First Order Twistor Lifts

Volume 2010, Article ID 594843,26pages doi:10.1155/2010/594843

Research Article

First-Order Twistor Lifts

Bruno Ascenso Sim ˜oes

1_{Centro de Matem´atica e Aplicac¸˜oes Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2,}

1649-003 Lisbon, Portugal

2_{Universidade Lus´ofona, N ´ucleo de Investigac¸˜ao em Matem´atica, Campo Grande, 376,}

1749-024 Lisbon, Portugal

Correspondence should be addressed to Bruno Ascenso Sim ˜oes,[email protected]

Received 30 December 2009; Accepted 30 March 2010

Academic Editor: Yuming Xing

Copyrightq2010 Bruno Ascenso Sim ˜oes. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The use of twistor methods in the study of Jacobi fields has proved quite fruitful, leading to a series of results. L. Lemaire and J. C. Wood proved several properties of Jacobi fields along harmonic maps from the two-sphere to the complex projective plane and to the three- and four-dimensional spheres, by carefully relating the infinitesimal deformations of the harmonic maps to those of the holomorphic data describing them. In order to advance this programme, we prove a series of relations between infinitesimal properties of the map and those of its twistor lift. Namely, we prove that isotropy and harmonicity to first order of the map correspond to holomorphicity to first order of its lift into the twistor space, relatively to the standard almost complex structuresJ1_and

J2_{. This is done by obtaining first-order analogues of classical twistorial constructions.}

1. Introduction

Harmonic maps are mappingsϕbetween Riemannian manifoldsMandNwhich extremize the energy functional

Eϕ 1 2

M

dϕ2ωg. 1.1

LettingTMdenote the tangent bundle ofM, one canlocallycharacterize harmonic maps as solutions of the nonlinear second-order differential equation

whereτϕdenotes thetension fieldofϕ,

τϕtrace∇dϕ

i

∇dϕXi, Xi, {Xi}orthonormallocalframe ofTM. 1.3

A bibliography can be found in1and for some useful summaries on this topic, see2,3. The infinitesimal deformations of a harmonic map are called Jacobi fields. More precisely, let ϕ : M → N be a smooth map and denote by Γϕ−1_{TN the set of smooth}

sections of the pull-back bundleϕ−1_{TN. Ifϕis harmonic andv∈Γϕ}−1_{TNis a vector field}

along it,vis said to be aJacobi fieldalongϕif it satisfies the linearJacobi equationJϕv 0, where theJacobi operatorJϕis defined by

Jϕv Δv−traceRN

dϕ, vdϕ. 1.4

Here,Δis the Laplacian onϕ−1_TN,

Δv−

i

∇Xi∇Xiv− ∇∇XiXiv

1.5

and, lettingRN_{denote the curvature tensor ofN,}

traceRNdϕ, vdϕ

i

RN_dϕX i, v

dϕXi

. _1.6

Jacobi fields are characterized as lying in the kernel of the second variation of the energy functional. Indeed, ifϕt,sis a two-parameter variation of a harmonic mapϕ0,0, then, writing

v∂ϕ/∂t|_0,0 andw ∂ϕ/∂s|_0,0, theHessianHϕofϕis the bilinear operator onΓϕ−1TN given by

Hϕv, w:

∂2_Eϕ t,s

∂t∂s

0,0

M

Jϕv, w

ωg 1.7

so that a Jacobi fieldvalongϕis characterized by the condition

Hϕv, w 0, ∀w. 1.8

A Jacobi field is calledintegrableif it is tangent to a deformation through harmonic maps. In4, 5, the question of whether all Jacobi fields are integrable is answered for the case where the domain is the two-sphere and the codomain the two-dimensional complex projective space or the three- and four-dimensional sphere. This was done by relating the deformations of the map associated with the Jacobi field and those of the twistor lift of the map. More precisely, given an oriented even-dimensional manifold N2n_{, we can construct its positive twistor} spaceΣN. This manifold admits two natural almost complex structuresJ1_andJ2_{. Given}

a mapϕ : M2 _{→ N}2n _{from a Riemann surfaceM}2_{, harmonicity is intimately related with}

the existence of aJ1_{-holomorphic lift ψ see6. On the other hand, Jacobi vector fields}

induce families of maps which are harmonic to first-order and, in some cases, isotropic to first order. The translation of these first order properties in terms of twistor lifts plays an important role on the study of the Jacobi fields and we shall exhibit how this translation can be established in general.

This work is divided as follows: in the next two sections, we recall some well-known results concerning twistor lifts of harmonic and isotropic maps. InSection 4, we show how this constructions generalize to their parametric versions and examine more closely the construction when the codomain is a four-dimensional manifold. We leave to the last section some technical proofs.

2. The Setup

2.1. Twistor Spaces

Let E2k _{be an oriented even-dimensional Euclidean space, equipped with a metric ,. A} Hermitian structureJonEisJ ∈glEwithJ2 _{−IdandJX, JYX, Yfor allX, Y ∈E.}

We say thatJ ispositive if there is a positive basis ofEof the form{e1, Je1, . . . , ek, Jek}and negativeotherwise. The set of all positive Hermitian structuresresp., all negative Hermitian structuresonEis denoted byΣEresp.,Σ−E. The Lie groupSOEacts transitively on

Σ_{Eby the formulaS·JSJS}−1_{and the isotropy subgroup atJis given by}

UJE {S∈SOE:SJJS}. 2.1

LettinguJE {λ∈soE:λJJλ}andmJE {λ∈soE:λJ −Jλ}, we easily conclude that

soE uJE⊕mJE. 2.2

In particular, the tangent space ofΣE SOE/UEatJ is given bymJE. In this vector space, we can introduce a complex structureJVdefining

JV_{λ:Jλ. 2.3}

When equipped withJV, the manifoldΣEis a complex manifold.

GivenEwith a Hermitian structureJ, we can consider onEC E⊗Cits1,0and

0,1-parts given as usual by

E10 {X−iJX, X∈E}, E01{XiJX, X∈E}. 2.4

These are isotropic subspaces, in the sense that X, Y 0 for all X, Y in E10 _{or in E}01_.

Let V2k_{, g,∇ be an oriented even-dimensional vector bundle over a manifold M} equipped with a connection∇and a parallel metricg. Then, we may take the bundle

Σ_{V SOV×}

SO2kΣR2kGriso

VC 2.5

whose fibre at x ∈ M is precisely ΣVx. If M2 is Riemann surface, the vector bundleVC has vanishing2,0-part of its curvature tensor and therefore admits a unique structure as a holomorphic bundle overM2_{, by a well-known theorem of Koszul and Malgrange7. This}

induces a holomorphic structureJKM_{on the bundle Gr}

isoVCΣVfor which a sectionsof

Gr_isoVCis holomorphic if and only if6 see8.

