Note. A second edition published in 1971 is much more readily available.

1.1. Review by: Anon.
The Two-Year College Mathematics Journal 4 (2) (1973), 75-81.

One-variable text, treating numbers as approximations; hence power, Taylor series are shown to boost computational knowhow. Differential equations introduced early, recur via spiral approach.

2.1. Review by: C Teleman.
Mathematical Reviews, MR0131842 (24 #A1689).

The purpose of this memoir is the determination of compact manifolds M3 with affine locally Euclidean, complete, for which the homogeneous holonomy groups H belong to the connected component of the identity of the pseudo-orthogonal group and are abelian groups.

3.1. From the Introduction.

During the academic year 1960-61, The National Science Foundation sponsored a conference on 'Analysis in the Large' at Yale University. This conference brought together for an extended period of time mathematicians from various parts of the country with different backgrounds. ... Markus and Hahn started things off with a search for examples of minimal flows on manifolds. Auslander suggested that flows on nilmanifolds induced by one parameter groups might be minimal. The results of these investigations are contained in Chapter IV and served as a starting point for almost all of the further material in this book.

3.2. Review by: Robert Ellis.
Amer. Math. Monthly 71 (6) (1964), 702.

This is a collection of research papers intended primarily for the expert in the field. ... This book is a valuable contribution to the subject not only because of the many results obtained but also for the many examples against which one may test various conjectures as to what happens in higher dimensions and for more general flows.

3.3. Review by: W R Utz.
Mathematical Reviews, MR0167569 (29 #4841).

This book is concerned primarily with recent work of the authors in which they study dynamical systems based on manifolds, especially solvmanifolds and nilmanifolds. Some of the results have been announced earlier ... The results provide a wide variety of behaviour. An application of nilflows is given to obtain a generalization of the Kronecker approximation theorem. Permanent regional transitivity is established for geodesic flows on frames based on three-dimensional compact manifolds of constant negative curvature.

4.1. From the Preface.

The book begins with a leisurely introduction to the general concept of a differentiable manifold. The path that we have chosen to this goal leads through a careful re-examination of the differentiable structure of Euclidean space. Once the reader has understood Euclidean space as a differentiable manifold, we define the general object and study its elementary properties. We have accompanied our abstract discussion of differentiable manifolds with a liberal dose of some of those historical examples which originally motivated their definition. Once these foundations have been laid, we have selected topics (see the Table of Contents for exact details) which illustrate further the historical background for differentiable manifolds and the directions in which the concepts have been applied. In particular, we have tried to sample the various techniques that have been found useful in handling differentiable manifolds. In a few places where the technicalities become oppressively intricate or repetitious, the arguments have been left incomplete, and the reader is urged to consult the references at the end of the book for the missing parts. Above all, we hope that the reader will be encouraged by this introduction to pursue the references well beyond the material which is presented here.

4.2. Review by: J R Munkres.
Amer. Math. Monthly 71 (9) (1964), 1059.

This is a very valuable book; it has been much needed. The notion of differentiable manifold is relevant to a number of different mathematical disciplines besides differential geometry - algebraic geometry, Lie groups, and differential topology among them - but until now a student has been hard-pressed to find a treatment of the subject which was not aimed well above his head. This book will fill his needs; it should prove accessible to one who is familiar with linear algebra, point-set topology, and what is usually called advanced calculus. It is not a book on differential geometry in the usual sense of the term; geodesics and curvature do not appear, and tensors are not defined until the final chapter. Instead, it is an introduction to a number of important topics in modern mathematics in which the concept of differentiable manifolds plays a role. ... The book is well-written and I have only one minor criticism in this regard. Too often the important definition of a section is submerged in the middle of an exposition which includes motivation, a geometric construction, and/or some subsidiary remarks. This is satisfactory for the reader who is plodding carefully in the authors' footsteps, but not so comfortable for others. Even such a reader may have trouble finding the crucial definition if he needs to refer to it a few pages later on.

