The volume of vector fields on Riemannian manifolds. Main results and open problems (Q6137473)

The book under review is an expository research monograph concerning the modern development of the extrinsic differential geometry of vector fields on Riemannian manifolds, c.f [\textit{S. Dragomir} and \textit{D. Perrone}, Harmonic vector fields. Variational principles and differential geometry. Amsterdam: Elsevier (2012; Zbl 1245.53002)]. The main attention is focused on minimality properties of unit vector fields arising in the context of specific geometrical variational problems. Evidently, the motivation comes from the seminal article by \textit{H. Gluck} and \textit{W. Ziller} [Comment. Math. Helv. 61, 177--192 (1986; Zbl 0605.53022)], where the volume functional for unit vector fields on the three-dimensional sphere \(\mathbb S^3\) was discussed and it was proved that the unit vector fields with globally minimal volume on \(\mathbb S^3\) are just the vector fields tangent to the fibers of the Hopf fibrations of \(\mathbb S^3\). The same problem for odd-dimensional spheres \(\mathbb S^m\) of dimension \(m\geq 5\) remains still open and inspired a large diversity of research papers on the subject. A bridge between global and local extrinsic geometrical properties of unit vector fields was built in a series of articles by O. Gil-Medrano and her co-authors. The approach is based on interpreting unit vector fields on a Riemannian manifold \(M\) as mappings / submanifolds in the unit tangent bundle \(T_1M\) equipped with the Sasaki metric. Then the minimality of unit vector fields on \(M\) in variational settings turns out to be related to extrinsic geometrical properties of associated submanifolds in \(T_1M\). This allows to apply powerful methods of the modern differential geometry of submanifolds for treating unit vector fields on Riemannian manifolds and results in a variety of fundamental theorems which are just exposed in the monograph. The main part of the book consists of four chapters. The chapter ``Minimal Sections of Tensor Bundles'' introduces a construction of the Sasaki metric on sections of a general vector bundle. Moreover, the first and second variation formulas for the volume functional within the space of unit vector fields or, more generally, vector fields of constant length \(r\) are derived in terms of the extrinsic geometry of associated submanifolds in the tangent bundle \(T_rM\). The chapter ``Minimal Vector Fields of Constant Length on the Odd-Dimensional Spheres'' presents a deep treatment of the Hopf vector fields on the odd-dimensional spheres \(\mathbb S^{2m+1}\) with particular emphasis on their stability as volume minimizing vector fields. The chapter ``Vector Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical Space Forms'' discusses the minimization of the volume for vector fields of fixed constant length on spaces of constant positive curvatures and leads to profound generalizations of Gluck-Ziller's results. Finally, the chapter ``Vector Fields of Constant Length on Punctured Spheres'' explores the volume minimization for vector fields on \(\mathbb S^n\) admitting singular points. Particular attention is paid to model examples of singular vector fields which are the parallel transport vector fields and radial vector fields. Thus, the monograph under review provides a quite complete survey of up-to-date results on the minimality of vector fields on odd-dimensional spheres and more general Riemannian manifolds. Besides, it can be recommended as a self-consistent textbook introducing to modern techniques for studying the extrinsic geometry of vector fields. Moreover, the monograph includes a lot of intriguing open questions and conjectures which may serve as a novel source of inspiration for further researches paving the way to solve the Gluck-Ziller problem asking to determine the infimum of the volume of unit vector fields on the odd-dimensional spheres of dimension greater than 3 and to know if this value is attained.