∇Xs⊆s ∀X∈T01M. 2.6

LetNbe an oriented Riemannian manifold with dimension 2n. We call(positive) twistor space ofNthe bundle whose fibre aty∈Nis preciselyΣTyN; that is,

Σ_NSON×

SO2nΣR2n SON×_USO_2n2nSO2n. 2.7

The Riemannian connection onNinduces a splitting of the tangent space toΣNintovertical and horizontal parts, TΣN V ⊕ H. Namely, if π : ΣN → N denotes the canonical projection defined byπy, Jy y, then

Vy,Jy kerdπy,Jy TJyΣTyN,

Hy,Jy

dσy

Xy

:Xy∈TyN, σsection of ΣNwith ∇Xyσ0 andσ

yy, Jy

.

2.8

With respect to this decomposition,dπy,Jy mapsHy,Jy isomorphically intoTyNand allows to

define an almost complex structureJH onHas the pull-back ofJyonTyN. Together with

2.3, this allows to define two almost complex structuresJ1_andJ2_{onΣNby the formulae}

J1 ⎧ ⎨ ⎩

JH _onH, JV _onV, J

2 ⎧ ⎨ ⎩

JH _onH,

−JV _onV. 2.9

When equipped withJ2_{,Σis never a complex manifold; as forJ}1_{, it is integrable if and only}

ifN2n_{is conformally flatn≥ 3or anti-self-dualn 2 for more details, see6,9,10; a} discussion on this topic can also be found in11and references therein.

Notice thatπ is a holomorphic map for any of these complex structures; that is, for a1 ora2, one has that

dπy,JyJ

a_{X J}

Definition 2.1. LetM, JandZ,Jbe two almost complex manifolds. LetTZ H ⊕ Vbe a decomposition ofTZintoJ-stable subbundles; that is,JH ⊆ HandJV ⊆ V. We shall call such a decomposition aJ-stable decomposition. Letψ:M → Zbe a smooth map. We shall say thatψisH-holomorphicif

dψJXH JdψXH, ∀X ∈TM. 2.11

Analogously we defineV-holomorphic maps.

A smooth map ψ : M → Z is holomorphic if and only if it is both H and V -holomorphic for some, and so any, stable decomposition TZ H ⊕ V. Taking Z ΣN, the decompositionTΣNH ⊕ Vis clearly stable for both the almost complex structuresJ1

andJ2_onΣN.

Remark 2.2. We can easily introduce a metric on the twistor spaceΣN: let y, Jy ∈ ΣN and consider the tangent space at this point,Ty,JyΣN H ⊕ V. We know that we have the identificationsH TyNandV mJTyN. To get a metric onH, transport the metric from that onTyN; that is,hX, Y dπX, dπY, for allX,Y ∈ H, where,denotes the metric on N at y πy, Jy. For the vertical space V mJyTyN ⊆ glTyN, we can

consider the restriction of the metric on the spaceglTyN. Finally, we declareHandV to be orthogonal under the metrich; that is,hX, V 0, for allX∈ H,V ∈ V. With this metric, the decompositionH ⊕ Vis orthogonal andJa_{-stablea 1,2,ΣN, h,J}a _{a 1,2are} almost Hermitian manifolds and the projection mapπis a Riemannian submersion.

2.2. Conformal and Isotropic Maps

Given a smooth mapϕ:M2 _{→ N,ϕis said to beweakly conformalatx∈Mif there isΛ} x∈R with

dϕxX, dϕxY

ΛxX, Y, ∀X, Y ∈TxM. 2.12

IfΛx/0, thenx is said to be aregular pointof ϕand the mapϕ is called conformalat x. Moreover, a map which is conformalresp., weakly conformalat all pointsx∈Mis said to be aconformal mapresp., aweakly conformal map.

IfN;,, Jis an almost Hermitian manifold, any holomorphicresp., antiholomor-phicmapϕ : M2 _{→ N isweaklyconformal as it mapsT}10_MtoT10_{Nresp., toT}01_N.

A stronger property than conformality is isotropy: ifϕ :M2 _{→ N is a smooth map from a}

Riemann surface,ϕisisotropicif12

∂r_zϕ, ∂s_zϕ0, ∀r, s≥1, 2.13

where∂r

zϕ∇∂z∂

r−1

z ϕ. Actually, the condition2.13can be weakened to

To check this, establish an induction onj |r−s|: ifj0 the result is trivial. Assuming now that2.14implies2.13for allj ≤nand takingr, s≥1 with|r−s|n1, we may assume without loss of generality thatr≥s,rsn1 and we get

∂s_zn1ϕ, ∂s_zϕ∂z

∂_zsnϕ, ∂s_zϕ−∂_zsnϕ, ∂s_z1ϕ. 2.15

Since|sn−s|nand|sn−s−1||n−1| ≤n, both terms in the above expression vanish. Moreover, lettingr s 1 in2.13, it is easy to check that an isotropic map from a Riemann surface is aweaklyconformal map.

Letϕ:M2 _{→ Nbe a smooth map from a Riemann surfaceM}2_{. We shall say thatz∈}

M2is anumbilic pointofϕif{∂zϕz, ∂2zϕz}is aC-linearly dependent set. Ifϕ:M2 → N is such that all pointsz∈M2_{are umbilic, we shall say thatϕistotally umbilicsee6.}

3. Nonparametric Twistorial Constructions

The following are well-known twistorial constructions13 see also6.

Theorem 3.1. Ifψ :M2 → ΣN,J1is holomorphic, the projection mapϕ π◦ψis isotropic. Conversely, ifϕ:M2 _{→ Nis a conformal totally umbilic immersion, there is (locally) a holomorphic} mapψ :M2 _{→ ΣN,J}1_{such thatϕπ◦ψ.}

If ψ : M2 → ΣN,J2 is holomorphic, the projection map ϕ π ◦ ψ is harmonic. Conversely, if ϕ : M2 _{→ N is a conformal harmonic map, there is (locally) a holomorphic map}

ψ:M2 _{→ ΣN,J}2_{such thatϕπ◦ψ.}

We shall sketch the proof of this result. We start by noticing that given a smooth map ϕ : M → Nobtained as the projection ofψ : M2 _{→ ΣN,ϕ π◦ψ, without requiring}

further conditions `a priorionψ, nothing guarantees thatϕ is holomorphic relatively to the induced almost Hermitian structure Jψ onTN; if it is, we shall say that the structureJψ is strictly compatible withϕ or that the mapψ is astrictly compatible twistor liftofϕ. Such a structure Jψ can exist if and only ifdϕT10M ⊆ T_J10_ψN is isotropic: in other words, if and only ifϕisweaklyconformal. IfJψ preservesdϕTMbut does not necessarily renderϕ holomorphic, we shall say thatJψ or the mapψiscompatiblewithϕ.

