L’INSTITUTFOURIER DE ANNALES

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A L

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E D L ’IN IT ST T U

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ANNALES

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L’INSTITUT FOURIER

Eric LOUBEAU & Radu PANTILIE

Harmonic morphisms between Weyl spaces and twistorial maps II Tome 60, n^o2 (2010), p. 433-453.

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HARMONIC MORPHISMS BETWEEN WEYL SPACES AND TWISTORIAL MAPS II

by Eric LOUBEAU & Radu PANTILIE (*)

This paper is dedicated to the memory of James Eells.

Abstract. — We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being a harmonic morphism naturally appears among the geometric properties of submersive twistorial maps between low- dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces of dimensions four and three. Also, we give a thorough description of the twistorial maps with one-dimensional fibres from four- dimensional Weyl spaces endowed with the almost twistorial structure of Eells and Salamon.

Résumé. — Nous définissons, sur les variétés lisses, les notions de structure presque twistorielle et d’application twistorielle, fournissant ainsi un cadre unifié pour tous les exemples d’espace de twisteurs. La condition de morphisme har- monique apparait naturellement dans les propriétés géométriques des applications twistorielles submersives entre espaces de Weyl de faible dimension, équipés d’une structure presque twistorielle non-intégrable due à Eells et Salamon. Ceci mène à la caractérisation twistorielle des morphismes harmoniques entre espaces de Weyl de dimension quatre et trois. De plus, nous donnons une description complète des applications twistorielles à fibres unidimensionelles d’un espace de Weyl de dimension quatre, équipé de la structure presque twistorielle non-intégrable due à Eells et Salamon.

Introduction

A predominant theme in the theory of harmonic maps is their rela- tionship to holomorphicity. Right from their inception, harmonic maps have been recognised by Eells and Sampson to include holomorphic maps between Kähler manifolds, and a few years later, Lichnerowicz distinguished two sub-classes of almost Hermitian manifolds for which this also holds.

Keywords:Harmonic morphism, Weyl space, twistorial map.

Math. classification:53C43, 53C28.

(*) Gratefully acknowledges that this work was partially supported by a CNCSIS grant, code 811, and by a PN II Idei grant, code 1193.

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A far harder task is to find conditions forcing harmonic maps to be holomorphic. The clearest situation is between two-spheres since harmonicity then implies conformality, but, in general, additional conditions are re- quired, on the map or on curvature.

From a Riemann surface, or a complex projective space, into a compact irreducible Hermitian symmetric space harmonic stability ensures holomorphicity [4], [12].

In the broader context of compact Kähler manifolds, Siu [19] used a∂∂- Bochner argument to prove, under strong negativity of the target curvature, the holomorphicity of rank four harmonic maps, from which he deduced the biholomorphicity of compact Kähler manifolds of the same homotopy type.

Whilst these results are of great importance in understanding harmonic maps, this scheme bears its own natural limits, if only because it requires the presence of complex structures, with their topological consequences.

To overcome this hurdle, one can replace the codomain with an adequate bundle admitting a natural complex structure, such that harmonicity of any map is given by holomorphicity of its lift to this bundle.

This strategy, which could be traced to Calabi and even Weierstrass, was put into effect by Eells and Salamon [7], who modified a well-known twistor construction of Atiyah, Hitchin and Singer (see Example 3.6, below) to define, in our terminology, an almost twistorial structure on any oriented four-dimensional Riemannian manifold (see Example 3.7). This yields a bĳective correspondence between conformal harmonic maps and holomorphic curves (see Proposition 4.7). These ideas were extensively pursued by Bryant [3], and Burstall and Rawnsley [5] for (even-dimensional) Riemann- ian symmetric spaces.

The success of this strategy leads naturally to considering lifts on both the domain and the codomain, hence removing any need of pre-existing almost complex structures. The objective is two-fold: firstly, show that the existence of a holomorphic lift implies harmonicity and, secondly, find natural conditions under which harmonic maps admit holomorphic lifts.

As holomorphic maps are closed under composition, it seems that harmonic morphisms will have an important role in this programme.

Besides, the very conformal nature of their characterization makes Weyl geometry an ideal framework for their study, but it also turns out to provide examples of twistor spaces in dimension two, three and four.

In the complex category, a twistor, that is, a point of the twistor space of a (complex) manifold M, determines an immersed submanifold of M. For example, the twistor space of an anti-self-dual complex-conformal

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four-dimensional manifold(M⁴, c)is the space of self-dual (immersed) surfaces of(M⁴, c); also, the twistor space of a three-dimensional Einstein–

Weyl space(M³, c, D)is the space of coisotropic surfaces of(M³, c)which are totally-geodesic with respect toD (see [16]).

A complex analytic map ϕ:M → N between manifolds endowed with twistorial structures is twistorial if it maps (some of the) twistors onM to twistors onN (see [16]).

In this paper we extend the notions of almost twistorial structure and twistorial map, to the smooth category. We show that,in the smooth cat- egory, a twistor on a manifoldM (endowed with a twistorial structure) is a pair (R, J) where R is an immersed submanifold of M and J is a lin- ear CR-structure on the normal bundle of R in M (Remark 3.2). These submanifolds may well just be points. For example, the twistor space of a four-dimensional anti-self-dual conformal manifold(M⁴, c) is formed of pairs(x, J_x)wherex∈M andJ_xis a positive orthogonal complex structure on(TxM, cx).

