First-Order Twistor Lifts
© Bruno Ascenso Simões. 2010
Received: 30 December 2009
Accepted: 30 March 2010
Published: 7 April 2010
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 structures and . This is done by obtaining first-order analogues of classical twistorial constructions.
A Jacobi field is called integrable if it is tangent to a deformation through harmonic maps. In [4, 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 , we can construct its (positive) twistor space . This manifold admits two natural almost complex structures and . Given a map from a Riemann surface , harmonicity is intimately related with the existence of a -holomorphic lift , whereas isotropy is related with the existence of a -holomorphic lift (see ). 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. In Section 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
These are isotropic subspaces, in the sense that for all in (or in ). Associating an Hermitian structure on with its -space gives a 1–1 correspondence between Hermitian structures and maximal isotropic subspaces. We say that a maximal isotropic subspace is positive if the corresponding orthogonal complex structure is positive and we denote the set of all such subspaces by .
When equipped with , is never a complex manifold; as for , it is integrable if and only if is conformally flat ( ) or anti-self-dual ( ) (for more details, see [6, 9, 10]; a discussion on this topic can also be found in  and references therein).
A smooth map is holomorphic if and only if it is both and -holomorphic for some, and so any, stable decomposition . Taking , the decomposition is clearly stable for both the almost complex structures and on .
We can easily introduce a metric on the twistor space : let and consider the tangent space at this point, . We know that we have the identifications and . To get a metric on , transport the metric from that on ; that is, , for all , , where denotes the metric on at . For the vertical space , we can consider the restriction of the metric on the space . Finally, we declare and to be orthogonal under the metric ; that is, , for all , . With this metric, the decomposition is orthogonal and -stable ( ), ( ) are almost Hermitian manifolds and the projection map is a Riemannian submersion.
2.2. Conformal and Isotropic Maps
If , then is said to be a regular point (of ) and the map is called conformal at . Moreover, a map which is conformal (resp., weakly conformal) at all points is said to be a conformal map (resp., a weakly conformal map).
Let be a smooth map from a Riemann surface . We shall say that is an umbilic point (of ) if is a -linearly dependent set. If is such that all points are umbilic, we shall say that is totally umbilic (see ).
3. Nonparametric Twistorial Constructions
We shall sketch the proof of this result. We start by noticing that given a smooth map obtained as the projection of , , without requiring further conditions à priori on , nothing guarantees that is holomorphic relatively to the induced almost Hermitian structure on ; if it is, we shall say that the structure is strictly compatible with (or that the map is a strictly compatible twistor lift of ). Such a structure can exist if and only if is isotropic: in other words, if and only if is (weakly) conformal. If preserves but does not necessarily render holomorphic, we shall say that (or the map ) is compatible with .
To proceed, we need the following result .
Notice that to guarantee the existence of the -holomorphic lift for , the important fact was that belongs to the -part of for any almost Hermitian structure strictly compatible with . This is guaranteed if is a totally umbilic map, but it is not strictly necessary. For instance, if is an isotropic map, the vectors , span an isotropic subspace. If this vectors are linearly independent, taking this space as the -space of , then is a -holomorphic lift of , although may be a map into ; on the other hand, if is totally umbilic, then we may take either as the unique strictly compatible map into or into and both these maps are -holomorphic.
4. First-Order Twistorial Constructions
4.1. Harmonicity and Isotropy to First Order
Let be a harmonic map, a vector field along , and a one-parameter variation of . We say that is tangent to if . The following result is a key ingredient in what follows :
We have seen in Theorem 3.1 that 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 -sphere, harmonicity implies (weak) conformality or even isotropy, the last case occurring if the target manifold is itself also a sphere or the complex projective space [12, 14].
As in the nonparametric case, harmonicity to first order implies conformality to first order for maps defined on the two-sphere and even isotropy when the codomain is itself a real or complex space form .
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 correspond to , and -holomorphicity of the twistor lift . As we shall see, these results do have a first-order version as follows. We start with a definition.
In contrast with the nonparametric case, it is not obvious that -holomorphicity to first order implies -holomorphicity to first order. As a matter of fact, from (4.6), it only follows that . However, we do have the following.