4.3. Review by: S Murakami.
Mathematical Reviews, MR0161254 (28 #4462).

This book is intended to introduce differentiable manifold theory to readers with some undergraduate background in mathematics and it is, to the reviewer's knowledge, the first book which treats the subject at the introductory level. ... The book is well organized and written with much consideration for the educational effect. The presentation is clear and unhurried. In each chapter basic notions are introduced after illustrating and discussing simple cases of the objects, and almost all sections end with a set of exercises which facilitate the understanding of the subject. It is very regrettable that the book contains nevertheless the following careless mistakes. ... the authors' way of treatment confines evidently the material to rather basic manifold theory, and no mention is made of advanced theories such as infinitesimal connection theory. After reading this book, the reader may quickly approach advanced theories by means of the many publications listed in the references.

5.1. From the Introduction.

In this monograph, we are going to treat some aspects of the unitary representation theory of solvable Lie groups. One of the fundamental questions one can raise about any locally compact group or C* algebra is that of determining when it is of type I and to try to determine the structure of the set of equivalence classes of its irreducible representations. It is this and closely related questions which will occupy us here. In the course of this investigation, we will have to make use of a rather elaborate theory of infinite dimensional representations of groups (and algebras) which has been developed, in large part, by G W Mackey.

6.1. From the Preface.

In this book we have tried to give a treatment of the differential geometry of surfaces that combines both the modern and classical approaches to the subject. With this goal in mind, we have organized the book as follows: Chapters I and II ... treat the algebraic and analytic prerequisites for the modern approach to manifold theory. {Chapter I is vector space theory, including inner product spaces, the orthogonal group, and tensor products. Chapter II is the theory of differentiable submanifolds of cartesian spaces.} Chapter III treats matrix Lie groups and has as its goal the equations of structure of a matrix Lie group. ... {Although interesting in themselves, these introductory chapters can hardly be called geometry. They form a rather pedantic barrier to those students whose goal is to learn the differential geometry of surfaces.} Chapters IV and V cover the local differential geometry of curves and surfaces. We have started with the modern approach in Chapter IV because we felt that it best motivates the basic definitions. ... In Chapter VI we have presented a fairly complete account of the surfaces of constant curvature and hyperbolic geometry. In Chapter VII we have contented ourselves with a brief introduction to the Gauss-Bonnet theorem.

6.2. Review by: L N Patterson.
Amer. Math. Monthly 76 (2) (1969), 213-214.

This book departs refreshingly from "classical" introductions to differential geometry by treating the subject consistently from the group theoretic point of view, essentially in the spirit of E Cartan and the method of "moving frames." The gist of this method is that if three orthogonal unit vectors ("frame") are attached to a point constrained to move along a surface so that two of the frame vectors are always tangent to the surface, then the direction changes of each of the frame vectors measure geometric properties of the surface (formally expressed by the "structure equations"). In the foreword the author describes his intention to "combine the modern and classical approach to the subject ..." One consequence of this compromise is that he dulls some of the tools he spends so much time sharpening. This happens, for instance, when he stays with the classical method of using the same symbols for the coordinates of a point and the coordinate functions. ... There is a nearly total lack of references, even where the author states theorems without proofs. This is a lost opportunity to bring home to the reader a taste of the vast span of differential geometry today: from the calculus of variations and topology to relativity. The book may also be criticized for its extreme lack of examples - only a handful or so. All problems and exercises are presumably incorporated into the text, but the impression remains that the author hurriedly published his lecture notes and forgot to read the galley proofs.

6.3. Review by: W B Houston Jr.
American Scientist 56 (3) (1968), 327A-328A.

A "complete" course in differential geometry would involve many areas of mathematics. Auslander chooses to use a lot of linear algebra, calculus, and group theory, a little point set topology, hardly anything else. This choice forces the omission of some topics and some proofs. While intrinsic geometry has not been neglected, the emphasis is extrinsic. But the (necessarily intrinsic) treatment of Bolyai's non-Euclidean plane geometry is the nicest I have seen. ... While a few misprints are probably inevitable, there are so many in this book as to make it unusable as a text - there are literally hundreds of misprints and other minor errors. One can only hope that the many good points in Auslander's book will be plagiarized by authors willing to proofread.

