) T (F) x, which yields T x = 0 by the nondegeneracy of T (F) x in (T x (M), G θ,x ), a contradiction. To prove the second statement in Proposition 1, let T (F) H(M) be the projection of T (F) to H(M), namely, T (F) H(M) = {X θ(x)t X T (F)}. Since [ T (F)H(M) ] R T H(M) R T = {0}, the sum in (2.1) is direct. To prove (2.1), first note that T T (F) 0. Next, if Z [ T (F) H(M) ], then Gθ (Z, Y ) = 0 for any Y T (F) H(M), which is written as Y = X θ(x)t with X T (F). Thus 0 = G θ (Z, Y ) = G θ (Z, X), and hence Z T (F) 0. To check the opposite inclusion, let Z T (F) 0 H(M) R T. Then Z = Y + ft for some Y H(M) and f C (M). Since G θ (Z, X) = 0 for any X T (F), it follows that G θ (Y, X θ(x)t ) = G θ (Z, X) = 0, which implies Y [ T (F) H(M) ]. Consider the bundle map σ 0 : ν(f) T (F) 0 defined by σ 0 (Π(Y )) = (Y θ(y )T ), Y T (M), where (Y θ(y )T ) is the [ T (F) H(M) ] -component of Y θ(y )T in H(M). To see that σ 0 (Π(Y )) is well-defined, suppose that Π(Y ) = Π(Z). Then Y Z T (F), and hence Y Z θ(y Z)T T (F) H(M) so that (Y Z θ(y Z)T ) = 0. Assume now that T T (F). The proof of (1) is immediate. To check that σ 0 is a bundle monomorphism, let σ 0 (Π(Y )) = 0, that is, Thus, by T T (F), Y θ(y )T T (F) H(M) = H(M) T (F) T (F). 0 = Π(Y θ(y )T ) = Π(Y ). The isomorphism claimed in (2) of Proposition 1 follows by a dimension argument. Indeed, since dim R ν(f) x = q, the fact that H(M) + T (F) H(M) + R T = T (M) implies that 2n + 1 = dim R H(M) x + dim R T (F) x dim R {H(M) x T (F) x }, [ and hence dim R T (F)H(M) = 2n q for any x M. ]x Let us now prove (3) of Proposition 1. As the image of σ 0 lies in H(M), H θ (r, r) = σ 0 (r) 2 0 and = 0 if and only if Y θ(y )T T (F) H(M) for each Y T (M) such that Π(Y ) = r. Therefore, H θ (r, r) = 0 if and only if r R Π(T ). In particular, if T T (F), then H θ is a Riemannian metric in ν(f). Proposition 1 is proved. 7

9 Remark 1. As the Webster metric g θ is a Riemannian metric on M, one may consider as well the normal bundle T (F) = {Y T (M) g θ (Y, X) = 0 for all X T (F)} with the corresponding bundle isomorphism σ : ν(f) T (F) given by σ(π(y )) = Y, where Y is the T (F) -component of Y T (M) = T (F) T (F), and the metric induced by g θ on ν(f) via σ. However, when T T (F), it follows that T (F) = [ ], T (F) H(M) σ = σ0, and the metric on ν(f) induced by g θ is precisely H θ. Indeed, let Y T (F), namely g θ (Y, X) = 0 for any X T (F). Since g θ (Y, T ) = 0 in particular, Y H(M). Therefore G θ (Y, X θ(x)t ) = G θ (Y, X) = g θ (Y, X) θ(y ) θ(x) = 0 }{{} =0 for any X T (F), which shows that T (F) [ T (F) H(M) ]. The opposite inclusion may be proved in a similar manner. Also, it is immediate to see σ(π(y )) = σ(π(y θ(y )T )) for any Y T (M). [as T T (F)] = (Y θ(y )T ) [as T (F) = [T (F) H(M) ] ] = σ 0 (Π(Y )) 2.3 The Fefferman metric The first statement in Proposition 1 shows that, under the natural assumption that T be tangent to the leaves of F, T (F) is degenerate in (T (M), G θ ). However, the pullback of F to the total space of the principal S 1 -bundle C(M) = {K(M) \ {zero section}} /R + M turns out to be nondegenerate in (C(M), F θ ), where F θ is the Fefferman metric of (M, θ). Here K(M) = Λ n+1,0 (M) is the canonical line bundle over M. To be more precise, a complex p-form ω on M is said to be a (p, 0)-form, or a form of type (p, 0) if T 0,1 (M) ω = 0, and Λ p,0 (M) M denotes the bundle of the (p, 0)-forms on M. We proceed by recalling a few notions regarding the Fefferman metric (cf., e.g., [29]). Consider the 1-form η on C(M) given by η = 1 n + 2 {dγ + π ( iω α α i 2 hαβ dh αβ )} R 4(n + 1) θ, where γ is the (local) fibre coordinate on C(M), π : C(M) M is the projection, h αβ are the (local) components of the Levi form with respect to a (local) frame {T α } of T 1,0 (M), i.e., h αβ = L θ (T α, T β ), ω α β are the connection 1-forms of the Tanaka-Webster connection of (M, θ), that is, T β = ω β α T α, 8

