Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 594843, 26 pages
Research Article

First-Order Twistor Lifts

1Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisbon, Portugal
2Universidade Lusófona, Núcleo de Investigação em Matemática, Campo Grande, 376, 1749-024 Lisbon, Portugal

Received 30 December 2009; Accepted 30 March 2010

Academic Editor: Yuming Xing

Copyright © 2010 Bruno Ascenso Simões. 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 structures J1 and J2. This is done by obtaining first-order analogues of classical twistorial constructions.