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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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