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First-order twistor lifts. (English) Zbl 1191.53031

Summary: 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 \(J^{1}\) and \(J^{2}\). This is done by obtaining first-order analogues of classical twistorial constructions.

MSC:

53C28 Twistor methods in differential geometry
53C43 Differential geometric aspects of harmonic maps
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References:

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