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Solvable Lie flows of codimension 3. (English) Zbl 1356.57022
In this paper flows mean orientable one-dimensional foliations. The author summarizes the techniques and methods in Lie foliations, Lie algebras, examples, diffeomorphism between Lie flows. Let $$G$$ be a simply connected Lie group and $$\widetilde G$$ a simply connected Lie group with a uniform lattice $$\Delta$$. The author considers a central exact sequence of Lie groups $1@>>>\mathbb R@>>> \widetilde G@>D_0>> G@>>> 1.\tag{$$*$$}$ The aim of this exact sequence is to demonstrate some interesting results.
Theorem: “Let $${\mathfrak g}$$ be a solvable Lie algebra and $$\mathcal F$$ be a Lie $${\mathfrak g}$$-flow on a closed manifold $$M$$. Suppose that $$\mathcal F$$ has a closed orbit.
(i)
If $${\mathfrak g}$$ is of type $$(R)$$ and unimodular, then $$\mathcal F$$ is diffeomorphic to the flow in example $$(*)$$;
(ii)
If the dimension of $${\mathfrak g}$$ is three and $${\mathfrak g}$$ is isomorphic to $${\mathfrak g}^0_3$$, then $$\mathcal F$$ is diffeomorphic to the flow in example $$(*)$$. In particular, if $${\mathfrak g}$$ is a 3-dimensional solvable Lie algebra and $$\mathcal F$$ has a closed orbit, then $$\mathcal F$$ is diffeomorphic to the flow in example $$(*)$$”.
##### MSC:
 57R30 Foliations in differential topology; geometric theory 53C12 Foliations (differential geometric aspects) 22E25 Nilpotent and solvable Lie groups
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