Ifϕis given as the projectionϕ π ◦ψ of anH-holomorphic mapψ : M, JM _→

Σ_N,Ja _{a1 or 2, thenϕis holomorphic with respect to the induced almost Hermitian} structureJψ onTN:

JψdϕX

dϕJM_{X ⇐⇒dπJ}H_dψXH_dπdψJM_XH_{⇐⇒ J}H_dψXH_dψJM_XH_.

3.1

In particular, ϕ is weakly conformal. Moreover, the above equivalence shows that any strictly compatible lift ofϕisH-holomorphic. On the opposite direction, letϕ:M2 _{→ Nbe}

a conformal map. LetV :TM⊥⊆ϕ−1_{TNdenote the normal bundle ofTMinTN. We may}

decompose the connection onϕ−1_{TNinto its tangent and normal parts,∇∇∇}⊥_{. Hence,}

we may take the bundleΣV overMwhich has theJKM_{-holomorphic structure. Sinceϕis}

conformal, we may transfer the Hermitian structureJM_ofM2_{intodϕTM. Hence, we have}

a natural map

η:ΣV −→ΣN,

J −→J

⎧ ⎨ ⎩

J onTM⊥,

JM _ondϕTM.

3.2

Taking any holomorphic section ofΣV,JKM_{and composing withη, we obtain a strictly}

compatible twistor liftψofϕ. Since any such lift isH-holomorphic, we have just proved the following result.

Proposition 3.2. Givenϕ:M2 _{→ N,ϕis conformal if and only ifϕis (locally) the projection of an}

H-holomorphic mapψ :M2 _{→ ΣN.}

To proceed, we need the following result13.

Proposition 3.3. Let ψ : M → ΣN and let ϕ π ◦ ψ. Take s the section of Gr_isoTNC

corresponding toψ. Then, the mapψisJ1_{-holomorphic if and only ifϕis holomorphic with respect to}

Jψ and

∇Xs⊂s for anyX∈T10M. 3.3

The mapψisJ2_{-holomorphic if and only ifϕis holomorphic with respect toJ} ψ and

∇_Xs⊂s for anyX∈T10M. 3.4

Letψ :M2 _{→ ΣNbe aJ}1_{-holomorphic map and letsbe the corresponding section}

of Gr_isoTNC. If ϕ π ◦ ψ is the projection map, we have that ϕ is Jψ-holomorphic. In particular,∂zϕ∈s. SinceψisJ1-holomorphic, we can write

∇∂z∂zϕ⊆s 3.5

and, inductively, it follows that∂r

zϕ ⊆ sfor all r ≥ 1 so thatϕ is isotropic. Conversely, let

ϕ:M2 _{→ Nbe a conformal totally umbilic map. In this case, we consider the manifoldM}2

equipped with the opposite holomorphic structure-JM_{and again construct the holomorphic} bundleΣV,JKM_{. For this structure, a section sis holomorphic if and only if∇}⊥

∂zs ⊆ s.

Sinceϕis conformal, we may define the mapηas in3.2. On the other hand, becauseϕis totally umbilic, we know that∂2

zϕlies in the span of∂zϕ; in other words, ifJ is any almost Hermitian structure onTNstrictly compatible withϕ,∂2

zϕ lies inTJ10N. We may therefore conclude thatηisJ1_{-holomorphic. As a matter of fact, given a holomorphic sectionsofΣV}

and consideringsη◦s, we have thatss⊕dϕT10_{M. We may write}

∇

∂zs, dϕ

T10_{M−s, ∇} ∂zdϕ

sincesand∂2

zϕlie ins. In particular,∇∂zs⊆sand so

∇∂zs⊆

∇

∂z∇

⊥

∂z

s∇∂zdϕ

T10M⊆ssspan∂zϕ, ∂2zϕ

⊆s 3.7

and therefore we conclude thatηisJ1_{-holomorphic.}

With a similar argument, letψ :M2 _{→ ΣNbe aJ}2_{-holomorphic map and letsbe}

the section of Gr_isoTNCcorresponding toψ. Then, taking the projection mapϕπ◦ψ, we get

τϕ01 Jψtraceg∇Jψ 01

4∇∂zdϕ∂z

01

. 3.8

Sinceϕ isJψ-holomorphic, dϕ∂z lies ins. Fromivin Proposition 3.3, we conclude that

∇∂zdϕ∂zlies insand therefore has vanishing0,1-part. From the reality ofτϕ, we have

thatτϕ 0 and soϕis harmonic. Conversely, ifϕ:M2 _{→ Nis conformal and harmonic, we}

may take the bundleΣV,JKM_{and the mapηin3.2. Sinceϕis harmonic,ηis holomorphic}

with respect to theJKM_andJ2_{structures6,8: lettingsdenote a holomorphic section ofΣV}

andsthe compositionη◦s, then,sdϕT10_M⊕sand

∇∂zs

∇⊥

∂z∇

∂z

s∇∂zdϕ

T10M. 3.9

Since ϕ is harmonic, ∇∂zdϕT10M ⊆ dϕT10M ⊆ s. On the other hand, since s is

holomorphic,∇⊥_∂

zs⊆s⊆s. Finally,

∇

∂zs, dϕ ∂z

−s, ∇∂zdϕ∂z

0 3.10

shows that∇∂zs⊆sand so thatηis holomorphic. Therefore, taking any holomorphic section

ofΣV and composing withηgive aJ2_{-holomorphic lift ofϕ.}

Remark 3.4. Notice that to guarantee the existence of the J1-holomorphic lift for ϕ, the important fact was that ∂2

zϕ belongs to the 1,0-part of TN for any almost Hermitian structure strictly compatible withϕ. This is guaranteed ifϕis a totally umbilic map, but it is not strictly necessary. For instance, ifϕ :M2 → N4 is an isotropic map, the vectors∂zϕ,

∂2

zϕspan an isotropic subspace. If this vectors are linearly independent, taking this space as the1,0-space ofψ, thenψis aJ1_{-holomorphic lift ofϕ, althoughψmay be a map intoΣ}−_N;

4. First-Order Twistorial Constructions

4.1. Harmonicity and Isotropy to First Order

LetIdenote an interval of the real line containing 0. Given afamily ofmapsϕ:I×M → N,t, x → ϕtx, we say thatϕis harmonic to first orderif

ϕ0is harmonic and ∂t|0τ

ϕt

0, 4.1

where∂t|0τϕt ∇ϕ

−1_TN

∂t|0 τϕtandτϕt trace∇dϕt∈ϕ

−1_TN.