On the other hand, the twistor space of a three-dimensional Einstein–

Weyl space (see Example 3.3) is formed of pairs(γ, J)whereγis a geodesic andJ is an orthogonal complex structure on the normal bundle of γ.

As in the complex category, a (smooth) mapϕ:M →N between manifolds endowed with twistorial structures is twistorial if it maps twistors on M to twistors on N (Definition 4.1, Remark 4.2). It follows that twistorial maps naturally generalize holomorphic maps. Examples of twistorial maps have been previously used to obtain constructions of Einstein and anti-self-dual manifolds (see [6], [15], [16] and the references therein).

In Section 1, we recall a few basic facts on harmonic morphisms between Weyl spaces. Section 2 is preparatory for Sections 3 and 4, where we give the definitions and some examples of almost twistorial structures and twistorial maps, on smooth manifolds.

In the first part [11] of this work, we introduced the notion of harmonic morphism between (complex-)Weyl spaces and we continued the study (ini- tiated in [16]) of the relations between harmonic morphisms and twistorial maps. As all of the known examples of complex analytic almost twistorial structures have smooth versions, all of the main results of [11] have

‘real’ versions. But not all smooth almost twistorial structures come from complex analytic almost twistorial structures. The first example of such an almost twistorial structure is the almost twistorial structure of Eells and Salamon, mentioned above. We introduce similar almost twistorial structures (Examples 3.5 and 5.2) with respect to which a map (between Weyl

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spaces of dimensions four and three) is twistorial if and only if it is a harmonic morphism (Theorem 5.4(i)); this implies thatthere exists a bĳective correspondence between one-dimensional foliations on(M⁴, c, D), which are locally defined by harmonic morphisms, and certain almost CR-structures on the bundle of positive orthogonal complex structures on(M⁴, c).

In Section 5, we also give the necessary and sufficient conditions for a map with one-dimensional fibres from a four-dimensional Weyl space endowed with the almost twistorial structure of Eells and Salamon to be twistorial (Theorem 5.6). It follows that, all such twistorial maps are harmonic morphisms. Also, a generalized monopole equation (Definition 4.4(1)) is naturally involved.

1. Harmonic morphisms

In this section all manifolds and maps are assumed smooth.

Let M^m be a manifold of dimension m. If m is even then we denote byL the line bundle associated to the frame bundle of M^m through the morphism of Lie groupsρ_m: GL(m,R)→(0,∞),ρ_m(a) =|deta|^1/m, a∈ GL(m,R)

; obviously,Lis oriented. Ifm is odd then we denote byL the line bundle associated to the frame bundle ofM^m through the morphism of Lie groupsρm: GL(m,R)→R^∗,ρm(a) = (deta)^1/m, a∈GL(m,R)

; obviously,L^∗⊗T M is an oriented vector bundle. We say thatListhe line bundle ofM^m(cf. [6]).

Similar considerations apply to any vector bundle.

Let ϕ : M → N be a submersion and let H be a distribution on M, complementary to the fibres of ϕ. Let L_H and LN be the line bundles ofH andN, respectively. AsL_H²ⁿ = Λⁿ(H)²

and L_N²ⁿ = Λⁿ(T N)² , wheren= dimN, the differential ofϕ induces a bundle mapΛfrom L_H² to L_N². If n is odd then we also have a bundle map λ from L_H to LN; obviously,Λ =λ². Furthermore, ifnis odd,dϕandλinduce a bundle map fromL^∗_H ⊗H to L^∗_N⊗T N, which will also be denoted bydϕ; note that, dϕ:L_H^∗ ⊗H →L_N^∗ ⊗T N is orientation preserving, on each fibre.

Let c be a conformal structure on M; that is, c is a section of L²⊗ ²T^∗M

which is ‘positive-definite’; that is, for any positive sections² of L² we have c =s²⊗g_s, where g_s is a Riemannian metric on M; theng_s is arepresentative ofc. Therefore,c corresponds to a Riemannian metric on the vector bundleL^∗⊗T M (see [6]). Obviously,calso corresponds to a reduction of the frame bundle ofM toCO(m,R), wherem= dimM; the total space of the reduction corresponding toc is formed of theconformal frameson(M, c).

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If dimM is odd then local sections of L correspond to oriented local representatives ofc; that is, (local) representatives ofc, on some oriented open set ofM. LetH be a distribution onM. Thencinduces a conformal structurec|_H onH and, it follows that, we have an isomorphism, which depends ofc, betweenL² andL_H² .

Definition 1.1(cf. [2]). — Let(M, cM)and(N, cN)be conformal man- ifolds. A mapϕ : (M, cM)→ (N, cN) is horizontally weakly conformal if at each pointx∈M, eitherdϕx= 0ordϕx|_{(ker dϕ}_x₎^⊥ is a conformal linear isomorphism from (ker dϕ_x)^⊥,(c_M)x|_{(ker dϕ}_x₎^⊥

onto T_ϕ(x)N,(c_N)ϕ(x)

. Let(M, c)be a conformal manifold. A connectionDonM isconformalif Dc= 0; equivalently,Dis the covariant derivation of a principal connection on the bundle of conformal frames on(M, c). IfDis torsion-free then it is called aWeyl connectionand(M, c, D)is aWeyl space.

