1.1. From the Preface.

According to its original definition, a symmetric space is a Riemannian manifold whose curvature tensor is invariant under all parallel translations. The theory of symmetric spaces was initiated by E Cartan in 1926 and was vigorously developed by him in the late 1920's. By their definition, symmetric spaces form a special topic in Riemannian geometry ; their theory, however, has merged with the theory of semisimple Lie groups. This circumstance is the source of very detailed and extensive information about these spaces. They can therefore often serve as examples for the testing of general conjectures. On the other hand, symmetric spaces are numerous enough and their special nature among Riemannian manifolds so clear that a properly formulated extrapolation to general Riemannian manifolds often leads to good questions and conjectures.

The definition above does not immediately suggest the special nature of symmetric spaces (especially if one recalls that all Riemannian manifolds and all Kähler manifolds possess tensor fields invariant under the parallelism). However, the theory leads to the remarkable fact that symmetric spaces are locally just the Riemannian manifolds of the form $\mathbb{R}^{n} \times G/K$ where $\mathbb{R}^{n}$ is a Euclidean $n$-space, $G$ is a semisimple Lie group which has an involutive automorphism whose fixed point set is the (essentially) compact group $K$, and $G/K$ is provided with a $G$-invariant Riemannian structure. É Cartan's classification of all real simple Lie algebras now led him quickly to an explicit classification of symmetric spaces in terms of the classical and exceptional simple Lie groups. On the other hand, the semisimple Lie group $G$ (or rather the local isomorphism class of $G$) above is completely arbitrary; in this way valuable geometric tools become available to the theory of semisimple Lie groups. In addition, the theory of symmetric spaces helps to unify and explain in a general way various phenomena in classical geometries. Thus the isomorphisms which occur among the classical groups of low dimensions are geometrically interpreted by means of isometries; the analogy between the spherical geometries and the hyperbolic geometries is a special case of a general duality for symmetric spaces.

On a symmetric space with its well-developed geometry, global function theory becomes particularly interesting. Integration theory, Fourier analysis, and partial differential operators arise here in a canonical fashion by the requirement of geometric invariance. Although these subjects and their relationship are very well developed in Euclidean space (Lebesgue integral, Fourier integral, differential operators with constant coefficients) the extension to general symmetric spaces leads immediately to interesting unsolved problems. The two types of non-Euclidean symmetric spaces, the compact type and the noncompact type, offer different sorts of function-theoretic problems. The symmetric spaces of the noncompact type present no topological difficulties (the spaces being homeomorphic to Euclidean spaces) and their function theory tics up with the theory of infinite-dimensional representations of arbitrary semisimple Lie groups, which has made great progress in recent years. For the symmetric spaces of the compact type, on the other hand, the classical theory of finite-dimensional representations of compact Lie groups provides a natural framework, but the geometry of the spaces enters now in a less trivial fashion into their function theory.

The objective of the present book is to provide a self-contained introduction to Cartan's theory, as well as to more recent developments in the theory of functions on symmetric spaces.

Chapter I deals with the differential-geometric prerequisites, and the basic geometric properties of symmetric spaces are developed in Chapter IV. From then on the subject is primarily Lie group theory, and in Chapter IX Cartan's classification of symmetric spaces is presented. Although this classification may be considered as the culmination of Cartan's theory, we have confined Chapter IX to proofs of general theorems involved in the classification and to a description of Cartan's list. The justification of this notable omission is first that the usefulness of the classification for experimentation is based on its existence rather than on the proof that it exhausts the class of symmetric spaces; secondly this omission enabled us to include Chapter X (on functions on symmetric spaces) where it is felt that more open questions present themselves. At some places we indicate connections with topics in classical analysis, such as Fourier analysis, theory of special functions (Bessel, Legendre), and integral theorems for invariant differential equations. However, no account is given of the role of symmetric spaces in the theory of automorphic functions and analytic number theory, nor have we found it possible to include more recent topological investigations of symmetric spaces.

Each chapter begins with a short summary and ends with an identification of sources as well as some comments on the historical development. The purpose of these historical notes is primarily to orient the reader in the vast literature and secondly they are an attempt to give credit where it is due, but here we must apologise in advance for incompleteness as well as possible inaccuracies.

This book grew out of lectures given at the University of Chicago 1958 and at Columbia University 1959-1960. At Columbia I had the privilege of many long and informative discussions with Professor Harish-Chandra; large parts of Chapters VIII and X are devoted to results of his. I am happy to express here my deep gratitude to him. I am also indebted to Professors A Korányi, K deLeeuw, E Luft and H Mirkil who read large portions of the manuscript and suggested many improvements. Finally I want to thank my wife who patiently helped with the preparation of the manuscript and did all the typing.

1.2. Review by: Richard L Bishop.
Pi Mu Epsilon Journal 3 (8) (1963), 419.

This book is intended for beginning graduate students and beyond. The first chapter of Helgason's book (of ten chapters) is concerned with the basic material in the geometry of manifolds. This includes the definition of a manifold, tensor fields on a manifold, vector fields and their integral curves, differentiable mappings, Riemannian metrics, and the exponential mapping. He has a discussion of affine connections and parallelism and some more special but important theorems on sectional curvature and totally geodesic submanifolds. The latter topic is particularly relevant to the remainder of his book.