Let ϕ0 : M → N be a harmonic map,v ∈ Γϕ−₀1TNa vector field along ϕ0, and

ϕ:I×M → Na one-parameter variation ofϕ0. We say thatϕistangenttovifv∂t|0ϕt. The following result is a key ingredient in what follows4:

Proposition 4.1. Letϕ0 :M → Nbe a harmonic map between compact manifoldsMandN. Let

v∈Γϕ−1

0 TNbe a vector field alongϕ0andϕ:I×M → Na one-parameter variation ofϕ0tangent tov. Then,

∂t|0τ

ϕt

−Jϕv. 4.2

In particular,vis Jacobi if and only if any tangent one-parameter variation is harmonic to first order.

We have seen inTheorem 3.1that harmonicity was not enough to establish a relation with possible twistor lifts of a map conformality and was also a key ingredient, as maps obtained as projections of twistorial maps must be holomorphic with respect to some almost Hermitian structure along the map. On the other hand, when the domain is the 2-sphere, harmonicity impliesweakconformality or even isotropy, the last case occurring if the target manifold is itself also a sphere or the complex projective space12,14.

LetM2_{be a Riemann surface andϕ:I×M → Na smooth map. The mapϕis said to}

beconformal to first orderif15

ϕ0 is conformal and ∂t|0

∂zφt, ∂zφt

0. 4.3

Analogously,ϕis said to beisotropic to first order,

φ0 is real isotropic and ∂t|0

∂r_zφt, ∂szφt

0, ∀r, s≥1. 4.4

As for the nonparametric case, one can prove16that condition4.4can be weakened to the following

φ0 is real isotropic and ∂t|0

∂r_zφt, ∂rzφt

0, ∀r ≥1. 4.5

4.2. Twistorial Constructions

As we have seen, Jacobi fields induce variations that are harmonic and, in some cases, conformal or even isotropic to first order. On the other hand, in Section 2 we have seen that conformality, harmonicity, and isotropy of the map ϕ correspond to H, J2 _{and J}1

-holomorphicity of the twistor liftψ. As we shall see, these results do have a first-order version as follows. We start with a definition.

Definition 4.2. LetM, Jbe an almost complex manifold andZ, h,Jan almost Hermitian manifold. Given a smooth mapψ :I×M → Z, we say thatψisholomorphic to first orderif ψ0:M → Zis holomorphic and

∇∂t|0

dψtJX− JdψtX

0 ∀X∈TM, 4.6

where∇is the Levi-Civita connection onZinduced by the metrich. Moreover, ifTZH ⊕ V is aJ-stable decomposition ofTZ, orthogonal with respect toh, we shall say thatψ isH -holomorphic to first orderifψ0isH-holomorphic and

∇∂t|0

dψtJX

H_{− J}H_dψ

tX H

0 ∀X∈TM, 4.7

where JH is the restriction of J to H. Changing H to V gives the definition of V -holomorphicity to first order.

In contrast with the nonparametric case, it is not obvious thatJ-holomorphicity to first order impliesH-holomorphicity to first order. As a matter of fact, from4.6, it only follows that∇∂t|0dψtJX− JdψtX

H _{0. However, we do have the following.}

Lemma 4.3. Let ψ : M, J → Z, h,Jbe a smooth map and let TZ H ⊕ V be aJ-stable decomposition ofZ, orthogonal with respect toh. Then,ψis holomorphic to first order if and only ifψ

is bothHandV-holomorphic to first order.

Proof. Assume thatψis holomorphic to first order. Then,ψ0isH-holomorphic. As for4.7,

lettingY denote an arbitrary section ofTZandYHits projection intoH, we have

4.7⇐⇒ ∇∂t|0

dψtJX _H

− JH_dψ

tX _H

, Y0

⇐⇒∂t|0

dψtJX _H

− JH_dψ

tX _H

, Y−dψ0JX _H

− JH_dψ

0X _H

,∇∂t|0Y

0

⇐⇒sinceψ0 isH-holomorphic

∂t|0

dψtJX− JdψtX, YH

0

⇐⇒ψ0 is holomorphic

∇∂t|0

dψtJX− JdψtX

H_0,

4.8

Remark 4.4. The importance of choosing the Levi-Civita connection on Z is illusory. In particular, we can define the concept ofholomorphicity to first order or H, V-holomorphicity to first orderfor maps defined between almost complex manifolds, not necessarily equipped with any metric.

Indeed, letψ :I×M, J → Z, h,Jbe a holomorphic to first-order map with respect to∇, so that4.6holds. Let{Yj}denote alocalframe forTZ. Then,

∇∂t|0

dψtJX− JdψtX

0

⇐⇒

j

∂t|0

dψtJX− JdψtX

jYj

0

⇐⇒

j

∂t|0

dψtJX− JdψtX

j

·Yj

dψ0JX− Jdψ0X

j· ∇ ψ−1

∂t|0Yj

0.

4.9

Sinceψis holomorphic to first order,ψ0is holomorphic, the above equation is equivalent to

∂t|0

dψtJX− JdψtX

j

0, ∀j. 4.10

Now, sinceψ0 holomorphicity does not depend on the chosen connection, we can deduce

that holomorphicity with respect to∇ reduces to the same condition 4.10. Thus,ψ being holomorphic to first order does not depend on the chosen connection. For H resp., V holomorphicity to first order, we use similar arguments, replacing{Yj}for a horizontalresp., verticalframe.

4.3. The

H

-Holomorphic Case

In the nonparametric case, given a conformal mapϕ:M2 _{→ N, we can always find a liftψ:}

M2 → ΣNsuch thatϕis holomorphic with respect toJψ. In other words,locally defined strictly compatible lifts always exist. In general, this lift may not beJ1_orJ2_{-holomorphic but}

it isH-holomorphic. Letϕtbe a variation ofϕ, conformal to first order. Then, if a liftψtto the twistor space that makesϕtholomorphic for all smalltexists,ϕtis necessarily conformal for all smallt, which may not be the case. So, even if conformality is preserved to first order, there might be no strictly compatible twistor lift for allt; hence, we should relax the condition on conformality. We shall say that a twistor liftψof a conformal to first order mapϕiscompatible to first orderwithϕif

ψ0 is strictly compatible withϕ0, ψtis compatible with ϕt, ∀t. 4.11

We start with a technical lemma, whose proof the reader can find inSection 5.

Lemma 4.5. Letϕ:I×M2 _{→ Nbe a conformal to first-order map. Letψbe a twistor lift compatible} to first order withϕ. Then for allX∈ΓTMthere is a functionaX

t and a vector fieldvXt ∈ϕ−t1TN withaX₀ 1,vX₀ 0and∂t|0aXt 0,∇∂t|0v

X

t 0such that

JψtdϕtXa

X

In particular,ϕisJψ-holomorphic to first order in the sense that

ϕ0 is holomorphic with respect toJψ0, ∇∂t|0

dϕtJX−JψtdϕtX

0. 4.13

Lemma 4.6. Letψ : I ×M2 _{→ ΣN beH-holomorphic to first order. Then,ϕ π◦ψ isJ}

ψ -holomorphic to first order.

