Harmonic morphisms between Weyl spaces and twistorial maps II

Loubeau, Eric; Pantilie, Radu

Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 433-453.

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 harmonique 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.

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.

Classification : 53C43, 53C28
Keywords: Harmonic morphism, Weyl space, twistorial map
Mot clés : morphisme harmonique, espace de Weyl, application twistorielle

[1] Atiyah, M. F.; Hitchin, N. J.; Singer, I. M. Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A, Volume 362 (1978), pp. 425-461 | DOI | MR | Zbl

[2] Baird, P.; Wood, J. C. Harmonic morphisms between Riemannian manifolds, London Math. Soc. Monogr. (N.S.), Oxford Univ. Press, Oxford, 2003 no. 29 | MR | Zbl

[3] Bryant, R. L. Lie groups and twistor spaces, Duke Math. J., Volume 52 (1985), pp. 223-261 | DOI | MR | Zbl

[4] Burns, D.; Burstall, F. E.; De Bartolomeis, P.; Rawnsley, J. Stability of harmonic maps of Kähler manifolds, J. Differential Geom., Volume 30 (1989), pp. 579-594 | MR | Zbl

[5] Burstall, F. E.; Rawnsley, J. H. Twistor theory for Riemannian symmetric spaces. With applications to harmonic maps of Riemann surfaces, Lecture Notes in Mathematics, 1424, Springer-Verlag, Berlin, 1990 | MR | Zbl

[6] Calderbank, D. M. J. Selfdual Einstein metrics and conformal submersions, Edinburgh University, 2000 (Preprint, http://people.bath.ac.uk/dmjc20/mpapers.html, math.DG/0001041)

[7] Eells, J.; Salamon, S. Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 12 (1985), pp. 589-640 | EuDML | Numdam | MR | Zbl

[8] Gauduchon, P.; Tod, K. P. Hyper-Hermitian metrics with symmetry, J. Geom. Phys., Volume 25 (1998), pp. 291-304 | DOI | MR | Zbl

[9] Hitchin, N. J. Complex manifolds and Einstein’s equations, Twistor geometry and nonlinear systems (Primorsko, 1980) (Lecture Notes in Math.), Volume 970, Springer, Berlin, 1982, pp. 73-99 | MR | Zbl

[10] LeBrun, C. R. Twistor CR manifolds and three-dimensional conformal geometry, Trans. Amer. Math. Soc., Volume 284 (1984), pp. 601-616 | DOI | MR | Zbl

[11] Loubeau, E.; Pantilie, R. Harmonic morphisms between Weyl spaces and twistorial maps, Comm. Anal. Geom., Volume 14 (2006), pp. 847-881 | MR | Zbl

[12] Ohnita, Y.; Udagawa, S. Stable harmonic maps from Riemann surfaces to compact Hermitian symmetric spaces, Tokyo J. Math., Volume 10 (1987), pp. 385-390 | DOI | MR | Zbl

[13] Pantilie, R. Harmonic morphisms between Weyl spaces, Modern Trends in Geometry and Topology (2006), pp. 321-332 (Proceedings of the Seventh International Workshop on Differential Geometry and Its Applications, Deva, Romania, 5-11 September, 2005) | MR | Zbl

[14] Pantilie, R. On a class of twistorial maps, Differential Geom. Appl., Volume 26 (2008), pp. 366-376 | DOI | MR | Zbl

[15] Pantilie, R.; Wood, J. C. A new construction of Einstein self-dual manifolds, Asian J. Math., Volume 6 (2002), pp. 337-348 | MR | Zbl

[16] Pantilie, R.; Wood, J. C. Twistorial harmonic morphisms with one-dimensional fibres on self-dual four-manifolds, Q. J. Math, Volume 57 (2006), pp. 105-132 | DOI | MR | Zbl

[17] Rawnsley, J. H. $f$-structures, $f$-twistor spaces and harmonic maps, Geometry seminar “Luigi Bianchi” II — 1984 (Lecture Notes in Math.), Volume 1164, Springer, Berlin, 1985, pp. 85-159 | MR | Zbl

[18] Rossi, H. LeBrun’s nonrealizability theorem in higher dimensions, Duke Math. J., Volume 52 (1985), pp. 457-474 | DOI | MR | Zbl

[19] Siu, Y. T. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2), Volume 112 (1980), pp. 73-111 | DOI | MR | Zbl

[20] Wood, J. C. Harmonic morphisms between Riemannian manifolds, Modern Trends in Geometry and Topology (2006), pp. 397-414 (Proceedings of the Seventh International Workshop on Differential Geometry and Its Applications, Deva, Romania, 5-11 September, 2005) | MR | Zbl

[21] Yano, K. On a structure defined by a tensor field $f$ of type $(1,1)$ satisfying $f^3+f=0$, Tensor (N.S.), Volume 14 (1963), pp. 99-109 | MR | Zbl