Abstract
In this paper, we present an overview of the theory of twistors on foliated manifolds, which has been developed in several papers of ours. We construct the twistor space of the normal bundle of a foliation. We demonstrate that the classical constructions of the twistor theory lead to foliated objects and permit us to formulate and prove foliated versions of some well-known results on holomorphic map**s. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated map**s. Finally, we recall suspension construction, which is used to construct examples of transversely harmonic maps between foliated manifolds and discuss the twistor methods in this case.
Dedicated to Vladimir Rovenski on his 70th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atiyah, M.F.: Some examples of complex manifolds, Bonn. Math. Schr. 6, 28 S. (1958)
de Bartolomeis, P.: Twistor Constructions for Vector Bundles. In: Complex Analysis and Geometry, eds. V. Ancona and A. Silva, Springer US, pp. 103–114 (1993)
Burstall, F. E.: Twistor methods for harmonic maps. In: Hansen V.L. (eds.). Differential Geometry. Lecture Notes in Mathematics, vol. 1263. Springer, Berlin, Heidelberg (1987)
Boyer, Ch.P., Galicki, K., Mann, B.M.: The geometry and topology of 3-Sasakian manifolds. J. Reine Angew. Math. 455, 183–220 (1994)
Boyer, C.P., Galicki, K.: Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)
Duggal, K.L., Ianus, S., Pastore, A.M.: Maps interchanging \(f\)-structures and their harmonicity. Acta Appl. Math. 67, 91–115 (2001)
El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compositio Math. 73, No. 1, 57–106 (1990)
Fujiki, A.: On the de Rham cohomology group of a compact Kähler symplectic manifold. Advanced Studies in Pure Mathematics 10, 105–165 (1987)
Haefliger, A., Girbau, J., Sundararaman, D.: On deformations of transversely holomorphic foliations. J. Reine Angew. Math. 345, 122–147 (1983)
Konderak, J.J., Wolak, R.A.: Transversely harmonic maps between manifolds with Riemannian foliations. Q. J. Math. 54, No. 3, 335–354 (2003)
Konderak, J.J., Wolak, R.A.: Some remarks on transversely harmonic maps. Glasg. Math. J. 50, No. 1, 1–16 (2008)
Lichnerowicz, A.: Applications harmoniques et variétés kähleriennes. Seminario Mat. e. Fis. di Milano. 39, 186–195 (1969)
Mohseni, R., Wolak, R.A.: Twistor spaces on foliated manifolds, Int. J. Math. 32, No. 8, 2150056 (2021)
Mohseni, R., Wolak, R.A.: Cohomology of quaternionic foliations and orbifolds, preprint, ar**v:2202.02733
Mohseni, R., Wolak, R.A.: Suspension and Grassmannians, preprint
O’Brian, N.R., Rawnsley, J.H.: Twistor spaces. Ann. Glob. Anal. Geom. 3, 29–58 (1985)
Penrose, R., MacCallum, M.A.H.: Twistor theory: An approach to the quantisation of fields and space-time. Physics Reports. 6, No. 4, 241–315 (1973)
Salamon, S.: Harmonic and holomorphic maps. In: Geometry Seminar “Luigi Bianchi” II: Lectures given at the Scuola Normale Superiore, ed. E. Vesentini, Springer-Verlag Berlin Heidelberg, pp. 161–224 (1984)
Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. U.S.A. 42, No. 6, 359–363 (1956)
Thurston, W.: The Geometry and Topology of Three-Manifolds. Princeton University Press, (1997)
Wolak, R.A.: Foliated and associated geometric structures on foliated manifolds. Ann. Fac. Sci. Toulouse Math. (5). 10, No. 3, 337–360 (1989)
Wolak, R.A.: Orbifolds, geometric structures and foliations. Applications to harmonic maps. Rend. Semin. Mat. Univ. Politec. Torino. 73, 173–187 (2016)
Acknowledgements
The second author acknowledges that the research cooperation was funded by the program Excellence Initiative—Research University at the Jagiellonian University in Kraków.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Mohseni, R., Wolak, R. (2024). Twistor Bundles of Foliated Riemannian Manifolds. In: Rovenski, V., Walczak, P., Wolak, R. (eds) Differential Geometric Structures and Applications. IWDG 2023. Springer Proceedings in Mathematics & Statistics, vol 440. Springer, Cham. https://doi.org/10.1007/978-3-031-50586-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-50586-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-50585-0
Online ISBN: 978-3-031-50586-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)