Proof. SinceψisH-holomorphic to first order,∇∂t|0dψtJX

H_{− J}H_dψ

tXH 0. Therefore, for allYH∈ H, sinceψ0isH-holomorphic,

0 ∇∂t|0

dψtJX

H_{− J}H_dψ

tX

H_{, Y}H _∂

t|0

dψtJXH− JH

dψtX

H_{, Y}H_.

4.14

Sinceπis aJψ-holomorphic Riemannian submersion, the above equation can be written as

0 ∂t|0

dπdψtJX

−Jψtdπ

dψtX

, dπYH. 4.15

Hence, for allY ∈TN, using the fact thatϕ0isJψ0holomorphic,

0 ∂t|0

dϕtJX−JψtdϕtX, Y

∇∂t|0

dϕtJX−JψtdϕtX

, Y, 4.16

showing that∇∂t|0dϕtJX∇∂t|0JψtdϕXand concluding our proof.

Proposition 4.7. Letψ:I×M2 _{→ ΣNbeH-holomorphic to first order map. Then, the projected}

mapϕ π◦ψ :I×M2 _{→ Nis conformal to first order. Conversely, letϕ :I×M}2 _{→ Nbe a} conformal to first order map. Then there is a (local)H-holomorphic to first order mapψ :I×M2 _→ Σ_{Nwhich is compatible to first order withϕ.}

Proof. Takeψ : I ×M2 _{→ ΣN anH-holomorphic to first order map and letϕ π ◦ψ.}

We know that ϕ0 is conformal Proposition 3.2. As for the first-order variation, using the

precedingLemma 4.6,

∂t|0dϕtJX2 2

∇∂t|0dϕtJX, dϕ0JX

2∇∂t|0JψtdϕtX, Jψ0dϕ0X

∂t|0

JψtdϕtX, JψtdϕtX

∂t|0

dϕtX, dϕtX

.

4.17

Using similar arguments, we can show that

∂t|0

dϕtJX, dϕtX

0, 4.18

concluding the first part of the proof.

Conversely, let ψ be any twistor lift of ϕ compatible to first order. Then, there is a functionaX

t and a vector fieldvXt verifying the conditions onLemma 4.5with

JψtdϕtXa

X

Now,ψ isH-holomorphic to first order if and only ifψ0 isH-holomorphicwhich follows

from Proposition 3.2and 4.6 holds. Using the same argument as inLemma 4.3, 4.6 is equivalent to

∂t|0

dϕtJX−JψtdϕtX, Y

0, ∀Y ∈TN. 4.20

But

∂t|0

dϕtJX−JψtdϕtX, Y

∇∂t|0

dϕtJX−aXtdϕtJX−vXt

, Y−dϕ0JX−aX0dϕ0JX−vX0,∇∂t|0Y

0,

4.21

from the given conditions onaX

t andvXt and thus concluding the proof.

4.4. The

J

Next, we give a useful characterization for maps to beJa_{-holomorphic to first ordera1} or 2, whose proof is inSection 5compare withProposition 3.3.

Lemma 4.8. Letψ :I×M2 _{→ ΣNbe a smooth map and letϕπ◦ψbe its projection. Then,ψ}

isJa_{-holomorphic to first order (a1or2) if and only if}

ϕisJψ-holomorphic to first order, 4.22

∀Y10 t ∈ϕ−t1

T10 J_ψtN

∃Z10 t ∈ϕ−t1

T10 J_ψtN

such that

∇∂t|0∇∂zY

10

t ∇∂t|0Z 10

t , ∇∂zY

10 0 Z010,

a1 4.23

or

∀Y_t10∈ϕ−_t1T_J10

ψtN

∃Z10_t ∈ϕ−_t1T_J10

ψtN

such that

∇∂t|0∇∂zY

10

t ∇∂t|0Z 10

t , ∇∂zY

10 0 Z100 .

a2 4.24

From the preceding lemma, we can also deduce the following.

Lemma 4.9. Let ψ : I ×M2 _{→ ΣN be a map J}1_{-holomorphic to first order and consider the} projected mapϕπ◦ψ. Then for allr ≥1there isZ10_t ∈ϕ−_t1T_J10

ψtNwith

∂r

zϕ0 Z100 , ∇∂t|0∂

r

zϕt∇∂t|0Z 10

t . 4.25

Proof. Forr1, we have∂zϕtdϕtX−idϕtJXso that using4.22we have

∇∂t|0∂zϕt∇∂t|0

dϕtX−idϕtJX

∇∂t|0

dϕtX−iJψtdϕtX

TakingZ_t10 dϕtX−iJψtdϕtX ∈ϕ−

1

t TJ10_ψtN, we obtain the result. To establish an induction, assume now that the result is valid forrk; that is, there isZ_t10,ksuch that

∇∂t|0∂

k

zϕt∇∂t|0Z 10,k

t , ∂kzϕ0Z₀10,k. 4.27

Takingrk1 and noticing that∂t|0ϕt, ∂zϕt dϕt∂z, ∂t|0 0,

∇∂t|0∂

k1

z ϕ∇∂t|0∇∂z∂

k

zϕtRN

∂t|0ϕt, ∂zϕt

∂k_zϕt∇∂z∇∂t|0∂

k

zϕt∇∂t|0ϕt,∂zϕt∂

k zϕt

RN∂t|0ϕt, ∂zϕ0

Z₀10,k∇∂z∇∂t|0Z 10,k t

RN_∂

t|0ϕt, ∂zϕ0

Z₀10,k∇∂t|0∇∂zZ

10,k t RN

∂zϕ0, ∂t|0ϕt

Z₀10,k ∇∂t|0∇∂zZ

10,k t ,

4.28

asRN _{is antisymmetric on the first two arguments. Now, sinceψ isJ}1_{-holomorphic,4.23}

holds so that there isZ10_t , k1such that

∇∂t|0∇∂zZ

10, k

t ∇∂t|0Z 10, k1

t , ∇∂zZ

10, k

0 Z

10, k1

0 . 4.29

But the second condition gives∂k1

z ϕ0 ∂z∂kzϕ0 ∇∂zZ

10, k

0 Z

10, k1

0 , whereas the first holds

precisely that∇∂t|0∂

k1

z ϕt∇∂t|0∇∂zZ

10, k

t ∇∂t|0Z 10, k1

t , as we wanted to show.

Proposition 4.10projections of mapsJ1_{-holomorphic to first order. Letψ:I×M}2 _{→ ΣN}

be a mapJ1_{-holomorphic to first order, whereM}2_{is any Riemann surface. Then, the projection map}

ϕπ◦ψis isotropic to first order.