10 and R = h αβ R αβ is the pseudohermitian scalar curvature (again cf. [29]). Also, R αβ is the pseudohermitian Ricci tensor. The Fefferman metric F θ of (M, θ) is the Lorentz metric on C(M) given by F θ = π Gθ + 2(π θ) η, where denotes the symmetric tensor product. Note that η is a connection 1-form on the principal S 1 -bundle π : C(M) M (cf. also [18], p. 855). Let then β z = {d z π : Ker(η z ) T x (M)} 1, z C(M) x, x M, be the horizontal lift with respect to η. For a tangent vector field X on M we adopt the notation X = β(x). Let S = / γ be the tangent vector filed to the S 1 -action. Then T S is timelike, and hence (C(M), F θ ) is time-oriented by T S, namely (C(M), F θ ) is a space-time (cf., e.g., [7], p. 17). Moreover, if M is compact, then (C(M), F θ ) is not chronological (cf. Proposition 2.6 in [7], p. 23). Let F be a foliation of M and π F the pullback of F to C(M), that is, T (π F) z = (d z π) 1 T (F) π(z), z C(M). The leaves of π F are connected components of the inverse images (via π) of the leaves of F. We may state the following Proposition 2. Let F be a foliation on the strictly pseudoconvex CR manifold M, carrying the contact form θ (with G θ positive definite). Let T (F) be the horizontal lift (with respect to η) of T (F), that is, T (F) z = β z ( T (F)π(z) ), z C(M). Then for the tangent bundle T (π F) of the pullback foliation π F on C(M) we obtain (2.2) T (π F) = T (F) Ker(dπ). Let T be the characteristic direction of (M, θ). If T is tangent to F, then the following hold: (1) T (π F) is nondegenerate in (T (C(M)), F θ ) and each leaf L of π F is a Lorentz manifold with the induced metric ι F θ, where ι : L C(M) denotes the inclusion. (2) The metric h θ induced by F θ on ν(π F) = T (C(M))/T (π F), the normal bundle of π F, is positive definite. (3) The Fefferman metric F θ is bundle-like for (C(M), π F) if and only if the Webster metric g θ is bundle-like for (M, F). Here, by slightly generalizing the definition in, e.g., [32], p. 79, given a semi- Riemannian manifold (N, g) (i.e., g is nondegenerate, of constant index) and a foliation F on N, we call g a bundle-like metric for (N, F) if 1) T (F) is nondegenerate in (N, g) and 2) the metric h induced by g on ν(f) is holonomy invariant, that is, L X h = 0 for any X T (F), where L X stands for the Lie differentiation with respect to X. 9

11 Proof. Let us first prove (2.2) in Proposition 2. Since η is a connection 1-form, it follows that T (F) Ker(dπ) Ker(η) Ker(dπ) = {0}. Therefore the sum in (2.2) is direct. definition of π F. Vice versa, if The inclusion holds by the very V T (π F) T (C(M)) = Ker(η) Ker(dπ), then V = X + fs for some X T (M) and f C (C(M)), where X = β(x) is the horizontal lift of X with respect to η. Also, V T (π F) yields that X = dπ(v ) T (F). Hence X T (F), that is, V T (F) + Ker(dπ). The identity (2.2) is thus proved. Assume now that T T (F). To see (1), consider V T (π F) such that F θ (V, W ) = 0 for any W T (π F). Then we have (π Gθ )(V, W ) + (π θ)(v )η(w ) + (π θ)(w )η(v ) = 0, which implies, by taking the decomposition V = V H + V V Ker(η) Ker(dπ) into account, that (2.3) Gθ (dπ(v H ), dπ(w H )) + θ(dπ(v H ))η(w V ) + θ(dπ(w H ))η(v V ) = 0 for any W T (π F). If W = S Ker(dπ) T (π F), then W H = 0. Since η = {dγ + π η 0 }/(n + 2) for some 1-form η 0 on M, which is determined in terms of θ, and dγ(s) = 1, it follows that η(w V ) = 1/(n + 2). Then from (2.3) we see that θ(dπ(v H )) = 0, which implies (2.4) dπ(v H ) H(M), with the corresponding simpler form of (2.3) as (2.5) Gθ (dπ(v H ), dπ(w H )) + θ(dπ(w H ))η(v V ) = 0. If W = V, then it follows from (2.4) that dπ(v H ) 2 = 0, that is, dπ(v H ) = 0 and hence V H Ker(dπ) Ker(η) = {0}. Substituting V H = 0 into (2.5), we then see (2.6) θ(dπ(w H ))η(v V ) = 0 for any W T (π F). Setting W = T T (F) T (π F) in (2.6) then yields that η(v V ) = 0, that is, V V = 0. Hence we conclude that V = 0, that is, T (π F) is nondegenerate in (T (C(M)), g θ ). Note now that F θ (S, S) = 0, and hence F θ is indefinite on T (π F). Since F θ is nondegenerate on T (π F), there F θ must have signature (2n+1 q, 1). Yet F θ is a Lorentz metric, therefore F θ is positive definite on T (π F). Consequently, the metric h θ (r, s) = F θ (ρ(r), ρ(s)), r, s ν(π F), induced by F θ on the normal bundle ν(π F) of π F is positive-definite, where ρ : ν(π F) T (π F) is the natural isomorphism. This proves (2). To prove (3), note first that L Xh θ = 0 if and only if (2.7) X(Fθ (V, W )) = F θ ([ X, V ], W ) + F θ (V, [ X, W ]) for any X T (π F) and V, W T (π F). We now need the following 10