Helgason's treatment of the basic material is extremely concise and restricted to the finite dimensional case. The emphasis is on the intrinsic formulation of concepts, but the expressions in terms of local coordinates are given and used whenever it will lead to a quickened exposition. Chapter two deals with the elementary theory of Lie groups in a manner which rivals the now classic volume I of Chevalley in both content and clarity. With a very good initial course in the theory of Lie groups and algebras is available here.

The remainder of the book deals with symmetric spaces, which are sufficiently general to include, e.g., all of the classical Euclidean and non Euclidean geometries, in all dimensions, the projective spaces, the Grassmann manifolds, and all compact Lie groups, but special enough so that a great deal can be said about their topology and analysis and so that they are completely classified. Their importance lies to a great extent in the extraction of counter-examples to conjectures, particularly in areas of synthesis between analysis and topology.

Helgason has included an extensive bibliography, a fair number of significant but difficult problems, and historical notes at the end of each chapter.

1.3. Review by: Carl B Allendoerfer.
The American Mathematical Monthly 71 (3) (1964), 336.

Symmetric spaces are Riemannian manifolds such that the covariant derivative of the curvature tensor is identically zero. They were defined by E Cartan in 1926, and their theory has been actively developed in the intervening years. The study of such spaces is valuable because of their connection with semi-simple Lie groups, because they can be used as testing grounds for conjectures about general Riemannian manifolds, and because they serve as a framework within which global function theory can be developed. This book provides a self-contained treatment of Cartan's theory and of the recent developments in the theory of functions on symmetric spaces. It thus provides access to a literature never before collected in a systematic fashion. Since an amazing amount of mathematics is crowded into its pages, the exposition is necessarily compact. This is an excellent book for the mathematician with an adequate background in differential geometry and Lie groups. Novices should be wary.

2.1. From the Publisher.

Sigurdur Helgason's Differential Geometry and Symmetric Spaces was quickly recognised as a remarkable and important book. For many years, it was the standard text both for Riemannian geometry and for the analysis and geometry of symmetric spaces. Several generations of mathematicians relied on it for its clarity and careful attention to detail.

Although much has happened in the field since the publication of this book, as demonstrated by Helgason's own three-volume expansion of the original work, this single volume is still an excellent overview of the subjects. For instance, even though there are now many competing texts, the chapters on differential geometry and Lie groups continue to be among the best treatments of the subjects available. There is also a well-developed treatment of Cartan's classification and structure theory of symmetric spaces. The last chapter, on functions on symmetric spaces, remains an excellent introduction to the study of spherical functions, the theory of invariant differential operators, and other topics in harmonic analysis. This text is rightly called a classic.

Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces.

Readership: Graduate students and research mathematicians interested in topological groups and Lie groups; mathematical physicists.

3.1. From the Preface.

The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "Groups and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.
...

Each chapter begins with a short summary and ends with references to source material. Given the enormity of the subject, I am aware that the result is at best an approximation as regards completeness and accuracy. Nevertheless, I hope that the notes will help the serious student gain a historical perspective, particularly as regards Cartan's magnificent papers on Lie groups and symmetric spaces, which are found in the two first volumes of his collected works. For example, he can witness Cartan's rather informal arguments in his climactic paper, written at the age of 58, leading him to a global classification of symmetric spaces ; an example of a different kind is Cartan's paper in Leipziger Berichte (1893) where he indicates models of the exceptional groups as contact transformations or as invariance groups of Pfaffian equations and which, to my knowledge, have never been verified in print. Being the first such models, they have distinct historical interest although simpler models are now known.

In addition to papers and books utilised in the text, the bibliography lists many items on topics that are at best only briefly discussed in the text, but are nevertheless closely related, for example, pseudo-Riemannian symmetric spaces, trisymmetric spaces, reflexion spaces, homogeneous domains, discrete isometry groups, cohomology and Betti numbers of locally symmetric spaces. This part of the bibliography is selective, and no completeness is intended ; in particular, papers on analysis and representation theory related to the topics of "Groups and Geometric Analysis" are not listed unless used in the present volume.

This book grew out of lectures given at the University of Chicago in 1958, at Columbia University in 1959-1960, and at various times at MIT since then. At Columbia I had the privilege of many long and informative discussions with Harish-Chandra, to whom I am deeply grateful. I am also indebted to A Koranyi, K de Leeuw, E Luft, H Federer, I Namioka, and M Flensted-Jensen, who read parts of the manuscript and suggested several improvements. I want also to thank H-C Wang for putting the material in Exercise A.9, Chapter VI at my disposal. Finally, I am most grateful to my friend and colleague Victor Kac who provided me with an account of his method for classifying automorphisms of finite order (§5, Chapter X) of which only a short sketch was available in print.