Notice that we could replaceΣNwithΣ−N, as real isotropyto first orderdoes not depend on the fixed orientation onN.

Proof. Thatϕ0 is isotropic follows from the nonparametric case. Therefore, we are left with

proving that

∂t|0

∂r_zϕ, ∂r_zϕ0, ∀r≥1. 4.30

UsingLemma 4.9, for fixedr ≥1, chooseZ10_t ∈ϕ−_t1T_J10

ψtNwith∂

r

zϕ0 Z₀10 and∇∂t|0∂

r zϕt

∇∂t|0Z 10

t . Then, the conclusion follows from

∂t|0

∂r_zϕ, ∂r_zϕ2∇∂t|0∂

r

zϕt, ∂rzϕ0

2∇∂t|0Z 10 t , Z100

∂t|0

Z10_t , Z10_t 0. 4.31

We now turn our attention to the existence of liftsJ1_{-holomorphic to first order for a}

given isotropic to first-order mapϕ : I ×M2 _{→ N}4_{. Recall that in the nonparametric case}

Theorem 4.11. Letϕ : I×M2 _{→ N}4 _{be an isotropic to first order with∂}

zϕ0 and∂2zϕ0 linearly independent. Then, there is either a (local) mapψ:I×M2 _{→ ΣN}4_{or a mapψ}−_:I×M2 _{→ Σ}−_N4 which isJ1_{-holomorphic to first order and compatible to first order withϕ.}

Before proving Theorem 4.11, we give the following lemma, which we prove in Section 5

Lemma 4.12. Letϕbe as in the precedingTheorem 4.11.

iSuppose that theJ1_{-holomorphic lift ofϕ}

0isψ₀ ∈ ΣN(resp.,ψ₀− ∈ Σ−N). TakeJψtthe

unique positive (resp., negative) almost Hermitian structure onTϕtNcompatible withϕt.

Then, taking

uX_t ∇XdϕtX− ∇JXdϕtJX, vXt −∇XdϕtJX− ∇JXdϕtX, 4.32

we have that

∇∂t|0Jψtu

X

t −∇∂t|0v

X

t. 4.33

iiThere areat,btsuch that

∇∂t|0∂ 3

zϕt∇∂t|0

at∂zϕtbt∂2zϕt

. 4.34

We are now ready to proveTheorem 4.11.

Proof ofTheorem 4.11. As before, takeψ₀orψ₀−theJ1_{-holomorphic lift ofϕ}

0. Assume, without

loss of generality, that it isψ₀. Then, at eachttakeJψt the unique positive almost Hermitian

structure compatible withϕtand let us prove that this mapψisJ1-holomorphic to first order. UsingLemma 4.5,ϕisJψ-holomorphic to first order and we are left with proving that4.23 holds. It is enough to prove that there is a basis {Y₁10

t, Y

10 2t} of ϕ

−1

t TJ10_ψtNfor which 4.23 holds. Now, takeY₁10

t dϕtX−iJψtdϕtXand Y

10 2t u

X t −iJψtu

X

t whereuXt is as in4.32. Then,

∇∂t|0∇∂zY

10 1t R

∂t|0ϕt, ∂zϕt

Y₁10

0 ∇∂z∇∂t|0

dϕtX−iJψtdϕtX

∇∂t|0∇∂z

dϕtX−idϕJX

∇∂t|0

uX_t ivX_t ∇∂t|0

uX_t −iJψtu

X t

Analogously,

∇∂t|0∇∂zY

10 2t R

∂t|0ϕt, ∂zϕ0

Y₂10

0 ∇∂z∇∂t|0

uX_t −iJψtu

X t

R∂t|0ϕt, ∂zϕ0

∂2_zϕ0R

∂zϕ0, ∂t|0ϕt

uX₀ ivX₀∇∂t|0∇∂z

uX_t ivX_t

∇∂t|0

a∂zϕb∂2zϕt

∂t|0aY1100 ∂t|0bY 10

20 a∇∂t|0∂zϕt∇∂t|0∂ 2 zϕt

∂t|0aY1100 ∂t|0bY 10

20 a∇∂t|0Y 10

1t ∇∂t|0Y 10 2t ∇∂t|0

aY₁10

t bY

10 2t

,

4.36

where we have usedY10 20 ∂

2

zϕ0,∇∂t|0Jψtu

X

t ∇∂t|0v

X

t ,uX0 ivX0 ∂2zϕ0, and∇∂zu

X t ivXt

∂3_zϕt. Hence,Y₁10_t andY₂10_t satisfy equation4.23, concluding our proof.

4.5. The

J

Holomorphic Case

Theorem 4.13. Letψ : I×M2 _{→ ΣN be a mapJ}2_{-holomorphic to first order. Then,ϕ π◦}

ψ : I ×M2 → N is harmonic to first order (and conformal to first order from Lemma 4.3 and

Proposition 4.7).

We first give the following characterization ofV-holomorphic to first order maps.

Lemma 4.14. Letψ : I×M2 → ΣN,Ja(a 1or a 2) be a smooth map. Then,ψ is a

V-holomorphic to first order map if and only ifψ0isV-holomorphic and

∇∂t|0

∇JXJψt −1

a_J ψt∇XJψt

0. 4.37

Proof. Ifψ :M → ΣNis any smooth map, then17 dψXV∇XJψ. Hence, from2.9,

Ja_dψXV ₋₁a1_J

ψ∇XJψ a1,2. 4.38

Thus, we can rephrase equation4.37as

∇∂t|0

dψtJX− JadψtX V

0, 4.39

Proof ofTheorem 4.13. Thatϕ0is harmonic follows fromTheorem 3.1. Hence, we are left with

proving that∂t|0τϕt 0. SinceψisJ2-holomorphic to first order, we deduce thatψis both

HandV-holomorphic to first orderLemma 4.3. FromV-holomorphicity4.37, we have

∇∂t|0

∇JXJψtJψt∇XJψt

0

⇒∇∂t|0

∇JXJψtJψt∇XJψt

dϕ0X

0

⇒ ∇∂t|0

∇JXdϕtJX∇XdϕtX

∇∂t|0

Jψt

∇JXdϕtX− ∇X

JψtdϕtX

⇒ ∇∂t|0τ

ϕt

∇∂t|0

Jψt

∇JXdϕtX− ∇X

JψtdϕtX

.

4.40

UsingLemma 4.6and4.13together with symmetry of the second fundamental form ofϕ0,

the right-hand side of the above identity becomes

∇∂t|0Jψt

∇JXdϕ0X− ∇XJψ0dϕ0X

Jψ0

∇∂t|0

∇JXdϕtX− ∇XJψtdϕtX

∇∂t|0Jψt

∇dϕ0JX, X− ∇dϕ0X, JX dϕ0

∇JXX− ∇XJX

Jψ0

∇∂t|0

∇JXdϕtX− ∇XdϕtJX

Jψ0

∇∂t|0

∇dϕtJX, X− ∇dϕtX, JX dϕt

∇JXX− ∇XJX

0,

4.41

so that∂t|0τϕt 0, concluding the proof.