12 Lemma 1. T (π F) Ker(η) and consequently (2.8) Ker(η) = T (F) T (π F). Moreover, dπ ( T (π F) ) H(M). Proof of Lemma 1. For any V T (π F) T (C(M)) = Ker(η) Ker(dπ), one has the decomposition V = V H + fs, V H Ker(η). On the other hand, since we have F θ (S, T ) = (π θ)(t )η(s) = θ(dπ(t ))/(n + 2) = 1/(n + 2), As T T (F), it follows that F θ (V, T ) = f/(n + 2) + F θ (V H, T ). T T (F) T (π F), so that T is orthogonal to V. Hence we obtain f = (n + 2)F θ (V H, T ) = (n + 2)(π Gθ )(V H, T ) = (n + 2) G θ (dπ(v H ), T ) = 0, that is, V Ker(η). Then the identity (2.8) follows, by (2.2), from the facts T (C(M)) = {T (F) Ker(dπ)} T (π F), T (F) T (π F) Ker(η). To prove the last statement in Lemma 1, let V T (π F) T (C(M)) = H(M) R T, that is, V = Y + ft for some Y H(M). Since S Ker(dπ) T (π F), S and V are orthogonal. Thus we have 0 = F θ (V, S) = F θ (Y, S) + f/(n + 2) = (π θ)(y )η(s) + f/(n + 2). Hence θ(y ) = 0 yields f = 0. Lemma 1 is proved. y Lemma 1, (2.7) holds if and only if it is satisfied for vector fields V, W of the form V = Y, W = Z for some Y, Z H(M). Also, (2.7) is identically satisfied when X Ker(dπ). Indeed, if this is the case, then (by a result in [25], Vol. I, p. 78) one has [ X, Y ] = [ X, Z ] = 0. Hence X(F θ (Y, Z )) = X(G θ (Y, Z) π) = 0, since dπ(x) = 0. Assume from now on that X T (F), that is, X = X for some X T (F). y Proposition 1.3 in [25], Vol. I, p. 65, [X, Y ] is the Ker(η)-component of 11

13 [X, Y ]. Then it follows from θ(y ) = θ(z) = 0 that the identity (2.7) is equivalent to (2.9) X( G θ (Y, Z)) = G θ ([X, Y ], Z) + G θ (Y, [X, Z]) for any X T (F) and Y, Z H(M) such that Y, Z T (π F). Finally, note that for each V = Y T (π F) with Y H(M) one has for any X T (F), and hence 0 = F θ (X, V ) = G θ (X, Y ) π Y H(M) T (F) = [ T (F) H(M) ]. Therefore, L Xh θ = 0 if and only if (2.9) holds for any X T (F) and Y, Z [ T (F)H(M) ], that is, if and only if LX F θ = 0. This completes the proof of Proposition Foliated Lorentz manifolds Let N be a C manifold and F a codimension q foliation of N. A differential p-form ω on N is called basic if X ω = 0, L X ω = 0 for all X T (F). Note that the exterior derivative d preserves basic forms, since X dω = L X ω d(x ω) = 0, L X dω = dl X ω = 0. Hence, denoting by Ω p (F) the set of basic p-forms, we obtain the basic complex of F (cf. [41], p. 119) Ω 0 (F) d Ω 1 (F) d d Ω q d (F) 0, where d = d Ω, and the corresponding basic cohomology of F H j (F) = Hj (Ω (F), d ), 0 j q. Also, we consider the following multiplicative filtration of the de Rham complex Ω (N) (a decreasing filtration by differential ideals, cf. [41], p. 120) F r Ω m = {ω Ω m (N) X 1 X m r+1 ω = 0 for X 1,, X m r+1 T (F)}. Let us now consider a codimension q foliation F on an n-dimensional connected Lorentz manifold (N, g) such that T (F) is nondegenerate in (T (N), g). The second fundamental form α of F in (N, g) is defined by α : T (F) T (F) ν(f), α(x, Y ) = Π( N XY ), X, Y T (N), where N is the Levi-Civita connection of (N, g). As in the Riemannian case, since N is torsion-free, the involutivity of T (F) implies that α is symmetric. Next, by mere linear algebra (cf., e.g., [34], p. 49), T (N) = T (F) T (F) and we have a bundle isomorphism σ : Q = ν(f) T (F), σ(s) = the T (F)-component of Y s, s ν(f), 12

