Geometric vector fields on Lie groups
On a Lie group \(G\) endowed with a left invariant metric, the author considers foliations consisting of left cosets of a connected Lie subgroup \(H\) such that the leaves are locally equidistant. In terms of the Lie algebras \(g\) and \(h\) of \(G\) and \(H\) respectively, the previous condition amounts to \(\langle[v,x],x\rangle=0\) for any \(v\in h\) and \(x\in h^\perp\). This kind of foliations are called ``left invariant''. They are called ``homogeneous'' when, locally, the leaves coincide with the orbits of some group of isometries acting freely on \(G\). If the foliation is one-dimensional, the vectors \(v\) in \(g\) giving rise to homogeneous flows are characterized in Proposition 2.1. Furthermore, in Theorem 2.4, it is shown that if \(G\) is unimodular, the directions producing homogeneous foliations have non-negative Ricci curvature. As a consequence, if \(v\) spans a one-dimensional ideal on a unimodular \(g\), the Ricci curvature in the direction of \(v\) is non-negative for any left invariant metric. Moreover, in Theorem 2.6, the author characterizes those directions on a Lie algebra \(g\) such that, for any left invariant metric, the Ricci curvature is non-negative. In Section 3, the author gives conditions on \(g\) and \(h\) in order to produce homogeneous foliations of higher dimensions with minimal leaves. Section 4 is devoted to the study of codimension one foliations in the Heisenberg group \(H_n\): it is shown that they are precisely the left invariants ones; they are generated by an ideal and have minimal leaves (Theorem 4.4).
scientific article; zbMATH DE number 1145938
scientific article