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On a class of twistorial maps. (English) Zbl 1149.53028
Summary: We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, (1,1)-geodesic immersions from (1,2)-symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold \((M,g)\), of dimension \(m\), a family of twistor spaces \(\{Z_r (M)\}_{1\leqslant r< \frac {1}{2} m}\) such that \(Z_r(M)\) parametrizes naturally the set of pairs \((P,J)\), where \(P\) is a totally geodesic submanifold of \((M,g)\), of codimension \(2r\), and \(J\) is an orthogonal complex structure on the normal bundle of \(P\) which is parallel with respect to the normal connection.

MSC:
53C28 Twistor methods in differential geometry
53C43 Differential geometric aspects of harmonic maps
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[1] Baird, P.; Wood, J.C., Harmonic morphisms between Riemannian manifolds, London math. soc. monogr. (N.S.), vol. 29, (2003), Oxford Univ. Press Oxford · Zbl 1055.53049
[2] Besse, A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer grenzgebiete (3), vol. 10, (1987), Springer-Verlag Berlin, New York · Zbl 0613.53001
[3] Brînzanescu, V., Pseudo-harmonic morphisms; applications and examples, An. univ. timişoara ser. mat.-inform., 39, 111-121, (2001), Special Issue: Mathematics · Zbl 1053.53041
[4] Calderbank, D.M.J., The Faraday 2-form in einstein – weyl geometry, Math. scand., 89, 97-116, (2001) · Zbl 1130.53303
[5] Eastwood, M.G.; Tod, K.P., Local constraints on einstein – weyl geometries, J. reine angew. math., 491, 183-198, (1997) · Zbl 0876.53029
[6] Eells, J.; Salamon, S., Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. scuola norm. sup. Pisa cl. sci. (4), 12, 589-640, (1985) · Zbl 0627.58019
[7] Kobayashi, S.; Nomizu, K., Foundations of differential geometry, I, II, Wiley classics library, (1996), Wiley-Interscience Publ., Wiley New York, Reprint of the 1963, 1969 original · Zbl 0175.48504
[8] Koszul, J.-L.; Malgrange, B., Sur certaines structures fibrées complexes, Arch. math., 9, 102-109, (1958) · Zbl 0083.16705
[9] LeBrun, C.R., Spaces of complex null geodesics in complex-Riemannian geometry, Trans. amer. math. soc., 278, 209-231, (1983) · Zbl 0562.53018
[10] LeBrun, C.R., Twistor CR manifolds and three-dimensional conformal geometry, Trans. amer. math. soc., 284, 601-616, (1984) · Zbl 0513.53006
[11] Loubeau, E.; Mo, X., The geometry of pseudo harmonic morphisms, Beiträge algebra geom., 45, 87-102, (2004) · Zbl 1055.53050
[12] Loubeau, E.; Pantilie, R., Harmonic morphisms between Weyl spaces and twistorial maps, Comm. anal. geom., 14, 847-881, (2006) · Zbl 1127.58010
[13] Loubeau, E.; Pantilie, R., Harmonic morphisms between Weyl spaces and twistorial maps II, Preprint, I.M.A.R., Bucharest, 2006 · Zbl 1127.58010
[14] O’Brian, N.R.; Rawnsley, J.H., Twistor spaces, Ann. global anal. geom., 3, 29-58, (1985) · Zbl 0526.53057
[15] R. Pantilie, Harmonic morphisms between Weyl spaces, in: Modern Trends in Geometry and Topology, Proceedings of the Seventh International Conference on Differential Geometry and Its Applications, Deva, Romania, 5-11 September, 2005, pp. 321-332
[16] R. Pantilie, Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds, Math. Proc. Cambridge Philos. Soc., in press · Zbl 1148.53047
[17] Pantilie, R.; Wood, J.C., Twistorial harmonic morphisms with one-dimensional fibres on self-dual four-manifolds, Quart. J. math., 57, 105-132, (2006) · Zbl 1117.53047
[18] Rawnsley, J.H., f-structures, f-twistor spaces and harmonic maps, (), 85-159
[19] B.A. Simoes, M. Svensson, Twistor spaces, pluriharmonic maps and harmonic morphisms, Preprint, University of Leeds, 2006 · Zbl 1186.53062
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