14 where Y s T (N) with Π(Y s ) = s. Let us consider the induced metric g Q (r, s) = g(σ(r), σ(s)), r, s ν(f). Set ind(f) = 1 if each leaf L of F is Lorentz, and ind(f) = 1 if each leaf L of F is Riemannian, with the induced metric g L = ι g, where ι : L N, respectively. We remark here that if g is a bundle-like metric for F, then no other possibility occurs. Indeed, let be the connection in Q N given by X s = Π[X, σ(s)] if X Γ (P ) and X s = Π N X σ(s) if X Γ (P ) for any X X (N) and s Γ (Q). A verbatim repetition of the proof of Theorem 5.11 in [41], p. 53, shows that g is bundle-like for F if and only if g Q = 0. Let x, y N and α(t) a piecewise smooth curve joining x and y. If g is bundle-like a standard argument based on -parallel translation along α shows that ind(g Q ) x = ind(g Q ) y, where ind(g Q ) is the index of g Q (in the sense of [34], p. 54), i.e. (Q, g Q ) is a semi-riemannian bundle. Let g P be the leafwise induced metric, i.e. if x N and L N/F is the leaf through x then g P,x := (ι g) x = g L,x. Clearly ind(g) = ind(g P ) + ind(g Q ) at each point of N, and hence g P has constant index. For any Z T (F), the Weingarten map W (Z) : T (F) T (F) of F is given by g P (W (Z)(X), X ) = g Q (α(x, X ), σ 1 (Z)), X, X T (F). Then W (Z) is self-adjoint. The mean curvature form of F in (N, g) is the 1-form κ Ω 1 (N) defined by κ(z) = trace W (Z), Z Γ (P ), X κ = 0, X Γ (P ). Assume from now on that g is bundle-like for F. Also, we assume that F is tangentially oriented, that is, F is equipped with a principal GL + (p, R)- subbundle N of the principal GL(p, R)-bundle L(P ) N, where p = n q and L(P ) x is the set of R-linear isomorphisms u : R p P x, x N. Let {E 1,, E p } be a local g P -orthonormal frame of P, adapted to, defined on an open set U N, satisfying g P (E i, E j ) = ɛ i δ ij with ɛ 2 i = 1 (thus ind(f) = ɛ 1 ɛ p ). The characteristic form of F is a p-form χ F Ω p (N) defined by χ F (Y 1,, Y p ) = det (g(y i, E j )), Y 1,, Y p Γ(T N). Note that P χ F = 0. The Lorentzian analogue of Rummler s formula (cf., e.g., [41], p. 68) still holds, namely, (2.10) Z dχ F + κ(z)χ F = 0 along P for any Z Γ (P ). Indeed, as χ F (E 1,, E p ) = ind(f), (L Z χ F )(E 1,, E p ) = = p χ F (E 1,, π ([Z, E i ]),, E p ) i=1 p ɛ i ind(f)g([z, E i ], E i ), i=1 13

15 where L Z is the Lie differentiation and π : T (N) P is the natural bundle morphism. On the other hand, by the definition of κ, we obtain κ(z) = p ɛ i g P (W (Z)(E i ), E i ) = i=1 p ɛ i g([z, E i ], E i ), and thus (2.10) is proved. Assume further that F is transversally oriented (i.e., P is oriented), and let ν be the characteristic form of P defined in a completely analogous manner with χ F. Let µ = d vol(g) be the Lorentz volume form on N. Assume also that N is oriented, and let {E A 1 A n} be an oriented local g-orthonormal frame, satisfying g(e A, E ) = ɛ A δ A, of T (N) such that {E i 1 i p} and {E α p + 1 α n} are frames in P and P, respectively, and denote its dual coframe by {ω A 1 A n}. Then for any α Ω r (N) ( α)(e A1,, E An r ) µ = ɛ A1 ɛ An r α ω A1 ω An r, where : Ω r (N) Ω n r (N) is the Hodge operator. In particular, for ν we have (2.11) ( ν)(e 1,, E p ) µ = ind(f) ν ω 1 ω p. Also, a calculation based on the identity leads to i=1 ν(y p+1,, Y n ) = det (g(y α, E β )) ν = ɛ p+1 ɛ n ω p+1 ω n. Hence ν is proportional to the transverse volume element ω p+1 ω n of F, and (2.11) may be written as ( ν)(e 1,, E p ) µ = ( 1) pq+1 ω 1 ω n. Since ν, χ F Ω p (N) and P ν = 0, P χ F = 0, there is a function f C (N) such that ν = fχ F. A calculation shows that f = ( 1) pq+1 ind(f) so that (2.12) ν = ( 1) pq+1 ind(f) χ F. As a corollary of (2.12), we have (2.13) ν χ F = ( 1) pq ind(f) µ. At this point we may prove the following Proposition 3. Let F be a transversally oriented foliation on a compact orientable Lorentz manifold (N, g). Assume that the transverse volume element ν of F is holonomy invariant; hence ν Ω q (F) and dν = 0. If F is harmonic (i.e., κ = 0), then [ν] 0 in H q (F). The proof is a verbatim repetition of the proof of Theorem 9.21 in [41], p. 124 (and Proposition 3 is the Lorentzian analogue of a result by Kamber and Tondeur [23]). Indeed, Rummler s formula (2.10) yields (when κ = 0) dχ F F 2 Ω p+1, 14