Theorem 4.15. Letϕ:I×M2 _{→ Nbe a harmonic and conformal to first order map. Then, there is} (locally) a mapψ:I×M2 → ΣNwhich isJ2_{-holomorphic to first order and withϕπ◦ψ.}

Since harmonicityto first orderdoes not depend on the orientation onN, we could have replacedΣNbyΣ−Nin both Theorems4.13and4.15

Proof. For eachtconsiderVtdϕtTM⊥ ⊆ϕ−t1TN, bundle overM2. SinceM2is a Riemann surface,R20

Vt0 and we can conclude that for eachtthere is a Koszul-Malgrange holomorphic

structure onΣVt. Hence16, Theorem I.5.1., there is a smooth sectionswithsta Koszul-Malgrange holomorphic section ofΣVt:∇⊥_∂zst⊆st. So,

Jψt

∇⊥

XiJX

vt−iJψtvt

i∇⊥_XiJXvt−iJψtvt

, ∀vt∈dϕtTM⊥, 4.42

equivalently,

Jψt

∇⊥

Xvt∇⊥JXJψtvt

−∇⊥

JXvt∇⊥XJψtvt,

Jψt

∇⊥

JXvt− ∇⊥XJψtvt

∇⊥

Xvt∇⊥JXJψtvt.

4.43

Takess⊕T10_whereT10

t is the1,0-part ondϕtTMCdetermined byJtrotation byπ/2 ondϕtTM.Notice that asϕtis not conformal we might not get a Hermitian structure by settingT10

orientation ondϕtTMimported fromTMviadϕt.Thensdefines a compatible twistor lift ofϕ. Let us check thatψisJ2_{-holomorphic to first order. Thatψ}

0is holomorphic is immediate.

From the proof ofProposition 4.7, we deduce thatψ isH-holomorphic to first order as it is compatible to first order withϕand the latter is conformal to first order. Hence, we are left with proving that4.37

∇∂t|0∇JXJψt−∇∂t|0Jψt∇XJψt 4.44

holds. We shall establish this equation by showing that both sides agree when applied to any vectorv ∈ TN. For that, we consider, in turn, the three casesv dϕ0X,dϕ0JX, and

v∈dϕ0TM⊥. The first two have similar arguments so that we prove only the first.

iv dϕ0X. From ψ0 holomorphicity, we have ∇JXJψ0 −Jψ0∇XJψ0. On the other

hand, asψisH-holomorphic to first order,4.13is satisfied. Finally, for allt,

∇JXdϕtX− ∇XdϕtJX∇dϕtJX, X− ∇dϕtX, JX dϕt

∇JXX− ∇XJX

0. 4.45

Sinceϕis harmonic to first order, our condition follows from

∇∂t|0∇JXJψt

dϕ0X−

∇∂t|0Jψt∇XJψt

dϕ0X

⇐⇒ ∇∂t|0∇JX

JψtdϕtX

− ∇∂t|0

Jψt∇JXdϕtX

⇐⇒ ∇∂t|0

∇JXdϕtJX∇XdϕtX

∇∂t|0Jψt

∇JXdϕ0X− ∇Xdϕ0JX

−∇∂t|0

Jψt∇X

JψtdϕtX

− ∇∂t|0∇XdϕtX Jψ0∇∂t|0

∇JXdϕtX− ∇XdϕtJX

⇐⇒ ∂t|0τ

ϕt

0.

4.46

iiv∈dϕ0TM⊥. We now have

∇∂t|0∇JXJψt

v−∇∂t|0Jψt∇XJψt

v

⇐⇒ ∇∂t|0∇JX

Jψtv

− ∇∂t|0

Jψt∇JXv

−∇∂t|0

Jψt∇X

Jψtv

− ∇∂t|0∇Xv

⇐⇒ ⎧ ⎪ ⎨ ⎪ ⎩

∇∂t|0∇

⊥

JX

Jψtv

− ∇∂t|0

Jψt∇⊥JXv

∇∂t|0

Jψt∇⊥X

Jψtv

∇∂t|0∇

⊥

Xv0,

∇∂t|0∇

JX

Jψtv

− ∇∂t|0

Jψt∇JXv

∇∂t|0

Jψt∇X

Jψtv

∇∂t|0∇

Xv0.

4.47

Now, the first condition follows from 4.43since s is Koszul-Malgrange holomorphic for eacht. As for the second, lettingLdenote its left-hand side, we shall prove thatL, w 0 for all w ∈ TN. We do this by establishing the three cases w dϕ0X, w dϕ0JX and

iiaWhenwdϕ0Xwe have

L, dϕ0X

−∂t|0

Jψtvt,∇JXdϕtX

vt,∇XdϕtX

∂t|0

−vt,∇JXdϕtJX

Jψtvt,∇XdϕtJX

−∂t|0

vt,∇XdϕtX∇JXdϕtJX

∂t|0

Jψtvt,∇XdϕtJX− ∇JXdϕtX

−∇∂t|0vt, τ

ϕ0

−v0,∇∂t|0τ

ϕt

0.

4.48

iibLetwt∈dϕtTM⊥. Then,

L, w ∂t|0

∇

JX

Jψtv

−Jψt∇

JXvJψt∇

X

Jψtv

∇

Xv, w

−∇

JX

Jψ0v

−Jψ0∇

JXvJψ0∇

X

Jψ0v

∇

Xv,∇∂t|0w

.

4.49

The first term on the right side of the above equation vanishes aswtlies indϕtTM⊥, whereas the second is zero fromψ0-holomorphicity, concluding our proof.

4.6. The 4-Dimensional Case

Theorem 4.16. Letϕ:I×M2 _{→ N}4_{be harmonic and isotropic to first-order map and with∂} zϕ0 and∂2_zϕ0being linearly independent. Then, (locally) there is either a mapψ:I×M2 → ΣNor a mapψ− :I×M2 _{→ Σ}−_{Nwhich is simultaneouslyJ}1_andJ2_{-holomorphic to first order and with}

ϕπ◦ψ. Conversely, ifψ:I×M2 _{→ ΣN}4_(orψ:I×M2 _{→ Σ}−_N4_{) isJ}1_andJ2_-holomorphic to first order, the projected mapϕπ◦ψ:I×M2 → N4is harmonic and isotropic to first order.