With the sequel to this book in mind I will be grateful to readers who take the trouble of bringing errors in the text to my attention.

3.2. Review by: V S Varadarajan.
American Scientist 68 (1) (1980), 94-95.

This is the first volume of an expanded two-volume treatment by Helgason of the themes covered in his well-known Differential Geometry and Symmetric Spaces (Academic Press, 1962).

The subject of Lie groups and their representations is at the heart of contemporary mathematics and physics. Especially important are the semisimple Lie groups, which are the groups of motions of the most important spaces that arise in geometry and analysis, namely the Riemannian symmetric spaces. Since, ultimately, one wants to do analysis, physics, and arithmetic on such spaces and others related to them naturally, it is necessary to understand their structure. Helgason gives a very thorough treatment of the structure of symmetric spaces, including their classification. This requires quite a substantial amount of the structure theory of semisimple Lie groups. Of course all of this can be done only when one knows some basic differential geometry and Lie theory, and the first two chapters are devoted to brief accounts of these subjects. The treatment of function theory on symmetric spaces in the 1962 book has been omitted in this volume, apparently because the next volume will be devoted almost entirely to this subject and its many ramifications.

Helgason maintains the high level of exposition he introduced in his 1962 book. This fact, together with the fundamental nature of the subject matter, will make this volume and its sequel the standard references in the subject for quite some time.

3.3. Review by: Robert Hermann.
SIAM Review 22 (4) (1980), 524-526.

When the Book Review Editor of this Review wrote me to enquire if I wanted to do this book, he also enclosed a note asking me if I even thought it was appropriate to review. I took this to mean that he was afraid that readers of this journal would not find Lie group theory relevant to their professional activities. Of course, I am hardly one to concede such a point - a good part of my own work has been concerned with applications of Lie group theory to physics and control theory - but the evident fact that some segments of the applied mathematical community might think this, if only in private, has inspired me to make several comments about the relations between "pure" and "applied" mathematics.

In the 1960's, when the pure mathematicians were prospering, there was a touch of "noblesse oblige" in their attitude toward applications, which reached its extreme in the COSRIMS report. In the 1970's there has been a tendency all through science to emphasise financially the "applied" over the "basic", and this has, slowly but inevitably, had an effect on the balance within mathematics. I, and many others whose education was in "pure" mathematics, welcomed this renewed emphasis on ties between mathematics and the outside world, but the pendulum is now swinging too far, and there is a danger that the health of the whole mathematical enterprise will be irreversibly affected.
...
Lie group theory itself is an excellent case-history for the study of the interaction between pure and applied mathematics. It was developed by Lie in the 1870's and 1880's, based on a close study of two seemingly different situations, which Lie's genius saw were interrelated: The curves in projective space that are acted on transitively by groups of projective transformations (Klein and Lie were students of Plücker, the founder of projective differential geometry, as well as an experimental physicist of first rank!), and the way that explicit solutions of certain differential equations are related to groups of symmetries. Lie correctly sensed that there was a vast new theory underlying these special situations-the excitement is very evident in his papers. (Lie was, according to both Klein and Poincaré, an intuitive person, presumably very much a product of the romantic nationalist movement of the times. He states many times that his goal is to carry over the ideas of his fellow Norwegian, Abel, from algebra to differential equations.) Now, there were certain subtleties in understanding how groups could act on differential equations. In order to elucidate them, Lie developed two superstructures, the theory of "infinitesimal transformations", which we call "Lie algebras", and "mapping elements", which C Ehresmann developed in the 1940's in terms of what he called the "theory of jets". Lie's ideas were picked up by Elie Cartan, who developed them brilliantly. Most of the basic Lie theory that we use today was completed by 1926. However, this material was almost completely unknown to the world, "pure" or "applied", until comparatively recently. It is still difficult to understand, especially for an applied person who has no natural feeling or sympathy for the version of geometry that underlies the work of both Lie and Cartan. For example, a recent monograph by two eminent applied mathematicians does not even display the structure which Lie gave the subject in 1884, not to mention the later work of Cartan and Ehresmann. I mention this not to berate the authors of this otherwise admirable book, but only to draw the moral that it is essential that applied mathematicians be prepared intellectually to keep up with the advances in all branches of mathematics which are related to their subject.

The book under review is a revision of Differential Geometry and Symmetric Spaces, first published in 1960, which finally made widely available to the mathematical-scientific world the work of Cartan. (Lie's work on the relation to differential equations played a very minor role - the equivalent book in this field is yet to be written.) It triggered a vast array of further research, and has been widely used by physicists in applications of group theory. What was so impressive about Helgason's 1960 book was its skill and force in developing the needed background in manifold theory and algebra, and pressing on to give crisp, but accessible expositions and proofs of many major results. It is still required reading for anyone seriously interested in Lie group theory, pure or applied. The first eight chapters of this new book have the same titles as the earlier work, with some additions. Chapters 9 and 10 are mostly new, containing a more extensive treatment of the classification theory utilising some unpublished work of V Kac. The old Chapter 9, devoted to the study of analysis on symmetric spaces, has disappeared; the author promises as a sequel which will treat this topic at book-length. Another valuable feature is a more extensive system of notes, guiding the reader through the history and further literature.