16 and the assumption that ν = d α for some α Ω q 1 (F) leads, by (2.13), to d(α χ F ) = ( 1) pq ind(f) µ, and then, by Green s lemma, to a contradiction. We may also establish the following Proposition 4. Let F be a foliation on a strictly pseudoconvex CR manifold M, and assume that F is tangent to the characteristic direction T of (M, θ), for some contact form θ on M. Then the following hold: (1) F is transversally oriented if and only if π F is transversally oriented and, if this is the case, the transverse volume element ν of F in (M, g θ ) is holonomy invariant if and only if the transverse volume element ν of π F in (C(M), F θ ) is holonomy invariant. (2) F is harmonic in (M, g θ ) if and only if π F is harmonic in (C(M), F θ ). Proposition 3 together with Proposition 4 then shows that for any transversally oriented codimension q foliation F on a compact strictly pseudoconvex CR manifold M, if 1) F is tangent to the characteristic direction T of (M, θ), 2) the transverse volume element ν of F in (M, g θ ) is holonomy invariant, and 3) F is harmonic in (M, g θ ), then [ν] 0 in H q (F). Indeed, if M is compact, then so is C(M) and, given a local coordinate system (U, x A ) on M, ( π 1 (U), u A = x A π, u 2n+2 = γ ) yields a local coordinate on C(M). Hence an orientation of M induces that of C(M). This is only illustrative of our ideas as to the use of the Fefferman metric. [The preceding statement also follows by directly applying the aforementioned result of Kamber and Tondeur (Theorem 9.21 in [23], p. 124) to F on (M, g θ ).] We may further exploit the relationship between pseudohermitian geometry and conformal Lorentzian geometry to prove the following Corollary 1. Let F be a transversally oriented codimension q foliation on a compact strictly pseudoconvex CR manifold M, which is tangent to the characteristic direction T of (M, θ) for a fixed contact form θ. Assume that the transverse volume element ν of F in (M, g θ ) is holonomy invariant and that the mean curvature form κ of F in (M, g θ ) is closed (i.e., dκ = 0). If [κ] = 0 in H 1 (F), then Hq (F) 0. Proof. We shall need the following Lemma 2. Let F be a transversally oriented codimension q foliation on an n-dimensional Lorentz manifold (N, g), and assume that T (F) is nondegenerate in (T (N), g). Then F is harmonic in (N, e 2u g), with u C (N), if and only if (2.14) du(z) = p 1 κ(z), Z T (F), where p = n q and κ is the mean curvature form of F in (N, g). Furthermore, the following are equivalent: (1) u is a basic function, i.e., u Ω 0 (F). 15

17 (2) The transverse volume element ˆν of F in (N, ĝ = e 2u g) is holonomy invariant if and only if the transverse volume element ν of F in (N, g) is holonomy invariant. The statement (1) in Lemma 2, that is, if du = p 1 κ, then the leaves of F are minimal in (N, e 2u g), was first discovered in [24] for the case of a Riemannian metric g (and our argument below follows closely the proof of Proposition 12.6 in [41], p. 151). The relationship between the Levi-Civita connections ĝ and g of Lorentz metrics ĝ = e 2u g and g, respectively, is given by ĝ = g + (du) I + I (du) g grad g u, which implies that the second fundamental forms ˆα and α of F in (N, ĝ) and (N, g), respectively, are related by ˆα = α g Π(grad g u). Hence, for the corresponding Weingarten maps, we have Ŵ (Z) = W (Z) du(z)i, Z Γ (P ), where I denotes the identity transformation. Consequently, the corresponding mean curvature forms satisfy ˆκ(Z) = κ(z) p Z(u), where p = n q. Therefore ˆκ = 0 if and only if u satisfies (2.14). The statement (2) in Lemma 2 follows from the formula L X ˆν = e qu {q du(x) ν + L X ν}, X Γ (P ). This completes the proof of Lemma 2. Let us go back to the proof of Corollary 1. Since [κ] = 0 in H 1 (F) by assumption, there is a basic function v Ω 0 (F) such that κ = dv. Set u = (p + 1) 1 v, where p = 2n + 1 q and dim M = 2n + 1. Then it is immediate from (2.17) in the proof of Proposition 4 that d(u π)(z ) = (p + 1) 1 κ(z ), Z T (F). Hence it follows from Lemma 2 that π F is harmonic in (C(M), e 2 u π F θ ), from which we see that π F is harmonic in (C(M), Fˆθ), since, by a result of Lee [29], the Fefferman metric changes conformally Fˆθ = e 2 u π F θ under a transformation ˆθ = e 2u θ. Now note that, by Proposition 4, the transverse volume element ν of π F in (C(M), F θ ) is holonomy invariant. Since u Ω 0 (F), it follows that u π Ω 0 (π F) so that, again by Lemma 2, the transverse volume element ˆν of π F in (C(M), Fˆθ) is holonomy invariant. In consequence, by Proposition 3, we may conclude that 0 H q (π F) H q (F). Proof of Proposition 4. First we note that ν(f) T (F) β [ T (F) ] = T (π F) ν(π F), 16

18 from which it is immediate that F is transversally oriented if and only if so is π F. We only need to justify here the equality in the sequence. To this end, let X T (π F) and write X = X + V for some X P = T (F) and V V = Ker(dπ). Then for any Y P we have F θ ( X, Y ) = (π Gθ )( X, Y ) + (π θ)(y )η( X) since P H(M). Therefore we see = G θ (X, Y ) + θ(y )η(v ) = G θ (π H (X) + θ(x)t, Y ), F θ ( X, Y ) = G θ (π H (X), Y ) = g θ (π H (X) + θ(x)t, Y ) = g θ (X, Y ) = 0, which shows [ P ] T (π F) and hence the desired equality holds, for both bundles have rank q. At this point we may relate the Weingarten maps of F and π F. Let C(M) be the Levi-Civita connection of (C(M), F θ ). Given Z P, the Weingarten map W (Z ) : T (π F) T (π F) of π F is given by F θ ( W (Z )( X), X ) = F θ ( C(M) X X, Z ) for any X = X + V and X = X + V, where X, X P and V, V V. As π : C(M) M is a principal S 1 -bundle, the projection π is a submersion. Recall, however, that for the vector field S = / γ tangent to the S 1 -action, F θ (S, S) = 0 and hence S is null, or isotropic, so that π is not a semi-riemannian submersion (according to the terminology adopted in [34], p. 212). Nevertheless, we may relate C(M) to M, in the spirit of [35]. Another difficulty is that Ker(η) and V are not orthogonal (with respect to the Fefferman metric F θ ), yet H(M) V does hold. Noting that [Y, V ] = 0 for any Y T (M), a calculation then leads to 2F θ ( W (Z )( X), X ) = Z( G θ (X, X )) G θ (X, [X, Z]) G θ (X, [X, Z]) + θ(x)ω(x, Z ) + θ(x )Ω(X, Z ) + dθ(x, Z)η(V ) + dθ(x, Z)η(V ), where Ω = Dη is the curvature 2-form of η. Let H(F) be the g θ -orthogonal complement of R T in P. In particular, for any X, X H(F) 2F θ ( W (Z )( X), X ) = Z(g θ (X, X )) g θ (X, [X, Z]) g θ (X, [X, Z]) + dθ(x, Z)η(V ) + dθ(x, Z)η(V ), or (by exploiting the explicit expression of M, cf., e.g., [25], p. 160) (2.15) 2F θ ( W (Z )( X), X ) = 2g θ (W (Z)(X), X ) + dθ(x, Z)η(V ) + dθ(x, Z)η(V ). Similarly, we also have (2.16) F θ ( W (Z )(T ), T ) = Ω(T, Z ). 17

