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Harmonic morphisms between Riemannian manifolds. (English) Zbl 1055.53049
London Mathematical Society Monographs. New Series 29. Oxford: Oxford University Press (ISBN 0-19-850362-8/hbk). xvi, 520 p. (2003).
Just like a journey in itself can be as much of a purpose as its final destination, picking up a book sometimes delivers far more than promised by its title. In this category I would, for example, put Besse’s “Einstein manifolds”, or casting a wider net, J. Saramago’s “Viagem a Portugal”, for a wealth of material reaching far beyond its official bailiwick.
The monograph under review deserves such illustrious company for the richness of discourse, its thoroughly thought-out arrangement of the content, its constant effort to illustrate by examples, topped by a “Notes and comments” section ending each chapter and giving an extra dimension to the exposition.
Given the awkward nature of harmonic morphisms, this certainly wasn’t an easy proposition. Conceived as a generalisation of holomorphic maps between Riemann surfaces via their property of preserving local holomorphic functions, harmonic morphisms were first introduced in potential theory as maps between Brelot spaces which pull back germs of an axiomatic harmonic sheaf onto germs of similar nature. While restricting this very general setting to continuous maps between Riemannian manifolds and local solutions of the Laplace-Beltrami operator required little or no effort, this sort of definition is wholly ungeometrical and totally sterile, apart, perhaps, from the remark that harmonic morphisms form a closed class.
In a striking case of mathematical coincidence, a potential theorist from Denmark and a geometer from Japan, not only introduced, at the same time and completely independently, the definition of harmonic morphisms between Riemannian manifolds but also proposed the same fundamental result. The Fuglede-Ishihara characterisation essentially identifies harmonic morphisms as possessing a dual nature, analytical, they must be harmonic maps, and geometrical, horizontal weak conformality (HWC), sometimes called, semi-conformality.
Here is perhaps the first of several pitfalls, the need to call on concepts, sometimes very well developed in their own right and the subject of lengthy volumes themselves, without losing the guiding light.
Trying to be as self-contained as possible while staying under the 500-page bar, has forced the authors to use a threefold strategy: first a showcase of a low-dimension situation, with all the computations explicitly carried out, where all the geometry is present but in a form that requires no heavy formal machinery. This is particularly true for the very first chapter of the book, which deals with harmonic morphisms from $${\mathbb R}^3$$ to $${\mathbb C}$$ and is really of undergraduate level, while challenging enough to be an introduction to research. To the trained eye, it sketches the underpinning philosophy and some of the results to come. In particular, it highlights the fundamentally geometric flavour of the methods used for harmonic morphisms, to solve what is, in all intends and purposes, an over-determined system of PDE’s.
This could also be said of the chapter called “Mini-twistor theory on 3-dimensional space forms” which, while already in the thick of the book, lays out the ground for more sophisticated tools and arguments.
This approach allows the authors to dedicate entire sections or chapters to subjects which are, at the same time, unavoidable and yet impossible to reduce to a few passing remarks and results. This way the flow of the exposition is not broken by side-issues, recalls or definitions, while allowing for an abundance of concepts and results gravitating around harmonic morphisms.
This is probably one of the strong suits of this work, a seemingly endless list of subjects, explored in more or less depth, such as: Riemannian geometry, including the Laplacian and conformal maps, orthogonal multiplication and Clifford systems, twistor theory, super-minimal surfaces, Hermitian structures on Euclidean spaces, Seifert fibre spaces and Thurston’s eight geometries, isoparametric maps, reduction techniques, semi-Riemannian manifolds and harmonic maps between them. To this list one could add a myriad of terms which perhaps hardly get more than a mention, but make this book a worthy reference.
This impression is reinforced by extensive index and bibliography, both benefiting from references from the relevant sections and a systematic closing section where are thrown in historical and accreditation notes, elaborations on further developments, pointers to the literature as well as some technicalities.
This is why this book has a lot more to offer than just a collection of results on harmonic morphisms, the little-known geometrical relation of the famous harmonic maps, and should endure a longer shelf-life than the average research book.
The architecture of the book is governed by the choice of having four parts. The first, after the aforementioned study of harmonic morphisms from $${\mathbb R}^3$$ to $${\mathbb C}$$ by elementary tools, a section on Riemannian geometry and a thirty-five-page crash course on harmonic maps, introduces harmonic morphisms and their fundamental properties, followed by a special study of polynomial harmonic morphisms, interesting on their own, e.g. a HWC polynomial is automatically harmonic, but also of general consequences, via the symbol map.
The second part, entitled “Twistor methods”, opens with the case of harmonic morphisms from three-dimensional space forms to surfaces. They have geodesic fibres which can be seen as elements of the space of oriented geodesics, the mini-twistor space. This enables a classification of harmonic morphisms, up to a weakly conformal map on the target, from $${\mathbb R}^3$$, as the orthogonal projection onto $${\mathbb R}^2$$, from $${\mathbb S}^3$$, as the Hopf map into $${\mathbb S}^2$$, and, from $${\mathbb H}^3$$, as either the orthogonal projection onto the disk or to the plane at infinity. This line of investigation is then pursued for maps between an Einstein manifold and a surface. In this set-up, harmonic morphisms are equivalent to holomorphic maps w.r.t. a certain Hermitian structure on the domain, the fibres being super-minimal. This allows constructions on $${\mathbb R}^4$$, $${\mathbb S}^4$$, $${\mathbb C}P^2$$ and $${\mathbb C}P^2 \# \overline{{\mathbb C}P^2}$$.