Definition 1.2 (cf. [2]). — (i) Let (M, c, D) be a Weyl space. Aharmonic function, on(M, c, D), is a functionf, (locally) defined onM, such thattracec(Ddf) = 0.

(ii) A map ϕ : (M, cM, D^M)→ (N, cN, D^N) between Weyl spaces is a harmonic mapiftracec(Ddϕ) = 0, whereDis the connection onϕ^∗(T N)⊗

T^∗M induced by D^M,D^N andϕ.

(iii) A map ϕ : (M, c_M, D^M) → (N, c_N, D^N) between Weyl spaces is a harmonic morphism if for any harmonic function f : V → R, on (N, cN, D^N), with V an open set of N such that ϕ⁻¹(V) is nonempty, f◦ϕ:ϕ⁻¹(V)→Ris a harmonic function, on(M, cM, D^M).

Obviously, any harmonic function is a harmonic map and a harmonic morphism, ifRis endowed with its conformal structure and canonical connection.

The following result is basic for the theory of harmonic morphisms (see [11], [2]).

Theorem 1.3. — A map between Weyl spaces is a harmonic morphism if and only if it is a harmonic map which is horizontally weakly conformal.

2. Complex distributions

Unless otherwise stated, all manifolds and maps are assumed smooth.

Definition 2.1. — Acomplex distributionon a manifoldM is a com- plex subbundleF ofT^CM such thatdim(Fx∩Fx),(x∈M), is constant.

IfF ∩F is the zero bundle thenF is analmost CR-structureonM, and (M,F)analmost CR-manifold.

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Example 2.2. — 1) LetF be an almostf-structure on a manifoldM; that is,F is a section ofEnd(T M)such thatF³+F = 0[21]. We denote by T⁰M, T^1,0M, T^0,1M the eigendistributions of F^C ∈ Γ End(T^CM) corresponding to the eigenvalues0,i,−i, respectively. ThenT⁰M⊕T^0,1M is a complex distribution onM and T^0,1M is an almost CR-structure on M. We say thatT⁰M ⊕T^0,1M is the complex distribution associated to F; similarly,T^0,1M isthe almost CR-structure associated toF.

2) LetM be endowed with a complex distributionF and letN ⊆M be a submanifold. Suppose thatdim(T_x^CN∩Fx)anddim(T_x^CN∩Fx∩Fx), (x ∈ N), are constant. Then T^CN ∩F is a complex distribution on N which we callthe complex distribution induced byF onN.

In particular, ifN is a real hypersurface in the complex manifoldM then T^CN∩T^0,1M is an almost CR-structure onN.

Next, we define the notion of holomorphic map between manifolds endowed with complex distributions.

Definition 2.3. — LetF^M andF^N be complex distributions on M and N, respectively. A map ϕ : (M,F^M) → (N,F^N) is holomorphic if dϕ(F^M)⊆F^N.

A map ϕ : (M, F^M) → (N, F^N) between manifolds endowed with f- structures isholomorphicifϕ: (M,F^M)→(N,F^N)is holomorphic where F^M and F^N are the complex distributions, on M and N, associated to F^M and F^N, respectively.

LetF be an almostf-structure onM andJan almost complex structure on N. Then, a map ϕ : (M, F) → (N, J) is holomorphic if and only if dϕ◦F =J◦dϕ.

Definition 2.4. — Let M be a manifold. A subbundle E of T^CM is integrable if for any sectionsX, Y of E we have that [X, Y] is a section ofE.

A CR-structureis an integrable almost CR-structure; a CR-manifoldis a manifold endowed with a CR-structure.

An almostf-structure isintegrableif its associated complex distribution is integrable; anf-structureis an integrable almostf-structure.

Note that a complex distributionF^M onM is integrable if and only if for any pointx∈M, there exists a holomorphic submersionϕ: (U,F^M|U)→ (N,F^N), from some open neighbourhood U ofx, onto some CR-manifold (N,F^N), such that ker dϕ = F^M ∩F^M

|U. If U = M and ϕ has

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connected fibres thenF^M is called simple; then, obviously, up to a CR- diffeomorphism,ϕis unique; we call it theholomorphic submersion corre- sponding toF^M.

Let E be a complex vector bundle over M endowed with a complex linear connection∇ (that is,∇ is a complex-linear mapΓ(E)→ Γ Hom_C(T^CM, E)

such that ∇(sf) = (∇s)f +s⊗

C

df for any section s of E and any complex-valued function f on M; equivalently, we have

∇^C =∇ ⊕ ∇, with respect to the decompositionE^C =E⊕E).

Let π : Grk(E) → M be the bundle of complex vector subspaces of complex dimension k of E; we shall denote by H ⊆ T(Gr_k(E))

the connection induced by ∇ on Grk(E). Note that, as Grk(E) is a bundle whose typical fibre is a complex manifold and its structural group a complex Lie group acting holomorphically on the fibre, we have(ker dπ)^C = (ker dπ)^1,0⊕(ker dπ)^0,1.

The following result, which we do not imagine to be new, will be useful later on. We omit the proof.

Proposition 2.5. — Let p be a section of Grk(E) (equivalently, p is a complex vector subbundle of complex rank k of E). Let X ∈T_x^C

0M for somex0∈M; denote byp0=p(x0).