3.4. Review by: Ravi S Kulkarni.
Bulletin of the American Mathematical Society 2 (3) (1980), 468-476.

Helgason develops differential geometry and the theory of Lie groups with the aim of classification of real semisimple Lie groups and symmetric spaces.
...

...the reader should get some feeling that presenting this magnificent theme in sufficient detail is a gigantic task. Helgason has accomplished it - and in a very competent way. This book, published in 1978, is a thoroughly revised and updated version of the author's well-known book published in 1962 under the title Differential geometry and symmetric spaces. The reader may note that the words "Lie groups" have been added in the title of the present book. The number of pages has gone up from 486 to 628 - and this is only the first volume. The author promises a sequel which will deal with function theory on symmetric spaces. The chapter on function theory in the '62 book has been dropped and a new chapter on the structure of semisimple Lie groups has been added. The '62 book dealt with the classification of semisimple Lie groups and symmetric spaces rather sketchily. The new book contains more details along the ideas in Kac-Moody algebras, which appeared in 1968 and which have significantly clarified and simplified the combinatorial aspects of the theory. Helgason has made a conscientious effort to make the book accessible to a wider public. A rather unusual feature for a book of this type is some 50 pages devoted to the solutions of the exercises in the book. All these are very welcome additions. The '62 book has served as a standard reference book for the last 16 years. It may be safely predicted that the new book will continue to do so. Many of Lie's and Cartan's intuitive assertions were given a rigorous treatment in the first edition of this book. Despite the appearance of many texts and notes during the past two decades, Helgason's book with the new additions remains the only complete and exhaustive reference on symmetric spaces.
...
... the mathematical community owes a great debt to Helgason for making this beautiful subject accessible in a book form with competence and with infectious enthusiasm. To be sure, the Lie theory has made more demands on Helgason than on us, who would be benefitting by reading his book. The reader should not miss the introduction, the notes, and the descriptive passages in each chapter. Also, from the beginning the reader should keep in touch with Chapter 10, which discusses the examples. That is the meat of this subject. Then the reader would better savour the gravy of the theory, which has made that meat digestible.

4.1. From the Publisher.

The study of homogeneous spaces provides excellent insights into both differential geometry and Lie groups. In geometry, for instance, general theorems and properties will also hold for homogeneous spaces, and will usually be easier to understand and to prove in this setting. For Lie groups, a significant amount of analysis either begins with or reduces to analysis on homogeneous spaces, frequently on symmetric spaces. For many years and for many mathematicians, Sigurdur Helgason's classic Differential Geometry, Lie Groups, and Symmetric Spaces has been - and continues to be - the standard source for this material.

Helgason begins with a concise, self-contained introduction to differential geometry. Next is a careful treatment of the foundations of the theory of Lie groups, presented in a manner that since 1962 has served as a model to a number of subsequent authors. This sets the stage for the introduction and study of symmetric spaces, which form the central part of the book. The text concludes with the classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over $\mathbb{C}$ and Cartan's classification of simple Lie algebras over $\mathbb{R}$, following a method of Victor Kac.

The excellent exposition is supplemented by extensive collections of useful exercises at the end of each chapter. All of the problems have either solutions or substantial hints, found at the back of the book. For this edition, the author has made corrections and added helpful notes and useful references.

Sigurdur Helgason was awarded the Steele Prize for Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis.

Readership: Graduate students and research mathematicians interested in differential geometry, Lie groups, and symmetric spaces.

5.1. From the Publisher.

The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representations, X-ray technology, nuclear magnetic resonance scanning, and tomography. This second edition, significantly expanded and updated, presents new material taking into account some of the progress made in the field since 1980. Aimed at beginning graduate students, this monograph will be useful in the classroom or as a resource for self-study. Readers will find here an accessible introduction to Radon transform theory, an elegant topic in integral geometry.

5.2. From the Preface.

The title of this booklet refers to a topic in geometric analysis which has its origins in results of Funk [1916] and Radon [1917] determining, respectively, a symmetric function on the two-sphere $S^{2}$ from its great circle integrals and a function on the plane $\mathbb{R}^{2}$from its line integrals. Recent developments, in particular applications to partial differential equations, X-ray technology, and radio-astronomy, have widened interest in the subject.

These notes consist of a revision of lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959-1965 on the Radon transform and some of its generalisations. (The term "Radon transform" is adopted from John [1955]). The viewpoint for these generalisations is as follows.

The set of points on $S^{2}$ and the set of great circles on $S^{2}$ are both homogeneous spaces of the orthogonal group $O(3)$. Similarly, the set of points in $\mathbb{R}^{2}$ and the set of lines in $\mathbb{R}^{2}$ are both homogeneous spaces of the group $M (2)$ of rigid motions of $\mathbb{R}^{2}$. This motivates our general Radon transform definition from [1965A, B] which forms the framework of Chapter II: Given two homogeneous spaces $G/K$ and $G/H$ of the same group $G$ the Radon transform $f$ → $\hat f$ maps functions $f$ on the first space to functions $\hat f$ on the second space. For $\zeta \in G/H$, $\hat f$ $(\zeta)$ is defined as the (natural) integral of $f$ over the set of points $x \in G/K$ which are incident to $\zeta$ in the sense of Chern [1942]. The problem of inverting $f$ → $\hat f$ is worked out in a few cases.
...

