Minimal unit vector fields on oscillator groups (Q6616247)

The minimality of the volume functional \(\mathrm{vol}(M,V^*g_\mathrm{S})\) on the set of unit vector fields \(V\) on a smooth Riemannian manifold \(M\) is the same as the Riemannian submanifold \(V(M)\subset T^1M\) of the unit tangent bundle with Sasaki metric \(g_\mathrm{S}\) being a minimal submanifold. This result and others by the author on the geometry of vector fields are first recalled in the present paper. The Riemannian geometry of submanifolds becomes essential for the study of the volume variational problem; here and again in the special case where \(M\) is an oscillator group.\N\NThe oscillator group \(G(\lambda)=G(\lambda_1,\ldots,\lambda_n)\simeq H(n,1)\rtimes\mathbb{R}\) admits left-invariant metrics depending on certain structural constants \(\lambda_1,\ldots,\lambda_n\). The most important parts of the geometry of \(G(\lambda)\) are recalled here from those in [\textit{N. Xu} and \textit{J. Tan}, Czech. Math. J. 69, No. 4, 907--924 (2019; Zbl 1513.53124)]. Theorem 3 of the paper gives the necessary and sufficient conditions for the left-invariant vector fields \(V\) to be minimal.\N\NKnowing previously from the cited reference all the left-invariant harmonic unit vector fields and in particular those which become harmonic maps \(V:G({\lambda})\longrightarrow T^1G({\lambda})\) and noticing that in certain geometries harmonic unit vector fields are minimal, but not always, the paper raises the question of deciding this for the oscillator group. The last main result proves that the harmonic vector fields are minimal when the \(\lambda_i\) are all equal.\N\NFor the entire collection see [Zbl 1532.53004].