(a) The following assertions are equivalent:

(i)(dp)^C(X)∈Hp^C₀.

(ii)∇^C_Xs∈p^C₀ for any sectionsofp^C. (b)The following assertions are equivalent:

(iii)(dp)^C(X)∈Hp^C₀ ⊕(ker dπ)^0,1_p

0. (iv)∇Xs∈p0for any sectionsofp.

Note that, ifX is real (that is,X ∈T M) then the assertions (i),. . . ,(iv), of Proposition 2.5, are equivalent.

3. Almost twistorial structures

In this section, we define, in the smooth category, the notion of (almost) twistorial structure.

Definition 3.1 (cf. [17], [16]). — An almost twistorial structure, on a manifold M, is a quadruple τ = (P, M, π,F) where π : P → M is a locally trivial fibre space andF is a complex distribution onP which induces almost complex structures on each fibre ofπ; ifF is induced by an almostf-structureF then, also,(P, M, π, F)is called an almost twistorial structure.

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The almost twistorial structureτ = (P, M, π,F)isintegrableifF is in- tegrable. Atwistorial structureis an integrable almost twistorial structure;

the leaf space ofF ∩F is called the twistor space ofτ.

A twistorial structureτ= (P, M, π,F)is calledsimpleifF is simple; if τis simple withϕ: (P,F)→Zthe corresponding holomorphic submersion thendϕ(F)is a CR-structure onZ.

Remark 3.2. — Letτ = (P, M, π,F)be a twistorial structure and let Z be its twistor space. Each z ∈ Z determines a pair (R_z, J_z) where R_z is an immersed submanifold ofM and Jz is a linear CR-structure on the normal bundle ofR_z.

Indeed, letRz=ϕ⁻¹(z), whereϕ:P →Zis the (continuous) projection whose fibres are the leaves ofF∩F. Thenπ|R_z :Rz→M is an immersion.

Also, let (ker dπ)^0,1 = F ∩(ker dπ)^C. Then the restriction to R_z of the quotient ofF throughT Rz⊕(ker dπ)^0,1 defines a linear CR-structureJz

on the normal bundle(π|Rz)^∗(T M)/T R_zofR_z inM.

If (M, J) is a complex manifold then, obviously, (M, M,IdM, J) is a twistorial structure whose twistor space is(M, J).

Also, the CR manifolds constructed in [10] and [18] can be easily seen to be natural examples of twistor spaces.

Next, we formulate the examples of almost twistorial structures with which we shall work.

Example 3.3 ([9],[17]). — Let(M³, c, D)be a three-dimensional Weyl space. Letπ:P →M be the bundle of nonzero skew-adjointf-structures on (M³, c). Obviously, P is also the bundle of nonzero skew-adjoint f- structures on the oriented Riemannian bundle(L^∗⊗T M, c). Therefore,P is isomorphic to the sphere bundle of(L^∗⊗T M, c). In particular, the typical fibre and the structural group ofP areCP¹ andPGL(2,C), respectively.

We could also definePas follows: firstly, note that, there exists a unique oriented Riemannian structure, on the vector bundle E of skew-adjoint endomorphisms on(M, c), with respect to which [A, B] =A×B, for any A, B∈E; thenP is the sphere bundle of E.

The bundle P is also isomorphic with the bundle of oriented two- dimensional subspaces onM³. Therefore, there exists a bĳective correspondence between one-dimensional foliations onM³, with oriented orthogonal complement, and almostf-structures on(M³, c). Furthermore, under this bĳection, conformal one-dimensional foliations correspond to (integrable) f-structures.

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Letkbe a section ofL^∗. We define a conformal connection ∇on(M, c) by∇XY =DXY +¹₂kX×Y for any vector fieldsX andY onM. We say that∇isthe connection associated toD andk.

Let H ⊆ T P be the connection induced by ∇ on P. We denote by H⁰, H^1,0 the subbundles of H^C such that, at each p ∈ P, the subspaces Hp⁰,Hp^1,0 ⊆ Hp^C are the horizontal lifts of the eigenspaces of p^C ∈End(T_π(p)^C M)corresponding to the eigenvalues0,i, respectively.

We define the almost f-structureF onP with respect to whichT⁰P = H⁰ andT^1,0P = (ker dπ)^1,0⊕H^1,0. Thenτ = (P, M, π,F)is an almost twistorial structure on M which is integrable if and only if (M³, c, D) is Einstein–Weyl (see [13])

Remark 3.4. — Let(M³, c)be a three-dimensional conformal manifold.

Let π : P → M be the bundle of nonzero skew-adjoint f-structures on (M³, c).

Let D be a Weyl connection on (M³, c)and let ∇ be a conformal connection on(M³, c). The following assertions are equivalent:

(i)∇andDinduce, by applying the construction of Example 3.3 (with H the connection onP induced by∇andD, respectively), the same almost f-structure onP.

(ii)∇ andD are projectively equivalent.

(iii) There exists a section k of L^∗ such that ∇ is the connection associated toD andk.

Example 3.5 (cf.[7]). — Under the same hypotheses as in Example 3.3, we define the almostf-structure F⁰ on P with respect to which we have T⁰P =H⁰ and T^1,0P = (ker dπ)^0,1⊕H^1,0. Then τ⁰ = (P, M, π,F⁰) is a nonintegrable (that is, always not integrable) almost twistorial structure onM.