19 Next, we may calculate κ(z ) = trace W (Z ). Let {E 1,, E p 1, T } be a g θ -orthonormal frame of T (F). Then { E 1,, E p 1, T + n S, T n + 2 } S 2 is an F θ -orthonormal frame of T (π F). Note that T ((n + 2)/2)S is timelike and ind(π F) = 1, in particular. Since θ(w (Z)(T )) = 0, a straightforward calculation based on (2.15) and (2.16) now leads to (2.17) κ(z ) = κ(z) π, Z P. In particular, F is harmonic in (M, g θ ) if and only if π F is harmonic in (C(M), F θ ). Finally, if ν Ω q (C(M)) is given by ( ) ν(ỹ1,, Ỹq) = det F θ (Ỹα, Eα) for some oriented g θ -orthonormal frame {E α 1 α q} of P, then ν = π ν and by a simple calculation we see that L X ν = π (L X ν) for any X = X + V with X P and V V. Proposition 4 is now proved. 3 Tangentially CR foliations Let (M, T 1,0 (M)) be a CR manifold and F a foliation on M. We say that F is a (tangentially) CR foliation if each leaf L of F is a CR submanifold of M, that is, L is a CR manifold and the inclusion ι : L M is a CR map, i.e. dι x (T 1,0 (L) x ) T 1,0 (M) x for each x L. A typical example of a CR foliation is illustrated by the following Example 1 (A CR foliation by level sets). Let C n+1 be the (n + 1)- dimensional complex Euclidean space with complex coordinates (z 1,, z n, w), w = u + iv, and α : R R a smooth function such that α(0) = 0 and α (t) < 0 for any t R. Define f : C n+1 R by f ( z 1,, z n, w ) = α ( z z n 2 v ) e u. Then f is a smooth submersion so that it defines a foliation F on C n+1 by the level sets of f. Note that {(z, w) Ω n+1 u = log(c/α(ρ))} if c > 0, f 1 (c) = Ω n+1 if c = 0, { (z, w) C n+1 \ Ω n+1 u = log(c/α(ρ)) } if c < 0, where ρ = n α=1 zα 2 v. Thus F is a CR foliation on C n+1, one of whose leaves is the Heisenberg group H n Ω n+1. 18

20 Now, let F be a CR foliation. Let H(F) M denote the subbundle of T (F) whose portion over a leaf L of F coincides with the Levi distribution H(L) of L. Similarly, let T 1,0 (F) M denote the complex subbundle of T (F) C whose portion over a leaf L of F coincides with the CR structure T 1,0 (L) of L. Assume from now on that M is a nondegenerate CR manifold (of hypersurface type), and fix a contact form θ on M and the corresponding characteristic direction T of (M, θ). It should be remarked that if M is strictly pseudoconvex, then θ is holonomy invariant if and only if H(F) = 0. Indeed, if X H(F) H(M), then θ(x) = 0. Hence, 0 = L X θ = X dθ = G θ (X, X) = 0 = X = 0. Recall that a (0, s)-form on M is a complex s-form ω on M such that T 1,0 (M) ω = 0 and T ω = 0. Let Λ 0,s (M) M be the bundle of (0, s)- forms on M and set Ω 0,s (M) = Γ (Λ 0,s (M)). We recall the tangential Cauchy- Riemann operator M, which is the first order differential operator M : Ω 0,s (M) Ω 0,s+1 (M) defined as follows. If ω is a (0, s)-form, then M ω is the unique (0, s + 1)- form which coincides with dω on T 0,1 (M) T 0,1 (M) (s + 1 terms). A smooth function f : M C is called a CR function if it satisfies the tangential Cauchy-Riemann equation M f = 0. The space of CR functions on M is denoted by CR (M). 3.1 The basic tangentially Cauchy-Riemann complex We say that ω Ω 0,s (M) is a basic (0, s)-form if it satisfies Z ω = 0, Z M ω = 0 for any Z T 1,0 (F). Let Ω 0,s (F) denote the space of all basic (0, s)-forms on (M, F). Since 2 M = 0, we see easily that M Ω 0,s (F) Ω0,s+1 (F). Let CR (F) be the space of smooth functions f : M C whose restriction f L to each leaf L of F is a CR function on L, namely f L CR (L). Note that CR (M) CR (F). Moreover, we have Ω 0,0 (F) = CR (F). Let be the restriction of M to Ω 0,s (F). Then we obtain a complex (3.1) Ω 0,0 (F) Ω 0,1 (F) Ω 0,k (F) 0, which is called the basic tangentially Cauchy-Riemann complex of (M, F). Here we suppose that dim M = 2N + 1 and F has codimension q = 2k. For the remainder of this section, we set n = N k and assume n 1. The cohomology of the complex (3.1) given by H 0,s (F) = Hs( Ω 0, (F), ) Ker { Ω 0,s = (F) Ω0,s+1 (F) } Ω 0,s 1, (F) 19

