Minimal foliations on Lie groups (Q5905516)

Let \(G\) be a connected Lie group, \(H\) a connected, not necessarily closed, Lie subgroup of \(G\). The authors discuss minimality of foliations \(F(G,H)\). In \textit{R. Takagi} and \textit{S. Yorozu} [Tohoku Math. J., II. Ser. 36, 541-554 (1984; Zbl 0559.53018)] it is proved that if a left invariant Riemannian metric \(g\) is \(\text{Ad}(H)\)-invariant then \(g\) is bundle-like for \(F(G,H)\) and \(F(G,H)\) is totally geodesic for \(g\). In this paper, the authors prove that there always exists a (bundle-like) Riemannian metric on \(G\) for which the leaves of \(F(G,H)\) are minimal submanifolds.