Example 3.6 ([1]). — Let (M⁴, c, D) be a four-dimensional oriented Weyl space. Letπ:P →M be the bundle of positive orthogonal complex structures on(M⁴, c). Obviously,P is also the bundle of positive orthogonal complex structures on the oriented Riemannian bundle(L^∗⊗T M, c).

LetE be the adjoint bundle of (L^∗⊗T M, c)and let ∗c be the involution ofE induced by the Hodge star-operator of (L^∗⊗T M, c), under the iso- morphismE= Λ²(L⊗T^∗M). ThenE=E+⊕E₋ whereE_± is the vector bundle, of rank three, formed of the eigenvectors of ∗_c corresponding to the eigenvalue ±1. There exists a unique oriented Riemannian structure

<·,·>onE_± with respect to whichAB=−< A, B >Id_{T M}±A×B for anyA, B∈E±. It follows thatP is the sphere bundle ofE+.

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Similarly to Example 3.3, there exists a bĳective correspondence between two-dimensional distributionsF onM⁴, with oriented orthogonal comple- ments, and pairs(J, K)of almost Hermitian structures on (M⁴, c), withJ positive andK negative, such thatJ|_F⊥=K|_F⊥.

LetH ⊆T P be the connection induced byDonP. We denote byH^1,0 the subbundle ofH^C such that, at eachp∈P, the subspaceHp^1,0⊆Hp^C

is the horizontal lift of the eigenspace ofp^C ∈End(T_π(p)^C M)corresponding to the eigenvaluei. We define the almost complex structure J on P with respect to whichT^1,0P = (ker dπ)^1,0⊕H^1,0. Then(P, M, π,J)is an almost twistorial structure onM which is integrable if and only if(M⁴, c)is anti- self-dual.

Example 3.7 ([7]). — Let (M⁴, c, D) be a four-dimensional oriented Weyl space. With the same notations as in Example 3.6, letJ⁰ be the almost complex structure onP with respect to whichT^1,0P= (ker dπ)^0,1⊕ H^1,0. Thenτ⁰ = (P, M, π,J⁰) is a nonintegrable almost twistorial structure onM.

4. Twistorial maps

We start this section with the definition of twistorial maps (cf. [17], [16]).

Definition 4.1. — Let τM = (PM, M, πM,F^M) and τN = (P_N, N, π_N,F^N)be almost twistorial structures and letϕ:M →N be a map. Suppose that there exist a locally trivial fibre subspaceπM,ϕ:PM,ϕ→ M ofπ_M :P_M →M and a mapΦ:P_M,ϕ→P_N with the properties:

1) F^M induces a complex distribution F^M,ϕ on PM,ϕ and almost complex structures on each fibre of πM,ϕ such that dπM(Fp^M) = dπM,ϕ(Fp^M,ϕ), for any p∈PM,ϕ.

2) ϕ◦π_M,ϕ=π_N◦Φ.

Then ϕ : (M, τM) → (N, τN) is a twistorial map (with respect to Φ) if the mapΦ : (PM,ϕ,F^M,ϕ) →(PN,F^N) is holomorphic. If F^M,ϕ and F^N are simple complex distributions, with (PM,ϕ,F^M,ϕ) → ZM,ϕ and (PN,F^N)→ZN, respectively, the corresponding holomorphic submersions onto CR-manifolds, then Φinduces a holomorphic map Z_ϕ :Z_M,ϕ →Z_N which is called thetwistorial representationofϕ.

Remark 4.2. — With the same notations as in Definition 4.1, we have that τM,ϕ = (PM,ϕ, M, πM,ϕ,F^M,ϕ) is an almost twistorial structure on M. Obviously,τ_M,ϕ is integrable if τ_M is integrable. Then from (1) it follows that the twistor spaceZM,ϕ of τM,ϕ is a topological subspace of the

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twistor spaceZ_M ofτ_M. Moreover, for anyz ∈Z_M,ϕ the pair(R_z, J_z) of Remark 3.2 applied toτM,ϕis equal to the pair determined byzas a point of Z_M. Furthermore, as Φ is holomorphic,we have that ϕ(R_z) ⊆R_Z_ϕ_(z) and the map between the normal bundles of Rz and R_Z_ϕ_(z), induced by the differential ofϕ, intertwinesJz andJ_Z_ϕ_(z)(here, we have identifiedRz

withπM,ϕ(Rz)and, similarly, forRZ_ϕ(z)).

If τM is simple then τM,ϕ is also simple andZM,ϕ is a submanifold of Z_M. Moreover, if we denote byCM andCM,ϕ the CR-structures ofZ_M and ZM,ϕ, respectively, thenCM,ϕ=CM ∩T^CZM,ϕ.

Let (M, JM) and (N, JN) be complex manifolds and let τM = (M, M,IdM, J_M)and τ_N = (N, N,IdN, J_N) be the corresponding twistorial structures. Then it is obvious that a map ϕ : (M, τM) → (N, τN) is twistorial (with respect to ϕ) if and only if ϕ : (M, J_M) → (N, J_N) is holomorphic.