21 where 0 s k, is called the basic Kohn-Rossi cohomology of (M, F). particular, we obtain H 0,0 (F) = Ker { Ω 0,0 (F) Ω0,1 (F)} = { f CR (F) f = 0 } = CR (F) CR (M) = CR (M). In Let H 0,s (M) = H s( Ω 0, (M), M ) be the ordinary Kohn-Rossi cohomology of the CR manifold M. For any CR foliation F on a nondegenerate CR manifold, there exists a natural injection of (F) into the Kohn-Rossi cohomology group H 0,1 (M), namely the map H 0,1 (3.2) H 0,1 (F) H0,1 (M), [ω] [ω] H 0,1 (M) is a monomorphism. Here ω Ω 0,1 (F) with ω = 0. Indeed, if ω, ω Ker{ : Ω 0,1 (F) Ω0,2 (F)} lie in the same Kohn-Rossi cohomology class, then ω ω = M f for some smooth function f : M C. Then 0 = Z ω = Z ω + Z M f }{{} =0 for any Z T 1,0 (F), and hence f Ω 0,0 (F). Thus ω ω = f, that is, [ω] = [ω ]. Remark 2. When M has CR codimension 0, that is, M is a complex manifold, Ω 0,s (M) is the space of all (0, s)-forms, which are locally spanned by monomials containing s anti-holomorphic differentials dz α, with respect to local complex coordinates z α on M. Note that H 0,s (M) is then the Dolbeaut cohomology, and given a foliation F on M by CR submanifolds, (3.2) still holds. Example 1 (continued). Set β(t) = α (t)/(α(t) iα (t)). Then T 0,1 (F) is spanned by (3.3) Z α = z α 2β(ρ)z α w, 1 α n, where we set z α = z α. Note that β(0) = i, and hence along the leaf Ω n+1 of F the vector fields (3.3) correspond, under the CR isomorphism Ω n+1 = Hn, to the Lewy operators. We remark that H 0,s (F) = 0 for s {1, 2}. Indeed, first it follows from (3.2) that H 0,1 (F) H0,1 (C n+1 ) = 0. On the other hand, if we define a (0, 1)-form Θ Ω 0,1 (C n+1 ) by Θ = dw + 2β(ρ)z α dz α, then we obtain that Ω 0,1 (F) = { λθ λ C ( C n+1), M λ(z α ) = i β (ρ) z α λ, 1 α n }, Ω 0,0 (F) = {( f/ w) Θ f CR (F)}, and ω = 0 for any ω Ω 0,1 (F) = 0 is that the system H 0,1 (F) as seen below. Thus the meaning of the fact f z α = 2β(ρ)z α λ, 20 f w = λ

22 admits a solution f C (C n+1 ), provided that λ satisfies the compatibility relations M λ(z α ) + iβ (ρ)z α λ = 0. To compute H 0,2 (F), let ω = ω αβ dz α dz β + ω α dz α dw be a basic (0, 2)- form. Then we see that the condition yields that Ω 0,2 (F) = {0}. 0 = Z α ω = 2(ω αβ + β(ρ)z α ω β )dz β + ω α dw Similar to the above, let Ω p,0 (M) denote the space of (p, 0)-forms ω such that T ω = 0, and consider the first order differential operator M : Ω p,0 (M) Ω p+1,0 (M) defined as follows. If ω Ω p,0 (M), then M ω is the unique element of Ω p+1,0 (M) which coincides with dω on T 1,0 (M) T 1,0 (M) (p+1 terms). Then 2 M = 0 in all degrees and one may consider the cohomology groups H p,0 (M) = H p( Ω, 0 (M), M ). Moreover, if F is a CR foliation on M, then one may define the space of basic (p, 0)-forms Ω p,0 (F), consisting of all elements ω Ωp,0 (M) satisfying and the corresponding cohomology T 1,0 (F) ω = 0, T 1,0 (F) M ω = 0, H p,0 (F) = Hp (Ω,0 (F), ), where denotes the restriction of M to Ω,0 (F). Then one sees that complex conjugation gives isomorphisms H p,0 (M) H 0,p (M), H p,0 (F) H0,p (F). Example 2 (The contact flow). Let (M, T 1,0 ) be a nondegenerate CR manifold of hypersurface type, and θ a contact form on M. Let T be the characteristic direction of (M, θ), and denote by F the flow defined by T (cf., e.g., [41], p. 132). Following [17], p. 160, let us consider the space Uh r of all horizontal r-forms on M, where an r-form ω on M is called horizontal if T ω = 0 and L T ω = 0. Thus Uh r is nothing but Ωr (F). Employing Kohn s solution (cf. [26]) to the Neumann problem for the M operator on a compact strictly pseudoconvex CR manifold, Gigante established the following Theorem 1 (Gigante [17]). Let M be a compact strictly pseudoconvex CR manifold and θ a contact form on M. Let T be the characteristic direction of (M, θ), and F the flow defined by T. If the Tanaka-Webster connection of (M, θ) has vanishing pseudohermitian torsion (τ = 0) and strictly positive definite pseudohermitian Ricci curvature, then H 1 (F) = 0. We may give a short proof of Theorem 1, based on a result of Lee [30], as well as on our previous considerations. Indeed, H 1 (F) = H0,1 (M) H 1,0 (M). Furthermore, by a result in [30], if R αβ ξ α ξ β > 0 for any ξ = (ξ 1,, ξ n ), then H 0,1 (M) = 0 (note that the assumption τ = 0 was removed). 21