The lecture notes indicated above have been updated a bit by the inclusion of a short account of some applications, by adding a few corollaries, and by giving indications in the bibliographical notes of some recent developments.

An effort has been made to keep the exposition rather elementary. The distribution theory and the theory of Riesz potentials, occasionally needed in Chapter I, is reviewed in some detail in §8.

Apart from the general homogeneous space framework in Chapter II, the treatment is restricted to Euclidean and isotropic spaces (spaces which are "the same in all directions"). For more general symmetric spaces the treatment is postponed (except for §4 in Chapter III) to another occasion since more machinery from the theory of semisinple Lie groups is required.

5.3. Review by: Lawrence Zalcman.
SIAM Review 25 (2) (1983), 275-278.

While very far from the definitive treatment the subject deserves, Professor Helgason's notes provide the most agreeable introduction to the Radon transform currently available.

Useful bibliographic notes follow each chapter. The list of references numbers 74 items (11 by the author), with dates of publication ranging from 1916 to 1980; more than a third of them appeared since 1966, when the lectures on which the book is based were originally delivered. Finally, as an appendix, Radon's original paper has been reprinted. The quality of the reproduction is mediocre at best, but it is good to have this hard-to-find classic once more easily available.

The discussion is at the level of an advanced graduate student. Familiarity with the Fourier transform is assumed in Chapter 1, and a certain acquaintance with the language of differential geometry and Lie groups (less than one might imagine) is required for reading the last three chapters.

5.4. Review by: Joseph A Wolf.
American Scientist 69 (5) (1981), 570.

There are many theoretical and applied situations in which one wants to compute a function (which measures something interesting) from various sorts of averages (which may be the raw data available). A fashionable medical example is the reconstruction for a CAT scan of a 2-dimensional density function, which represents a cross section of the person being scanned, from its averages along lines, which are the x-ray data. In fact, this sort of reconstruction is an old story in mathematics. It started with work of Radon published in 1917, found application in the theory of differential equations, was established in a geometric setting by Helgason in the 1960s, and has since found further application in the representation theory of semisimple Lie groups. The book under review is an excellent introduction to the analytic and geometric setting for that Radon transform theory. It is accessible to any careful reader with a solid standard mathematical background that includes some differential geometry.

6.1. From the Preface.

The first edition of this book has been out of print for some time and I have decided to follow the publisher's kind suggestion to prepare a new edition. Many examples of the explicit inversion formulas and range theorems have been added, and the group-theoretic viewpoint emphasised. For example, the integral geometric viewpoint of the Poisson integral for the disk leads to interesting analogies with the X-ray transform in Euclidean 3-space. To preserve the introductory flavour of the book the short and self-contained Chapter V on Schwartz' distributions has been added. Here §5 provides proofs of the needed results about the Riesz potentials while §§3-4 develop the tools from Fourier analysis following closely the account in Hörmander's books [1963] and [1983]. There is some overlap with my books [1984] and [1994b] which, however, rely heavily on Lie group theory. The present book is much more elementary.

I am indebted to Sine Jensen for a critical reading of parts of the manuscript and to Hilgert and Schlichtkrull for concrete contributions mentioned at specific places in the text. Finally I thank Jan Wetzel and Bonnie Friedman for their patient and skilful preparation of the manuscript.

7.1. From the Publisher.

This volume, the second of Helgason's impressive three books on Lie groups and the geometry and analysis of symmetric spaces, is an introduction to group-theoretic methods in analysis on spaces with a group action.

The first chapter deals with the three two-dimensional spaces of constant curvature, requiring only elementary methods and no Lie theory. It is remarkably accessible and would be suitable for a first-year graduate course. The remainder of the book covers more advanced topics, including the work of Harish-Chandra and others, but especially that of Helgason himself. Indeed, the exposition can be seen as an account of the author's tremendous contributions to the subject.

Chapter I deals with modern integral geometry and Radon transforms. The second chapter examines the interconnection between Lie groups and differential operators. Chapter IV develops the theory of spherical functions on semisimple Lie groups with a certain degree of completeness, including a study of Harish-Chandra's c-function. The treatment of analysis on compact symmetric spaces (Chapter V) includes some finite-dimensional representation theory for compact Lie groups and Fourier analysis on compact groups. Each chapter ends with exercises (with solutions given at the end of the book!) and historical notes.

This book, which is new to the AMS publishing program, is an excellent example of the author's well-known clear and careful writing style. It has become the standard text for the study of spherical functions and invariant differential operators on symmetric spaces.