Example 4.3 ([9],[6]). — Let(M⁴, cM)be a four-dimensional oriented conformal manifold and let (N³, c_N, D^N) be a three-dimensional Weyl space. Let τM = (PM, M, πM,J) be the almost twistorial structure of Example 3.6, associated to(M⁴, cM), and letτN = (PN, N, πN,F)be the almost twistorial structure of Example 3.3, associated to(N³, cN, D^N).

Letϕ:M⁴→N³be a submersion. LetV = ker dϕandH =V^⊥. Then the orientation of M⁴ corresponds to an isomorphism, which depends of cM, betweenV and the line bundle ofH. Therefore(V^∗⊗H, c|_H)is an oriented Riemannian vector bundle. We defineΦ:P_M →P_N by

Φ(p) =_||_dϕ(V∗¹⊗p(V))|| dϕ(V^∗⊗p(V)),

where{V} is any basis ofVπ_M(p)and{V^∗}its dual basis,(p∈PM).

Let I^H be the V-valued two-form on H defined by I^H(X, Y) =

−V[X, Y], for any sectionsXandY ofH. Then∗_HI^H is a horizontal one- form onM⁴, where∗_H is the Hodge star-operator of(V^∗⊗H, c|_H). De- note byD₊the Weyl connection on(M⁴, c_M)defined byD₊=D+∗_HI^H whereD is the Weyl connection of(M⁴, cM,V)(see [11]).

The following assertions are equivalent:

(i)ϕ: (M⁴, τM)→(N³, τN)is twistorial (with respect toΦ).

(ii)ϕ: (M⁴, c_M)→(N³, c_N)is horizontally-conformal andϕ^∗(D^N) = HD+ as partial connections onH, overH.

Let ϕ : (M⁴, cM) → N³ be a submersion from a four-dimensional oriented conformal manifold to a three-dimensional manifold. Let L be the line bundle ofN³. As the orientation ofM⁴corresponds to an isomorphism

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betweenV andϕ^∗(L), the pull-back byϕof any section ofL^∗is a (vertical) one-form onM⁴. Similarly, ifE is a vector bundle overN³, the pull-back byϕof any section ofL^∗⊗Eis aϕ^∗(E)-valued one-form onM⁴.

Next, we recall the following definitions (see [16] and the references therein).

Definition 4.4. — LetP be a principal bundle onN³ endowed with a (principal) connectionΓ and letAbe a section ofL^∗⊗AdP.

1) Let N³ be endowed with a conformal structure c_N and a Weyl con- nectionD^N. The pair(A,Γ)is called a monopole on(N³, cN, D^N)if

R=∗_N(D^N ⊗ ∇)(A)

whereR is the curvature form ofΓ and∇ is (the covariant derivation of) the connection induced byΓ onAdP.

2) Let(M⁴, cM)be a four-dimensional oriented conformal manifold and letϕ:M⁴→N³ be a submersion. The connectionΓe onϕ^∗(P)defined by

Γ =e ϕ^∗(Γ) +ϕ^∗(A) is calledthe pull-back by ϕof (A,Γ).

Next, we prove the following (cf. [16] and the references therein):

Proposition 4.5. — Let(M⁴, cM)be a four-dimensional oriented con- formal manifold and let(N³, c_N, D^N)be a three-dimensional Weyl space;

denote byτM andτN the almost twistorial structures of Examples 3.6 and 3.3, associated to(M⁴, c_M)and(N³, c_N, D^N), respectively.

LetPbe a principal bundle overN³endowed with a connectionΓand let Abe a nowhere zero section ofL^∗⊗AdP. Also, letϕ: (M⁴, cM)→(N³, cN) be a surjective horizontally conformal submersion with connected fibres.

Then any two of the following assertions imply the third:

(i)ϕ: (M⁴, τ_M)→(N³, τ_N)is twistorial.

(ii)(A,Γ)is a monopole on(N³, cN, D^N).

(iii)Γe is anti-self-dual.

Proof. — LetRebe the curvature form ofeΓ. A straightforward calculation shows that, up to an anti-self-dual term, the following equality holds (4.1) Re=ϕ^∗(R) +ϕ^∗ (D^N ⊗ ∇

(A)

+ ϕ^∗(D^N)−HD+

∧ϕ^∗(A)

where we have used the isomorphism Λ²(T^∗M) = Λ² ϕ^∗(T^∗N)

⊕ ϕ^∗(T N)⊗V^∗

, induced byc_M. The proof follows.

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Remark 4.6. — The fact that (4.1) holds, up to an anti-self-dual term, does not requireϕbe horizontally conformal; moreover, this relation char- acterizesHD₊among the partial connections on V, overH.

It follows thata submersion from a four-dimensional oriented conformal manifold to a three-dimensional Weyl space is twistorial, as in Example 4.3, if and only if it pulls-back any (local) monopole to an anti-self-dual con- nection; the ‘only if’ part is essentially due to [16] whilst the ‘if’ part is an immediate consequence of Proposition 4.5 (see also [16] and the references therein).

We end this section with the following result which follows quickly from Proposition 2.5.

Proposition 4.7(cf. [7]). — Let(M^m, c, D)be a Weyl space,m= 3,4, and letkbe a section of the dual of the line bundle ofM^m; ifm= 4assume M⁴oriented andk= 0. Letτ_M⁰ be the almost twistorial structure onM^m given by Examples 3.5 or 3.7 according tom= 3orm= 4, respectively.