The importance of choosing the Levi-Civita connection on is illusory. In particular, we can define the concept of holomorphicity to first order (or , -holomorphicity to first order) for maps defined between almost complex manifolds, not necessarily equipped with any metric.
Now, since holomorphicity does not depend on the chosen connection, we can deduce that holomorphicity with respect to reduces to the same condition (4.10). Thus, being holomorphic to first order does not depend on the chosen connection. For (resp., ) holomorphicity to first order, we use similar arguments, replacing for a horizontal (resp., vertical) frame.
We start with a technical lemma, whose proof the reader can find in Section 5.
Let be -holomorphic to first order map. Then, the projected map is conformal to first order. Conversely, let be a conformal to first order map. Then there is a (local) -holomorphic to first order map which is compatible to first order with .
concluding the first part of the proof.
From the preceding lemma, we can also deduce the following.
Before proving Theorem 4.11, we give the following lemma, which we prove in Section 5
We are now ready to prove Theorem 4.11.
Proof of Theorem 4.11.
where we have used , , , and . Hence, and satisfy equation (4.23), concluding our proof.
We prove the following.
Proof of Theorem 4.13.
holds. We shall establish this equation by showing that both sides agree when applied to any vector . For that, we consider, in turn, the three cases , , and . The first two have similar arguments so that we prove only the first.
Now, the first condition follows from (4.43) since is Koszul-Malgrange holomorphic for each . As for the second, letting denote its left-hand side, we shall prove that for all . We do this by establishing the three cases , and (since the first two cases have similar arguments, we prove only the first).
4.6. The 4-Dimensional Case
Let be harmonic and isotropic to first-order map and with and being linearly independent. Then, (locally) there is either a map or a map which is simultaneously and -holomorphic to first order and with . Conversely, if (or ) is and -holomorphic to first order, the projected map is harmonic and isotropic to first order.
The converse is obvious from Proposition 4.10 and Theorem 4.13. As for the first part, in Theorem 4.11 we saw that we can lift the map to a map -holomorphic to first order. Moreover, this lift could be defined as the unique positive or negative almost complex structure compatible with . On the other hand, in Theorem 4.15 we have seen that there is a map -holomorphic to first order with and for which is compatible. From the comment after Theorem 4.15, there is also a twistor lift of into . Therefore, from the dimension of , we conclude that the lifts constructed in both cited results are the same and, therefore, simultaneously and -holomorphic to first order.
We would now like to guarantee the uniqueness to first order of our twistor lift. Before stating such a result, we start with a lemma, proved in Section 5.
Notice that in Lemma 4.12, we were given and defined the twistor lift as the unique lift compatible with . Now, we are given the twistor map but nothing guarantees that projecting the map to makes compatible; that is, may not preserve .
Let be two -holomorphic to first-order maps such that and the variational vector fields induced on are the same; that is, writing , , we have . Then, at all points for which and are linearly independent, writing , , we have .
Hence, the twistor lifts constructed in Theorem 4.16 are unique to first order, in the sense that the vector field induced on (or ) by the map , depends only on the initial projected map and on the Jacobi field along . Moreover, taking the -sphere or the complex projective plane, letting be a harmonic map, and a Jacobi field, isotropy to first order is immediately guaranteed. Hence, the previous construction allows a (local) unified proof of the twistor correspondence between Jacobi fields and twistor vector fields that are tangent to variations on which are simultaneously and -holomorphic (infinitesimal horizontal holomorphic deformations in ). We can also conclude which different properties (namely, conformality, real isotropy or harmonicity) are related with those of the twistor lift (resp., , or -holomorphicity).
5. Additional Proofs
Proof of Lemma 4.5.
Proof of Lemma 4.8.
finishing the first part of our proof.
As is compatible with , Lemma 4.5 guarantees that is -holomorphic to first order. On the other hand, since and are linearly independent, we deduce that , and form a basis for . Hence, (4.33) will be satisfied if and only on evaluating the inner product of and with which one of these four vectors one obtains the same result. We shall only prove for the first and fourth vectors, the other two cases being similar.
Proof of Lemma 4.17.
The argument to establish the second and third identities in (4.52), will now be similar to the one in Lemma 4.12(i).
The author is grateful to Professor J. C. Wood for helpful comments and stimulating discussions during the preparation of this work. The author would also like to thank the referee for valuable comments.
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