23 3.2 The filtration {F r Ω 0, } r 0 We define a multiplicative filtration of the Cauchy-Riemann complex by setting F r Ω 0,m = { ω Ω 0,m (M) Z 1 Z m r+1 ω = 0 for Z 1,, Z m r+1 T 1,0 (F) }. Note that we have Ω 0,m (M) = F 0 Ω 0,m F 1 Ω 0,m F m Ω 0,m F m+1 Ω 0,m = {0} for any 0 m N. Also, the following diagrams are commutative Indeed, let Ω 0,m (M) F r Ω 0,m F r+1 Ω 0,m M M M Ω 0,m+1 (M) F r Ω 0,m+1 F r+1 Ω 0,m+1 ω F r Ω 0,m Ω 0,m (M) M Ω 0,m+1 (M). Then, since T 0,1 (F) is involutive, it follows that for any Z j T 1,0 (F). Thus we have Z 1 Z m r+2 ( M ω ) = 0 M F r Ω 0,m F r Ω 0,m+1. Now, setting we obtain the following N F r Ω 0, = F r Ω 0,m, m=0 Proposition 5. Let F be a CR foliation on the nondegenerate CR manifold M. Then {F r Ω 0, } r 0 is a decreasing filtration of Ω 0, (M) by differential ideals. Also, dim C T 1,0 (F) x = n, x M, implies that F r Ω 0,n+r = Ω 0,n+r (M) and dim C T 1,0 (M) x /T 1,0 (F) x = k, x M, yields that (3.4) F k+1 Ω 0,m = {0}. Proof. Since we have seen that M F r Ω 0, F r Ω 0,, it remains to check that Ω 0, (M) F r Ω 0, F r Ω 0,. To this end, let ω = ω ω N with ω m F r Ω 0,m. Then we have α ω m F r Ω 0,m+s for any α Ω 0,s (M). Indeed, it is easy to see that Z 1 Z m+s r+1 (α ω m ) = 0 22

24 for any Z j T 1,0 (F), because at most s of the Z j s enter α, so that there are enough Z j s left to kill ω m. Hence we obtain Ω 0,s (M) F r Ω 0,m F r Ω 0,m+s, from which the desired inclusion follows. To prove (3.4), we need some local considerations. Let {T 1,, T N } be a local frame of T 1,0 (M) such that {T 1,, T n } is a local frame of T 1,0 (F). Let {θ 1,, θ N } be a local dual frame determined by θ i (T j ) = δ i j, θ i (T j ) = 0, θ i (T ) = 0. Each ω F r Ω 0,m is then locally a sum of monomials of the form θ α1 θ αp θ j1 θ jq, 1 α 1,, α p n, n + 1 j 1,, j q N, with C (M)-coefficients, where 0 p m r and q = m p. If r = k + 1, then 0 p m k 1 so that q k + 1. Hence θ j1 θ jq = 0, and (3.4) is proved. For a given CR foliation F on the nondegenerate CR manifold M with a fixed contact form θ, we set Z, W = L θ (Z, W ) and define T 1,0 (F) = {Z T 1,0 (M) Z, W = 0 for any W T 1,0 (F)}. An argument of mere linear algebra then shows that T 1,0 (F) is nondegenerate in (T 1,0 (M),, ) and T 1,0 (F) T 1,0 (F) = T 1,0 (M). Proposition 6. Let F be a CR foliation on the nondegenerate CR manifold M. Let {E r,s i } i 0 be the spectral sequence associated with the filtered differential space ( ) Ω 0, (M), M, {F r Ω 0, } r 0. Then we have the following isomorphisms of linear spaces: E r,s 0 Hom ( Λ s T 0,1 (F), Λ r [T 0,1 (F) ] ), E r,0 1 Ω 0,r (F), where T 0,1 (F) = T 1,0 (F) T 0,1 (M). Proof. We set and 2 H 0,r (F), Er,0 Z r,m i = {ω F r Ω 0,m M ω F r+i Ω 0,m+1 }, D r,m i = ( F r Ω 0,m) M ( F r i Ω 0,m 1), E r i Ω 0,m = Z r+1,m i 1 Z r,m i (Cf., e.g., [22], Vol. III, p. 21.) Also, we set E r,s i + D r,m i 1. = E r i Ω0,r+s. Then (3.5) E r,s 0 = F r Ω 0,r+s F r+1 Ω 0,r+s Hom ( Λ s T 0,1 (F), Λ r [T 0,1 (F) ] ). 23