LetN²be an oriented surface in M^m. LetτN = (N, N,IdN, J)whereJ is the positive Hermitian structure of(N², c|N).

The following assertions are equivalent:

(i)(N², τN)→(M^m, τ_M⁰ )is twistorial (with respect to pN).

(ii)N² is a minimal surface in(M^m, c, D)andk|_N = 0.

5. Harmonic morphisms and twistorial maps between Weyl spaces of dimensions four and three

We start this section by introducing a natural generalization of the almost twistorial structure of Example 3.5.

Example 5.1. — Let(M³, c)be a three-dimensional conformal manifold endowed with two Weyl connectionsD⁰ and D⁰⁰. Let kbe a section of L^∗ and let∇ be the connection associated toD⁰⁰andk.

Let π: P →M be the bundle of nonzero skew-adjointf-structures on (M³, c). We denote byH⁰the subbundle ofT^CPsuch that, at eachp∈P, the subspaceHp⁰ ⊆T_p^CP is the horizontal lift, with respect to D⁰, of the eigenspace ofp^C ∈End(T_π(p)^C M)corresponding to the eigenvalue 0. Also, we denote byH^1,0 the subbundle of T^CP such that, at each p∈P, the subspace Hp^1,0 ⊆ T_p^CP is the horizontal lift, with respect to ∇, of the eigenspace ofp^C ∈End(T_π(p)^C M)corresponding to the eigenvalue i.

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We define the almostf-structureF⁰⁰ onP with respect to whichT⁰P = H⁰andT^1,0P = (ker dπ)^0,1⊕H^1,0. Thenτ⁰⁰= (P, M, π,F⁰⁰)is a nonintegrable almost twistorial structure onM (the nonintegrability ofτ⁰⁰ follows easily from the proof of the integrability result presented in [14]).

We shall also need the following.

Example 5.2. — Let (M⁴, c, D) be a four-dimensional oriented Weyl space. Letπ:P →M be the bundle of positive orthogonal complex structures on(M⁴, c).

Letϕ:M⁴→N³be a submersion; denote byV = ker dϕandH =V^⊥. For anyp∈P letl⁰_p andl⁰⁰_p be the lines inT_π(p)^C M spanned by V −ip(V) andX−ip(X), respectively, for anyV ∈Vπ(p)andX∈p(Vπ(p))^⊥∩Hπ(p). We define the almost CR-structureCϕ⁰ onP which, at eachp∈P, is the direct sum of(ker dπp)^0,1 and the horizontal lift, with respect toD, ofl_p⁰.

Similarly, we define the almost CR-structure Cϕ⁰⁰ on P which, at each p∈P, is the direct sum of(ker dπ_p)^0,1and the horizontal lift, with respect toD, ofl_p⁰⁰.

Then τ_ϕ⁰ = (P, M, π,Cϕ⁰) and τ_ϕ⁰⁰ = (P, M, π,Cϕ⁰⁰) are nonintegrable almost twistorial structures onM⁴(the nonintegrability ofτ_ϕ⁰ andτ_ϕ⁰⁰follows easily from the proof of the integrability result presented in [14]).

Remark 5.3. — Let (N³, cN) be a three-dimensional conformal manifold. At least locally, we may assume that(L^∗⊗T N, c)is the adjoint bundle of a rank two complex vector bundleEwith groupSU(2). Furthermore, under this identification,P_N is isomorphic toP E. AsSO(3,R) = SU(2)/Z2= PSU(2), any connection on (L^∗⊗T N, c)corresponds to a connection on the PSU(2)-bundle PN. Also, (L^∗ ⊗T N, c) is the adjoint bundle of the PSU(2)-bundlePN.

Similarly, if(M⁴, cM)is a four-dimensional oriented conformal manifold then the bundleP_M of positive orthogonal complex structures on(M⁴, c_M) is aPSU(2)-bundle.

Furthermore, a submersion ϕ : (M⁴, c_M) → (N³, c_N) is horizontally conformal if and only ifΦ :PM → PN is a morphism ofPSU(2)-bundles (in general,Φ is a morphism ofSL(3,R)-bundles). Also, note that, ifϕ is horizontally conformal thenPM =ϕ^∗(PN)asPSU(2)-bundles.

The following result shows the importance of the almost twistorial structures of Example 5.2.

Theorem 5.4. — Let (M⁴, cM, D^M) be a four-dimensional oriented Weyl space and letP_M be the bundle of positive orthogonal complex struc- tures on(M⁴, cM).

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Let(N³, c_N, D^N)be a three-dimensional Weyl space and letkbe a sec- tion of the dual of the line bundle of N³; denote by ∇ the connection associated toD^N and k. Let τ_N⁰ be the almost twistorial structure of Ex- ample 3.5, associated to(N³, cN, D^N, k).

Letϕ:M⁴→N³be a submersion; denote byV = ker dϕandH =V^⊥. Letτ_ϕ⁰ andτ_ϕ⁰⁰be the almost twistorial structures of Example 5.2, associated to(M⁴, c_M, D^M)andϕ.

(i)The following assertions are equivalent:

(i1) ϕ: (M⁴, τ_ϕ⁰)→(N³, τ_N⁰ )is twistorial.