Sigurdur Helgason was awarded the Steele Prize for Groups and Geometric Analysis and the companion volume, Differential Geometry, Lie Groups and Symmetric Spaces.

Readership: Graduate students and research mathematicians interested in analysis on homogeneous spaces, differential geometry, and topological groups, Lie groups.

7.2. From the Preface.

This volume is intended as an introduction to group-theoretic methods in analysis on spaces that possess certain amounts of mobility and symmetry. The role of group theory in elementary classical analysis is a rather subdued one; the motion group of $\mathbb{R}$) enters rather implicitly in standard vector analysis, and the conformal groups of the sphere and of the unit disk become important tools primarily after the Riemann mapping theorem has been established.

In contrast, our point of view here is to place a natural transformation group of a given space in the foreground. We use this group as a guide for the principal concepts (like that of an invariant differential operator) and as motivation for the leading problems in analysis on the space. In contrast, our point of view here is to place a natural transformation group of a given space in the foreground. We use this group as a guide for the principal concepts (like that of an invariant differential operator) and as motivation for the leading problems in analysis on the space. ...

The present volume is intended as a textbook and a reference work on the three topics in the subtitle. Its length is ca used by a rather leisurely style with many applications and digressions, sometimes in the form of exercises (with solutions).

Since this book is in part intended as a textbook, we now give so me description of its level. The first third of the book is introductory and has on occasion been used as a textbook for first-year graduate students without background in Lie group theory. The remainder of the book requires some standard functional analysis results. The Lie-theoretic tools needed can for example be found in my book "Differential Geometry. Lie Groups, and Symmetric Spaces", of which the present book can be considered an "analytic continuation." Thus, our aim has been to provide complete proofs of all the result s in the book. Although this process of unification and consolidation has at times led to some simplifications of proofs. we have in the exposition been more concerned with clarity than brevity.

Each chapter begins with a short summary and ends with historical notes giving references to source material: an effort has been made to give appropriate credit to authors of individual results. At the same time we have tried to make these notes reflect the fact that the logical order of the exposition often differs drastically from the order of the historical development.

Much of the material in this book has been the subject of lectures at the Massachusetts Institute of Technology in recent years; some of the content of my lecture notes [1980, 1981] has been incorporated in the Introduction and in Chapter I.

7.3. Review by: Adam Koranyi.
American Scientist 73 (5) (1985), 488.

The author's Differential Geometry and Symmetric Spaces appeared in 1962 and became a classic instantly. It was the first treatment of the basic geometric and algebraic theory of Riemannian symmetric spaces in book form, and it also had a long chapter about harmonic analysis on symmetric spaces containing results that were quite recent at the time. This branch of mathematics has been developing rapidly ever since. Its aim is no less than the generalisation of all that has to do with the Fourier integral and the Fourier series to functions defined on arbitrary semisimple Lie groups and on their quotients, the symmetric spaces. Some of these groups and their harmonic analysis play a fundamental role in modern physical theories. The applications within pure mathematics, in particular in number theory, are omnipresent. The author's contributions are of the first importance both in the development of the general theory and in improving our understanding of what has been achieved before.

The present book is essentially volume two of a greatly expanded new version of the 1962 book. (Volume one appeared under the title Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978, and contains the basic theory; volume three is being written, its tentative table of contents is given in the present volume.) The principal achievement of the book is a beautiful exposition of the theory of the spherical Fourier transform and all that leads up to it. There is also a very complete chapter on the compact case, and another one on integral geometry and the Radon transform. In an extremely useful introductory chapter the simplest groups are directly treated by elementary methods. This, as well as many other parts of the book, can be read independently of the rest. The book is excellent both as a text and as a reference work; it will clearly become another instant classic.

8.1. From the Publisher.

Group-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group. The introductory chapter is a self-contained account of the analysis on surfaces of constant curvature. Later chapters cover general cases of the Radon transform, spherical functions, invariant operators, compact symmetric spaces and other topics. This book, together with its companion volume, Geometric Analysis on Symmetric Spaces (AMS Mathematical Surveys and Monographs series, vol. 39, 1994), has become the standard text for this approach to geometric analysis. Sigurdur Helgason was awarded the Steele Prize for outstanding mathematical exposition for Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces.

8.2. From the Preface.

The manuscript for this book was completed in late 1982 and was published by Academic Press in 1984. The book has been unavailable for a long time and I am happy to have this new printing published by the American Mathematical Society.

The book is the forerunner to my book "Geometric Analysis on Symmetric Spaces" published in 1994 by the American Mathematical Society as Vol. 39 in the Series "Mathematical Surveys and Monographs".

In this new printing I have taken the opportunity to correct a few inaccuracies and also to add a number of footnotes which are collected towards the end of the book in a section entitled "Some Details". The place of the footnotes is indicated by a dagger. Some of these footnotes furnish alternative, more elementary proofs and some are clarifications of occasional tough spots in the exposition. In addition, an errata list appears after the Index.

Many of these corrections and footnotes are based on suggestions by readers and I would like to express my appreciation to A Korenyt, A Mattuck, F Rouvtere, H Schlichtkrull, G Shimura, H Stetkaer, and particularly P Kuchment who translated the book into Russian.

