Solvable Lie flows of codimension 3 (Q341765)

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 NEWLINE\[NEWLINE1@>>>\mathbb R@>>> \widetilde G@>D_0>> G@>>> 1.\tag{\(*\)}NEWLINE\]NEWLINE The aim of this exact sequence is to demonstrate some interesting results.NEWLINENEWLINE 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.{\parindent=6mm \begin{itemize}\item[(i)] If \({\mathfrak g}\) is of type \((R)\) and unimodular, then \(\mathcal F\) is diffeomorphic to the flow in example \((*)\); \item[(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 \((*)\)''.\end{itemize}}