Abstract
In this paper, we study on the following two problems of realization for Lie foliations. (1) Which pair of Lie algebras \((\mathfrak {g},\mathfrak {h})\) can be realized as a Lie \(\mathfrak {g}\)-foliation in a closed manifold with the structure Lie algebra \(\mathfrak {h}\)? (2) Which pair \((\mathfrak {g},m)\) can be realized as a Lie \(\mathfrak {g}\)-flow in a closed manifold with the structure Lie algebra \({\mathbb {R}}^m\)? We give a complete answer to (1) in the case where \(\mathfrak {g}\) is a nilpotent Lie algebra and give a complete answer to (2) in the case where \(\mathfrak {g}\) is a nilpotent Lie algebra which has a rational structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Caron, P., Carrière, Y.: Flots transversalement de Lie \({\mathbb{R}}^n\), flots de Lie minimaux. C. R. Acad. Sci. Paris 280(9), 477–478 (1980)
Chao, C.Y.: Uncountably many nonisomorphic nilpotent Lie algebras. Proc. Am. Math. Soc. 13, 903–906 (1962)
Corwin, L., Greenleaf, F.: Representations of Nilpotent Lie Groups and their Applications. Cambridge University Press, Cambridge (1990)
Fedida, E.: Sur les feuilletages de Lie. C. R. Acad. Sci. Paris 272, 999–1002 (1971)
Gallego, E., Reventós, A.: Lie Flows of Codimension \(3\). Trans. Am. Math. Soc. 326, 529–541 (1991)
Haefliger, A.: Pseudogroups of Local Isometries, Differential Geometry (Santiago de Compostela, 1984), Res. Notes in Math, vol. 131. Pitman, Boston (1985)
Herrera, B., Llabrés, M., Reventós, A.: Transverse structure of Lie foliations. J. Math. Soc. Japan 48, 769–795 (1996)
Llabrés, M.: Sobre les foliations de Lie. Thesis , U. A. B (1988)
Llabrés, M., Reventós, A.: Unimodular Lie foliations. Ann. Fac. Sci. Toulouse 95, 243–255 (1988)
Mal’cev, A.I.: On a class of homogeneous spaces. Trans. Am. Math. Soc. 9, 276–307 (1962)
Meigniez, G.: Feuilletages de Lie résolubles. Ann. Fac. Sci. Toulouse (6) 4, 801–817 (1995)
Molino, P.: Feuilletages riemanniens sur les variétés compactes; champs de killing transverses. C. R. Acad. Sci. Paris 289, 421–423 (1979)
Molino, P.: Géométrie globale des feuilletages riemanniens. Proc. Kon. Nederl. Akad, Ser. A1 85, 45–76 (1982)
Molino, P.: Riemannian Foliations, Progress in Math, vol. 73. Birkhäuser, Boston (1988)
Molino, P., Sergiescu, V.: Deux remarques sur les flots riemanniens. Manuscripta Math. 51, 145–161 (1985)
Raghunathan, M.S.: Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb, vol. 68. Springer, Berlin (1988)
Acknowledgments
The author would like to express his gratitude to Professor Takashi Tsuboi for helpful suggestions and encouragement.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kato, N. Lie foliations transversely modeled on nilpotent Lie algebras. Geom Dedicata 179, 21–37 (2015). https://doi.org/10.1007/s10711-015-0065-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-015-0065-9