Solvable Lie flows of codimension $3$

Abstract

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