(i2) ϕ: (M⁴, cM, D^M)→(N³, cN, D^N)is a harmonic morphism.

(ii)The following assertions are equivalent:

(ii1) ϕ: (M⁴, τ_ϕ⁰⁰)→(N³, τ_N⁰ )is twistorial.

(ii2)ϕ: (M⁴, cM)→(N³, cN)is horizontally conformal,V(D^M−D) =

1

2kandϕ^∗(D^N) =HD^M+¹₂∗_HI^H as partial connections, overH, where Dis the Weyl connection of (M⁴, cM,V).

(ii3)ϕ: (M⁴, cM)→(N³, cN)is horizontally conformal and the partial connections onPM, overH, induced byD^M andϕ^∗(∇)are equal.

Proof. — (i) By Remark 5.3, if (i1) holds thenϕ: (M⁴, cM)→(N³, cN) is horizontally conformal.

Letτ_ϕ⁰ = (PM, M, πM,Cϕ⁰)and letτ_N⁰ = (PN, N, πN,F⁰). By associating to each orthogonal complex structure on (M⁴, cM) its eigenspace corresponding to −i, we identify π_M : P_M → M with the bundle of self-dual spaces on(M⁴, cM).

Similarly, by associating to each skew-adjoint f-structure on (N³, cN) the sum of its eigenspaces corresponding to 0 and −i, we identify πN : PN →N with the bundle of (complex) two-dimensional degenerate spaces on(N³, c_N).

Under these identifications the natural lift Φ :PM →PN ofϕ is given byΦ(p) = dϕ(p), for anyp∈P_M.

Let q be a local section of PN, over some open set V ⊆ N, and let p be the local section ofPM, overϕ⁻¹(V), such thatΦ◦p=q◦ϕ. At least locally, we may assume thatpis generated byY =U+ iX andZ, withU vertical, andX and Z basic. Thus,q is generated bydϕ(X)anddϕ(Z).

We shall assume that, for somex₀∈ϕ⁻¹(V), we havedq(dϕ(X_x₀))horizontal, with respect to the connection induced by∇onPN. Then Proposi- tion 2.5, the fundamental equation (see [11]) and a straightforward calculation show thatdp(Yx₀)∈(Cϕ⁰)

p(x0)if and only ifcM tracec_M(Ddϕ), Ze

= 0, whereDe is the connection induced byD^M andD^N onϕ^∗(T N)⊗T^∗M.

This proves that (i2)=⇒(i1).

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To prove the converse, assume that (i1) holds. Then there exists a unique A ∈ (Cϕ⁰)_p(x

0) such that dΦ(A) = dq(dϕ(Xx₀)). From the fact that dπM (C_ϕ⁰)_p(x

0)

is the line spanned byYx₀, it follows quickly thatdπM(A) =

−Yx0. Together with dΦ(A) = dΦ(dp(−Yx0)) this gives A = dp(−Yx0).

Hence,dp(Y_x₀)∈(Cϕ⁰)

p(x0)and, therefore,c_M trace_c_M(Ddϕ), Ze

= 0.

(ii) By Remark 5.3, if (ii1) holds thenϕ: (M⁴, c_M)→(N³, c_N)is horizontally conformal.

Similarly to above, let p be a basic local section of P_M; that is, there exists a local sectionq ofPN such thatΦ◦p=q◦ϕ. At least locally, we may assume thatp is generated byY =U + iX and Z, with U vertical, andX andZ basic. Thus,qis generated bydϕ(X)anddϕ(Z).

To prove (ii1) ⇐⇒ (ii2), we shall assume that p is horizontal, at some pointx₀∈M, with respect to the connection induced byD^M onP_M (this could be done as follows: firstly, definepover some hypersurface, containing x₀ and transversal to the fibres ofϕ, such thatpis horizontal atx₀; then, extendp, to an open neighbourhood ofx0, so that to be basic); in particular, dp(Zx₀)∈(Cϕ⁰⁰)

p(x0).

Assertion (ii1) is equivalent to the fact that, for any such p, we have dq(dϕ(Xx₀))contained in the eigenbundle ofF⁰corresponding to the eigenvaluei. By using Proposition 2.5, it quickly follows that (ii1)⇐⇒(ii2).

Let hbe an oriented representative of cN and let g be a representative of c_M such that ϕ : (M⁴, g)→ (N³, h) is a Riemannian submersion. Let α^M and α^N be the Lee forms ofD^M and D^N with respect to g and h, respectively.

The equivalence (ii2)⇐⇒(ii3) follows from the fact that any two of the following assertions imply the third:

(a) dp(Hx0)is horizontal, with respect to the connection induced by D^M onPM.

(b)qis horizontal atϕ(x₀), with respect to the connection induced by

∇onPN.

(c) At x₀ we haveα^M|_V = ¹₂k andα^N =α^M|_H +¹₂∗_HI^H, where we have identified k and α^N with their pull-backs by ϕ and, in the first equality, we have used the isomorphism betweenV and the pull-back byϕ of the line bundle ofN³ induced byϕand the orientation ofM⁴.

The proof is complete.

Remark 5.5. — Let (M⁴, c, D) be a four-dimensional oriented Weyl space andPthe bundle of positive orthogonal complex structures on(M⁴, c).