Harmonic and minimal invariant unit vector fields on homogeneous Riemannian manifolds (Q2744634)

Let \((M,g)\) be a Riemannian manifold and denote by \((T_1M,g_S)\) its unit tangent sphere bundle equipped with the Sasaki metric \(g_S\). Furthermore, let \(\chi^1(M)\) denote the space of smooth unit vector fields on \(M\) which is supposed to be nonempty. Every \(\xi\in{\mathfrak X}^1(M)\) determines a mapping between \((M,g)\) and \((T_1M,g_S)\) embedding \(M\) into its tangent unit sphere bundle \(T_1M\). When \(M\) is compact and orientable, this leads to the definition of two functionals on \(\chi^1 (M)\): the energy of \(\xi\) which is the energy of the corresponding map and the volume of \(\xi\) which is the volume of the immersion. \(\xi\) is called a harmonic vector field if it is critical for the energy functional and it is said to be a minimal vector field if it is critical for the volume functional.NEWLINENEWLINENEWLINEThe main purpose of this paper is to consider both classes of vector fields for \(G\)-invariant unit vector fields on Riemannian homogeneous spaces \((M=G/G_0,g)\). Locally and globally homogeneous spaces may be treated by means of homogeneous structures and correspond to infinitesimal models. These notions and their study also provide the framework and tools for the treatment by the authors. This leads to the construction of a lot of new examples of unit vector fields which are minimal or harmonic or which determine a harmonic map form \((M,g)\) into its unit tangent sphere bundle equipped with the Sasaki metric. For several cases they obtain the complete list of such vector fields, in particular for low dimensions.