Harmonic vector fields. Variational principles and differential geometry (Q2884696)

The term `harmonic vector field' is currently adopted in the literature with at least two different meanings, the simpler being a vector field that is locally the gradient of some harmonic (scalar) function. The harmonic vector fields considered in this book arise instead in connection with the theory of harmonic maps between Riemannian manifolds. If indeed a vector field \(X\) on a Riemannian manifold \(M\) is viewed (as usual) as a map \(M\to TM\), and giving \(TM\) the Sasaki metric, one could wonder when it becomes harmonic. In the case when \(M\) is compact and oriented, the answer has been known for a long time: \(X\) is a harmonic map \(M\to TM\) if and only if \(X\) is a parallel vector field on the Riemannian manifold \(M\), i.e., it is parallel along every path see [\textit{O. Nouhaud}, C. R. Acad. Sci., Paris, Sér. A 284, 815--818 (1977; Zbl 0349.53015)]. Hence, under such strict conditions there would be little to say (at least in the compact oriented case). However, there are weaker harmonicity conditions that are well worth taking into consideration. For instance, the Sasaki metric could be replaced by some of the so-called \(g\)-natural metrics (\(g\) being the metric on \(M\)). To state the `appropriate' harmonicity notion, the authors assume a variational viewpoint and, like in the theory of harmonic maps, the Dirichlet energy functional is considered. Then, the search for the appropriate notion may be reduced to the search for the appropriate functional space from which critical points should be taken. It turns out to be the space \(\Gamma^\infty\left(S(M)\right)\) of unit vector fields. The resulting theory is supported by a considerable amount of work by many authors for (roughly) the past two decades, of which the book gives a careful account.NEWLINENEWLINEAccording to the book's preface, the text is `mostly confined to the study of differential geometric property of harmonic vector fields' and of their geometric background, mainly within contact Riemannian and pseudo-Hermitian geometry, and with a strong relationship to CR geometry and subelliptic systems appearing in the theory of Hörmander systems of vector fields. Variational aspects, such as the study of weak solutions of PDE systems associated with some functionals (related to the Dirichlet energy), are also treated, though necessarily to a lesser extent. Results in the semi-Riemannian (in particular, the Lorentzian) framework are reported in the final part of the book. The book also contains five appendixes which, though motivated by issues in the main text, look to be digressions of their own interest; they cover twisted Dolbeaut cohomology, (a corollary of) Stokes' theorem, complex Monge-Ampère equations, exceptional orbits of the highest dimension and Reilly's formula.NEWLINENEWLINEAlthough the authors sometimes attend to minute details, I believe that the expository style is more suitable for experts in related fields than for beginners (unless they benefit from the guidance of scholars acquainted with the subject). On the other hand, minute technicalities could sometimes distract the advanced reader, even though they have the merit of giving a very clear account of the authors' choice of basic conventions.NEWLINENEWLINEThe book is certainly a valuable reference source as well. The bibliography appears both extensive and carefully selected, but I found the index a bit less useful than it could be; the search for notions of one's interest may sometimes be uncomfortable. The style of formal statements is clear and helpful when browsing for specific results, but the possibility of some context-dependent standing assumptions, stated aside, still has to be taken into account.NEWLINENEWLINETo summarize my view on the book, I would describe it as mainly an ample and reasonably organised, yet not systematic, survey of its target topic.