This interaction between harmonicity and holomorphicity is the subject of an entire chapter, particularly developed in the case of Euclidean domains, where examples of complex-valued harmonic morphisms, holomorphic w.r.t. no Kähler structure, are constructed. This part concludes with “Multivalued harmonic morphisms”, a construction inspired by complex analytic functions and implicitly defined. This approach is a bit involved, especially the examination of branch sets, but leads to several interesting classes of examples.
The third part considers the consequences, on curvatures and topologies, of the existence of a harmonic morphism between two Riemannian manifolds, and is perhaps the heart of the matter. At first sight one might suspect the first chapter to be expository, as for the first two parts, given the restriction of the domain to compact 3-manifolds. This is not the case, since, in dimension three, the authors, after concise but very informative descriptions of Seifert spaces and Thurston’s eight geometries, give a link between harmonic morphisms and Seifert fibred spaces, yielding a characterisation and repercussions, e.g. the non-existence of non-constant harmonic morphisms from Sol. The geometrical condition satisfied by harmonic morphisms, being a generalisation of Riemannian submersions, one can extend O’Neill’s formulas and deduce necessary curvature relations for the existence of horizontally weakly conformal maps. In fact, one can also use the rather general and quite powerful Walczak formula to find restrictions in terms of sectional curvature. Further curvature conditions are given, this time for harmonic morphisms, using either the O’Neill tensors or the Weitzenböck formula, and, in the case of one-dimensional fibres, they even imply a link between total scalar curvatures.
This last case, one-dimensional fibres, has proved a fruitful one. Indeed, Baird, Pantilie and Wood found such fibres to be the orbits of an $${\mathbb S}^1$$-action, implying strong topological obstructions. Even more interesting is that between Einstein manifolds of dimension four and three, harmonic morphisms can only be of three types: the Killing type (the fibres are tangent to Killing vector fields); warped product type (the map is the orthogonal projection of a warped product), type (T) (the vertical part of the gradient of the dilation is, in norm, a nowhere zero function of $$\lambda$$, away from critical points). This line of investigation actually started in 2000, when Bryant showed that from manifolds of constant sectional curvature, harmonic morphisms with one-dimensional fibres are either of Killing or warped product type.
The proofs of this section are not easy, but given in extenso, sometimes reworked by the authors. This third part ends with a chapter dedicated to the well-tested method of reduction. The idea for harmonic morphisms is that since the geometrical condition is a conformal one, it is possible to combine reduction techniques, often used for harmonic maps, and conformal changes of the metrics to “render” the map harmonic, once horizontal weak conformality is achieved. While the theoretical aspects are given in details, it is really the worked-out examples that justify this chapter.
Apart from an appendix where three technical results (the local existence of harmonic functions, regularity for a class of elliptic equations and a condition on the order of a smooth map), the fourth and last part is a showcase of how to adapt definitions in semi-Riemannian geometry and what remains or disappears. This is particularly true of harmonic morphisms between semi-Riemannian manifolds. While the definition of harmonicity carries over for non-definite positive metrics, the geometrical definition requires a slight modification to enable a recovering of the Fuglede-Ishihara characterisation verbatim.
Unfortunately, and in spite of its potential importance in mathematical physics, work in this direction almost stops here, except perhaps for Pambira’s treatment of the degenerate metric case [see also the article “Harmonic morphisms and shear-free ray congruences” by the authors]. Yet the possibility of having fibres of different causal types, of trapped submanifolds is surely of a great appeal to physicists. This concluding chapter is theirs for the taking, in the hope that they can raise harmonic morphisms to another level.
While work in harmonic morphisms continues, as witness recent new lines of investigation, e.g. Pantilie and Wood’s suggestion of working in Weyl geometry, generating hopes of another link with mathematical physics, or Svensson’s results on holomorphic foliations [cf. Gudmundsson’s up-to-date bibliography at:
http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/publications.html], Baird and Wood did not allow themselves, perhaps due to a lack of space, perhaps to excessive modesty, to conjecture new results or even draw up a list of open problems, which could have served as motivation and direction for (new) workers in the field.
Probably similar reasons limited two very promising directions to passing mentions: discrete harmonic morphisms, initiated by Urakawa, and probability, in the picture right from the start but always underused in Riemannian geometry, with the notable exception of Duheille’s work. Maintenance works are on J. C. Wood’s web page, where a corrections and typos list is kept up-to-date [http://www.amsta.leeds.ac.uk/Pure/staff/wood/BWBook/Corrections.html].

##### MSC:
 53C43 Differential geometric aspects of harmonic maps 58E20 Harmonic maps, etc. 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry