Algebraic & Geometric Topology

Solvable Lie flows of codimension $3$

Naoki Kato

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In Appendix E of Riemannian foliations [Progress in Mathematics 73, Birkhäuser, Boston (1988)], É Ghys proved that any Lie g–flow is homogeneous if g is a nilpotent Lie algebra. In the case where g is solvable, we expect any Lie g–flow to be homogeneous. In this paper, we study this problem in the case where g is a 3–dimensional solvable Lie algebra.

Article information

Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2751-2778.

Received: 11 February 2015
Revised: 13 November 2015
Accepted: 11 April 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R30: Foliations; geometric theory
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 22E25: Nilpotent and solvable Lie groups

foliations Lie foliations homogeneous spaces solvable Lie algebras solvable Lie groups


Kato, Naoki. Solvable Lie flows of codimension $3$. Algebr. Geom. Topol. 16 (2016), no. 5, 2751--2778. doi:10.2140/agt.2016.16.2751.

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  • A Andrada, M,L Barberis, I,G Dotti, G,P Ovando, Product structures on four dimensional solvable Lie algebras, Homology Homotopy Appl. 7 (2005) 9–37
  • P Caron, Y Carrière, Flots transversalement de Lie ${\mathbb R}^{n}$, flots transversalement de Lie minimaux, C. R. Acad. Sci. Paris Sér. A-B 291 (1980) A477–A478
  • Y Carrière, Flots riemanniens, from “Transversal structure of foliations”, Astérisque 116, Société Mathématique de France, Paris (1984) 31–52
  • E Fedida, Sur les feuilletages de Lie, C. R. Acad. Sci. Paris Sér. A-B 272 (1971) A999–A1001
  • E Gallego, A Reventós, Lie flows of codimension $3$, Trans. Amer. Math. Soc. 326 (1991) 529–541
  • É Ghys, Flots d'Anosov sur les $3$–variétés fibrées en cercles, Ergodic Theory Dynam. Systems 4 (1984) 67–80
  • É Ghys, Riemannian foliations: examples and problems (1988) Appendix E to, Progress in Mathematics 73, Birkhäuser, Boston (1988)}
  • A Haefliger, Groupoï des d'holonomie et classifiants, from “Transversal structure of foliations”, Astérisque 116, Société Mathématique de France, Paris (1984) 70–97
  • A Hattori, Spectral sequence in the de Rham cohomology of fibre bundles, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960) 289–331
  • B Herrera, M Llabrés, A Reventós, Transverse structure of Lie foliations, J. Math. Soc. Japan 48 (1996) 769–795
  • B Herrera, A Reventós, The transverse structure of Lie flows of codimension $3$, J. Math. Kyoto Univ. 37 (1997) 455–476
  • M Llabrés, A Reventós, Unimodular Lie foliations, Ann. Fac. Sci. Toulouse Math. 9 (1988) 243–255
  • S Matsumoto, N Tsuchiya, The Lie affine foliations on $4$–manifolds, Invent. Math. 109 (1992) 1–16
  • H Tsunoda, Complete affine flows with nilpotent holonomy group, Hokkaido Math. J. 28 (1999) 515–533