9.1. From the Preface.

Among Riemannian manifolds the symmetric spaces in the sense of E Cartan form an abundant supply of elegant examples whose structure is particularly enhanced by the rich theory of semisimple Lie groups. The simplest examples, the classical 2-sphere $S^{2}$ and the hyperbolic plane $H^{2}$, play familiar roles in many fields in mathematics.

On these spaces, global analysis, particularly integration theory and partial differential operators, arises in a canonical fashion by the requirement of geometric invariance. On $\mathbb{R}^{n}$ these two subjects are related by the Fourier transform. Also harmonic analysis on compact symmetric spaces is well developed through the Peter-Weyl theory for compact groups and Cartan's refinement thereof. For the noncompact symmetric spaces, however, we are presented with a multitude of new and natural problems.

The present monograph is devoted to geometric analysis on noncompact Riemannian symmetric spaces $X$.
...
The length of this book is a result of my wish to make the exposition easily accessible to readers with some modest background in semisimple Lie group theory. In particular, familiarity with representation theory is not needed. To facilitate self-study and to indicate further developments each chapter is concluded with a section "Exercises and Further Results". Solutions and references are given towards the end of the book. The harder problems are starred. Occasionally, results and proofs rely on material from my earlier books, "Differential Geometry, Lie groups, and Symmetric Spaces" and "Groups and Geometric Analysis".

Some of the material in this book has been the subject of courses at MIT over a number of years and feedback from participants has been most beneficial. I am particularly indebted to Men-chang Hu, who in his MIT thesis from 1973 determined the conical distributions for $X$ of rank one. His work is outlined in Chapter II, §6, No. 5-6, following his thesis and in greater detail than in his article. I am also deeply grateful to Adam Koranyi for his advice and generous help with the material in Chapter V,
Many people have read at least parts of the manuscript and have furnished me with helpful comments and corrections; of these I mention Fulton Gonzalez, Jeremy Orloff, An Yang, Werner Hoffman, Andreas Juhl, Frangois Rouviere, Sonke Seifert, and particularly Frank Richter. I thank them all. Finally, I thank Judy Romvos for her expert and conscientious TEX-setting of the manuscript.

A good deal of the material in this monograph has been treated in earlier papers of mine. While subsequent consolidation has usually led to a rewriting of the proofs, texts of theorems as well as occasional proofs have been preserved with minimal change.

10.1. From the Publisher.

This book gives the first systematic exposition of geometric analysis on Riemannian symmetric spaces and its relationship to the representation theory of Lie groups. The book starts with modern integral geometry for double fibrations and treats several examples in detail. After discussing the theory of Radon transforms and Fourier transforms on symmetric spaces, inversion formulas, and range theorems, Helgason examines applications to invariant differential equations on symmetric spaces, existence theorems, and explicit solution formulas, particularly potential theory and wave equations. The canonical multitemporal wave equation on a symmetric space is included. The book concludes with a chapter on eigenspace representations - that is, representations on solution spaces of invariant differential equations. Known for his high-quality expositions, Helgason received the 1988 Steele Prize for his earlier books Differential Geometry, Lie Groups and Symmetric Spaces and Groups and Geometric Analysis. Containing exercises (with solutions) and references to further results, this revised edition would be suitable for advanced graduate courses in modern integral geometry, analysis on Lie groups, and representation theory of Lie groups.

Readership: Graduate students and research mathematicians interested in analysis on symmetric spaces and the representation theory of Lie groups.

10.2. From the Preface.

This book has been unavailable for some time and I am happy to follow the publisher's suggestion for a new edition.

While a related forthcoming book, "Integral Geometry and Radon Transforms" deals with several examples of homogeneous spaces in duality with corresponding Radon transforms, the present work follows the direction of the first edition and concentrates on analysis on Riemannian symmetric spaces $X = G/K$. We develop further the theory of the Fourier transform and horocycle transform on $X$, also taking into account tools developed by Eguchi for the Schwartz space $S (X)$. These transforms provide the principal methods for analysis on $X$, existence and uniqueness theorems for invariant differential equations on $X$, explicit solution formulas, as well as geometric properties of the solutions, for example the harmonic functions and the wave equation on $X$. On the space $X$ there is a canonical hyperbolic system on $X$, introduced by Semenov-Tian-Shansky, which is multitemporal in the sense that the time variable has dimension equal to the rank of $X$. The solution has remarkable analogies to the classical wave equation on $\mathbb{R}^{n}$, summarised in a table in Chapter V, §5.

My intention has been to make the exposition easily accessible to readers with some modest background in Lie group theory which by now is rather widely known. To facilitate self-study and to indicate further developments each chapter concludes with a section "Exercises and Further Results". Solutions and references are collected at the end of the book. The harder problems are starred. Occasionally results and proofs rely on material from my previous books "Differential Geometry, Lie Groups and Symmetric Spaces" and "Groups and Geometric Analysis".

