Examples of Lie foliations. (Exemples de feuilletages de Lie.) (French) Zbl 1143.53027
D. Tischler [Topology 9, 153–154 (1970; Zbl 0177.52103)] has shown that every \(G\)-foliation, where \(G\) is an abelian Lie group, can be deformed into a \(G\)–foliation with discrete holonomy. D. Lehmann [C. R. Acad. Sci., Paris, Sér. A 286, 251–254 (1978; Zbl 0379.57004)], showed this result is false if \(G\) is the Heisenberg group. In this paper, the authors generalize Lehmann’s counterexample showing that, for any simply-connected non-abelian nilpotent group \(G\), there exists a \(G\)-foliation which cannot be deformed into a \(G\)-foliation with trivial holonomy. Similarly, they prove that for any (almost) simple Lie group \(G\), there exists a \(G\)-foliation which has dense holonomy and is structurally stable. Finally, if \(G\) is the Heisenberg group they show that Lehmann’s example can be obtained by deforming an \({\mathbb R}^3\)-foliation.
Reviewer: James Hebda (St. Louis)
MSC:
| 53C12 | Foliations (differential geometric aspects) |
| 57R30 | Foliations in differential topology; geometric theory |
| 22E25 | Nilpotent and solvable Lie groups |
References:
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