No man should carve runes
unless he can read them well;
many a man go astray
around those dark letters.
These lines come from the 73d chapter of Egil’s Saga. It is the oldest translated fragment of the saga, dating back to the early thirteenth century. In their original Icelandic they read:
Skalat madr rúnir rista,
nema ráda vel kunni,
pat verdr mörgum manni,
es of myrkvan staf villisk,
and it is is this poetic fragment, verbatim, which Sigurdur Helgason, a native Icelander, presents us with on the page facing his preface to Integral Geometry and Radon Transforms. The poem’s content is indeed a most apposite warning to any mathematician setting out to carve runes, or prove theorems, on his own: how dreadfully easy it is to go astray if the particular magic attached to what he is working with is not properly handled. Or, on a more prosaic level, how dreadfully easy it is to go at a theme in one’s research, exploring the unknown, and go astray because of missteps, ignorance of the literature, insufficient preparation, and so on: the stuff our nightmares are made of.
In view of this wisdom, the book under review is evidently offered as a means whereby to prepare properly for work on a theme in geometric analysis going back to the work of Funk and Radon nearly a century ago, namely that of “determining, respectively, a symmetric function on the two-sphere… from its great circle integrals and an integrable function on [the plane] from its straight line integrals.” This theme was later taken up by Fritz John, who “found significant applications to differential equations.”
Helgason’s own perspective on the matter goes back to his 1966 MIT lectures (based on his papers from the earlier 1960s), aiming at addressing a generalization of the foregoing. Specifically, “[t]he set of points … and the set of great circles on [the 2-sphere] are both acted on transitively by … O(3). Similarly, the set of points in [the Cartesian plane] and the [projective plane] are both homogeneous spaces of of the group … of rigid motions of [the plane] … [which] motivates our general Radon transform definition … [forming] the framework of Chapter II” [of the book under review]. To wit, after a compact first chapter on the Radon transform, the second chapter weighs in: titled “A Duality in Integral Geometry,” it deals with such central topics as duality for homogeneous spaces, the Radon transform for the double fibration, and orbital integrals. Helgason adds ten case studies (“Examples for Radon Transforms for Homogeneous Spaces in Duality”) that display the wide range of application of this part of integral geometry: X-rays, the Poisson integral, Grassmann manifolds, and (yes!) theta series and cusp forms, for example.
After this, Helgason turns his attention to what is possibly best characterized as truly hard-core geometric analysis, with the Radon transform on two-point homogeneous spaces and the X-ray transform on symmetric spaces featured next. Maximal tori are discussed in the latter context (the fourth chapter of the book). Riemannian geometry is next, launching a chapter on orbital integrals and the wave operator on Lorentz spaces. Interestingly, here too the close connection between integral geometry and certain themes in physics is brought out by means of a discussion of Huygens’ Principle. However there is no trace of physics, as such, to be found in Helgason’s discussion: it is all pure analysis from beginning to end, as Huygens’ Principle is presented as a theorem about the delta function (or distribution) and the wave operator as a Laplace-Beltrami operator for a Lorentzian structure. (Well, all right, Hendrik Antoon Lorentz was a Dutch physicist and, yes the delta function goes back to P. A. M. Dirac, but still… ) Helgason goes on to address mean value operators and Fritz John’s identities.
The last three chapters of Integral Geometry and Radon Transforms are largely autonomous and serve “to make the book self-contained” (and so it is wise to refer to these appendices often, given that the material Helgason presents is sophisticated and dense). These chapters deal with, respectively, “Fourier transforms and distributions, relying heavily on the concise treatment in Hörmander’s books… a short treatment of basic Lie group theory assuming only minimal familiarity with the concept of a manifold… [and] a short exposition of the basics of the theory of Cartan’s symmetric spaces.” Given the masterful exposition Helgason evinces throughout his opus, these three chapters can possibly serve as introductory mini courses in their own right.
Finally, says Helgason toward the end of his preface, “[m]ost chapters end with some Exercises and Further Results with explicit references.” Accordingly, Integral Geometry and Radon Transforms is truly a pedagogical contribution by a master in this field, aiming at providing proper background — and then some — for entry into real work on the topics he brings into the spotlight. It is, as I indicated already, a dense (if beautifully written) text and should be covered slowly and carefully: every page is filled with serious mathematics, and Helgason provides a lot of commentary and references that ought to be pursued by those wishing to enter the field. It is a marvelous example of fine scholarship.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.