Once again I wish to express my gratitude to my friends and collaborators, Adam Koranyi, Gestur Olafsson, Frangois Rouviere and Henrik Schlichtkrull and especially to my long-term colleague David Vogan for significant help at specified spots in the text. Finally, I thank Brett Coonley and Jan Wetzel for their invaluable help in the production and the editor Dr Edward Dunne for his interest in the work and his patient and accommodating cooperation.

11.1. From the Publisher.

In this text, integral geometry deals with Radon's problem of representing a function on a manifold in terms of its integrals over certain submanifolds-hence the term the Radon transform. Examples and far-reaching generalisations lead to fundamental problems such as: (i) injectivity, (ii) inversion formulas, (iii) support questions, (iv) applications (e.g., to tomography, partial differential equations and group representations). For the case of the plane, the inversion theorem and the support theorem have had major applications in medicine through tomography and CAT scanning. While containing some recent research, the book is aimed at beginning graduate students for classroom use or self-study. A number of exercises point to further results with documentation.

11.2. From the Preface.

This book deals with a special subject in the wide field of Geometric Analysis. The subject has its origins in results by Funk [1913] and Radon [1917] determining, respectively, a symmetric function on the two-sphere $S^{2}$ from its great circle integrals and an integrable function on $\mathbb{R}^{2}$ from its straight line integrals. The first of these is related to a geometric theorem of Minkowski [1911].
...

This book includes and recasts some material from my earlier book, "The Radon Transform", Birkhäuser (1999). It has a large number of new examples of Radon transforms, has an extended treatment of the Radon transform on constant curvature spaces, and contains full proofs for the antipodal Radon transform on compact two-point homogeneous spaces. The X-ray transform on symmetric spaces is treated in detail with explicit inversion formulas.

In order to make the book self-contained we have added three chapters at the end of the book. Chapter VII treats Fourier transforms and distributions, relying heavily on the concise treatment in Hörmander's books. We call particular attention to his profound Theorem 4.9, which in spite of its importance does not seem to have generally entered distribution theory books. We have found this result essential in our study in 1999 of the Radon transform on a symmetric space. Chapter VIII contains a short treatment of basic Lie group theory assuming only minimal familiarity with the concept of a manifold. Chapter IX is a short exposition of the basics of the theory of Cartan's symmetric spaces. Most chapters end with some Exercises and Further Results with explicit references.

Although the Bibliography is fairly extensive no completeness is attempted. In view of the rapid development of the subject the Bibliographical Notes can not be up to date.

11.3. Review by: Fulton Gonzalez and Eric Todd Quinto.
Bulletin of the American Mathematical Society 50 (4) (2013), 663-674.

The book under review is an excellent introduction to the group theoretical and analytic aspects of the field by one of its pioneers. Before reviewing the book, we will provide an overview of the field.

Integral geometry draws together analysis, geometry, and numerical mathematics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The fundamental problem in integral geometry is to determine properties of a function $f$ in the plane or three-dimensional space or other manifolds from knowing the integrals of $f$ over lines, planes, hyperplanes, spheres, or other submanifolds.
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Integral geometry is an important subject in the large field of geometric analysis, and this very readable book serves as an essential introduction to the topic. Throughout the book, the group-theoretic point of view, which the book's author helped to introduce, is emphasised. The wide range of examples in the book presented serves to demonstrate the power and effectiveness of the use of group techniques.
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The author's two previous books on the subject, which appeared in 1980 and 1999, are already considered classics in the field. This book adds many more results and examples, including a fuller treatment of integral transforms on constant curvature spaces, and a full proof of the inversion formula for the antipodal transform on a rank one compact symmetric space. Further significant improvements over the two previous editions are the inclusion of helpful exercises at the end of each chapter, as well as the addition of the extensive appendix mentioned above.

Beginning graduate students and interested non-specialists will gain from the book because of its clear exposition and comprehensive nature. Practitioners of integral geometry will find it a valuable reference with complete and clear proofs as well as specialised items of interest, such as orbital integrals, generalised Riesz potentials, and the group-theoretical basis for inversion formulas using shifted dual transforms. Because of its group-theoretical emphasis, this book does not include topics for which the reliance on groups is less important or which require a more specialised background. Among such excluded topics are integral transforms on differential forms; the $\kappa$ operator and universal inversion formulas; the Penrose transform and the relation of integral geometry to twistor theory and cohomology; the relation of integral geometry to representation theory; Radon transforms and microlocal analysis; and computed tomography. Rather than aiming to be comprehensive, the book focuses on important topics in integral geometry which any beginner in the field ought to know, and it presents the material in a lucid and appealing fashion.

The author of this book is one of the pioneers of integral geometry, and his mathematics has deeply influenced the pure and applied parts of the field. This book is a well-written and beautiful introduction to integral geometry from the perspective of group actions. It has valuable thought provoking exercises. It demonstrates the richness of the subject and provides new examples as well as clear and complete proofs of the fundamental theorems in the field. The book answers questions posed at the start of this review for the most important classical Radon transforms. It should be read by anyone who would like to learn more about integral geometry.