The vector field problem for homogeneous spaces
In this very interesting and relevant survey, the author considers the vector field problem, mainly on the class of compact homogeneous spaces. The vector field problem consists in asking what is the maximal number \(r\) such that there exist vector fields \(v_1, \ldots, v_r\) on a smooth connected manifold \(M\), which are everywhere linearly independent. The number \(r\) is called the span of \(M\) and is denoted by span\((M)\). The author explains the problem for the spheres \(S^{n-1}\) to give some idea of how to develop the theory in more general spaces, like homogeneous spaces for a compact connected Lie group. In many sections, the author presents yet unpublished results. Also he considers the problem for homogeneous spaces for non-compact Lie groups. This survey besides being very interesting and well written, deserves to be read by all who are interested in this area. For the entire collection see [Zbl 1411.55001].
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