Proof. The converse is obvious fromProposition 4.10andTheorem 4.13. As for the first part, inTheorem 4.11we saw that we can lift the mapϕto a mapJ1_{-holomorphic to first order.}

Moreover, this lift could be defined as the unique positive or negative almost complex structure compatible with ϕ. On the other hand, in Theorem 4.15we have seen that there is a mapJ2_{-holomorphic to first order withϕ π◦ψ and for whichϕis compatible. From}

the comment afterTheorem 4.15, there is also a twistor lift ofϕintoΣ−N. Therefore, from the dimension ofN, we conclude that the lifts constructed in both cited results are the same and, therefore, simultaneouslyJ1_andJ2_{-holomorphic to first order.}

We would now like to guarantee theuniqueness to first orderof our twistor lift. Before stating such a result, we start with a lemma, proved inSection 5.

Lemma 4.17. Letψ :I×M2 _{→ ΣNbe a mapJ}1_{-holomorphic to first order. Consider the twistor} projectionϕtπ◦ψand the vectorsutandvtdefined by

so that

ut∇XdϕtX− ∇JXdϕtJX, vt−∇XdϕtJX− ∇JXdϕtX. 4.51

Then, for allz0for which∂zϕ0z0and∂2zϕ0z0are linearly independent

∇∂t|0dϕtJX∇∂t|0JψtdϕtX, ∇∂t|0Jψtut−∇∂t|0vt, ∇∂t|0Jψtvt∇∂t|0ut. 4.52

Notice that inLemma 4.12, we were givenϕanddefinedthe twistor lift as the unique lift compatible with ϕ. Now, we are given the twistor mapψ but nothing guarantees that projecting the map toϕtmakesϕcompatible; that is,Jψtmay not preservedϕtTM.

Proposition 4.18. Letψ1_{, ψ}2 _{:I ×M}2 _{→ ΣN}4 _{be twoJ}1_{-holomorphic to first-order maps such} thatψ1

0 ψ02 and the variational vector fields induced onN4 are the same; that is, writing ai :

∂t|0π ◦ ψi,i1,2, we havea1 a2. Then, at all pointsz0for which∂zϕ0z0and∂2zϕ0z0are linearly independent, writingwi∂t|0ψti,i1,2, we havew1w2.

Proof. Letϕi_π◦ψ

ii1,2denote the projection maps. From our hypothesis, it follows that

w₁H wH₂ . Hence, the only thing left is to prove that the vertical parts coincide. Now, from the proof ofLemma 4.14,∂t|0ψtiV ∇∂t|0Jψti so that our result follows if∇∂t|0Jψt1 ∇∂t|0Jψt2.

We prove this identity showing thatQY 0 for allY, where

QY ∇∂t|0Jψ1t

Y−∇∂t|0Jψ2t

Y. 4.53

We consider the four possible cases forY; namely, whenY is equal todϕ0X,dϕ0JX,u0orv0,

whereu0andv0 are as in the preceding lemmanotice that, sinceψ₀1 ψ₀2, thenu1₀ u2₀and

v1 0v20.

iWhenY dϕ0XY dϕ0JXuses similar arguments, we have

Qdϕ0X

∇∂t|0

J_ψ1 tdϕ

1 t

−J_ψ1 0

∇∂t|0dϕtX

− ∇∂t|0

Jψ2 tdϕ

2 t

Jψ2 0

∇∂t|0dϕ 2 tX

∇∂t|0dϕ 1

tJX− ∇∂t|0dϕ 2

tJX−Jψ0∇Xa1− ∇Xa2 ∇JXa1− ∇JXa2 0,

4.54

where we have usedLemma 4.17, as well as the fact thatJψ2

0 Jψ10and∇∂t|0dϕ

i

tX∇Xai.

iiTakingY u0Y v0uses similar arguments, we have

Qu0 ∇∂t|0

J_ψ1 tu

1 t

−J_ψ1 0

∇∂t|0u 1 t

− ∇∂t|0

J_ψ2 tu

2 t

J_ψ2 0

∇∂t|0u 2 t

−∇∂t|0v 1

t ∇∂t|0v 2 t −Jψ0

∇∂t|0u 1

t− ∇∂t|0u 2 t

.

But

∇∂t|0v 1

t −∇∂t|0∇Xdϕ 1

tJX− ∇∂t|0∇JXdϕ 1 tX

−RN∂t|0ϕ1t, dϕ1tX

dϕ1_tJX∇X∇∂t|0dϕ 1 tJX

RN_∂

t|0ϕ1t, dϕ1tJX

dϕ1

tX∇JX∇∂t|0dϕ 1 tX

−RN_a 1, dϕ0X

dϕ0JX∇X∇JXa1RN

a1, dϕ0JX

dϕ0X∇JX∇Xa1.

4.56

Asa1a2, we deduce∇∂t|0v 1

t ∇∂t|0v 2

t; analogously,∇∂t|0u 1

t ∇∂t|0u 2

tso thatQu0 0.

Hence, the twistor lifts constructed inTheorem 4.16areunique to first order, in the sense that the vector field w induced on ΣN4 _{or Σ}−_N4 _{by the map ψ, w ∂}

t|0ψt depends only on the initial projected map ϕ0 and on the Jacobi field valong ϕ0. Moreover, taking

N4 _{the 4-sphere or the complex projective plane, letting ϕ : M}2 _{→ N}4 _{be a harmonic}

map, and v ∈ ϕ−1_{TN a Jacobi field, isotropy to first order is immediately guaranteed.}

Hence, the previous construction allows alocalunified proof of the twistor correspondence between Jacobi fields and twistor vector fields that are tangent to variations onΣN4_which

are simultaneouslyJ1_andJ2_{-holomorphicinfinitesimal horizontal holomorphic deformationsin} 5. We can also conclude which different propertiesnamely, conformality, real isotropy or harmonicityare related with those of the twistor liftresp.,H,J1_orJ2_{-holomorphicity.}

5. Additional Proofs

Proof ofLemma 4.5.Sinceψis compatible to first order,JψtpreservesdϕtTMfor allt. Hence,

there areaX_t andb_tXsuch that

JψtdϕtXa

X

tdϕtJXbXtdϕtX. 5.1

TakevX

t bXtdϕtX. Since, att 0,Jψ0dϕ0X dϕ0JXwe deducev

X

0 0 andaX0 1. Now,

sincedϕtXandJψtdϕtXform an orthogonal basis fordϕtTM, we have

dϕtJX

dϕtJX, dϕtX dϕtX2

dϕtX

dϕtJX, JψtdϕtX

JψtdϕtX

2 JψtdϕtX

⇒JψtdϕtX, dϕtJX

2

dϕtJX2dϕtX2−

dϕtJX, dϕtX 2

.

5.2

Differentiating with respect totat the pointt0, the above identity yields

∂t|0

JψtdϕtX, dϕtJX

∂t|0

dϕtX, dϕtX