6.4. Review by: R L Bishop.
Mathematical Reviews, MR0211326 (35 #2208).

The material from Chapter IV onward is chosen very well; these are just the sort of things that ought to be included in an initial course on curves and surfaces. However, the work is seriously marred by numerous errors; there are over 150 lines in which some error occurs. Most of these are of a trivial typographical sort and do little harm, but one computational error has led the author to state a theorem which is obviously false from a geometrical point of view.

7.1. From the Preface.

This book has as its goal a treatment of the real numbers that is at the same time intuitively appealing and rigorous. While writing this book we have always tried to present the material for as broad an audience as possible. We feel that it can be read either before a course in the Calculus or after an intuitive Calculus course and before a rigorous Calculus course.

7.2. Review by: Francine Abeles.
Amer. Math. Monthly 81 (7) (1974), 796-797.

This book is one that ought to be listed under the heading, Caveat Studiosus. According to the author, it is suitable for pre-calculus students as well as those in teacher-training programs or engineering courses. For any of these audiences the results might be disastrous for four reasons, one of which is the source of the book's strength. Auslander presents one of the most beautiful constructive treatments of the real numbers that I have ever seen. The approach uses sequences of finite decimals to approximate real numbers (written as infinite decimals). This is the heart of the text and the subject of the middle chapter of this three chapter work. The problem is that the treatment is not elementary. ... This book was used with a class of sophomores most of whom had intentions of majoring in mathematics or computer science. I chose it because I thought the formal approach to real numbers would be a good introduction to the required abstract algebra courses of the upper division, while the discussions of how errors behave when finite decimals are used as approximations for real numbers are important for computer science courses. The better students emerged with a healthy scepticism for the printed word; the average student was confused and frustrated; the poor student simply turned off.

8.1. Review by: Leon Jay Gleser.
Journal of the American Statistical Association 69 (346) (1974), 582.

A major problem which many colleges and universities face is how to deal with the student with severe deficiencies in mathematical background. Do you force the students to retake high school (and grade school!) courses, and thus delay their progress toward a degree; or do you put the students in the normal program, expecting most of them to sink and a few, with special tutoring, to swim? The present textbook offers the possibility of a compromise between these extremes - a course which on one hand reviews addition of large columns of numbers, fractions, long division, positive and negative numbers, decimals, percentages, square roots, graphing, and linear equations; while on the other hand introducing the students to the methods of descriptive statistics, the rudiments of discrete probability, the normal and binomial distributions, and a little bit of hypothesis testing. Despite its multiple authorship, the book has a consistent and generally clear style. ... this textbook is a welcome addition to the literature, filling a pressing need for remedial mathematics courses that move the students ahead while making up their mathematical deficiencies. It might also be used as supplementary reading for introductory statistics courses, although it is perhaps too elementary for most students. Finally, if textbooks like the present one were used in the high schools, the problem of remedying mathematical deficiencies now faced by our higher education system would be far less severe, and as a bonus, we might start to produce a statistically more sophisticated citizenry.

9.1. From the Preface.

These notes are concerned with the inter-relationship between abelian harmonic analysis, theta functions and functional analysis on a certain nilmanifold. Some of the results in these notes are new and some are old. However, our approach, because it puts a certain nilmanifold and its function theory at centre stage, often leads to new proofs of standard results. For example, we view theta functions as the analogue on a nilmanifold of the spherical functions on the sphere, where the Heisenberg group plays the role of the orthogonal group. Thus the classical theta identities will follow from basic group theoretic results. Historically, there are many names that can be associated with the topics treated in these notes. Because of the informal nature of these notes we have not made any effort at giving complete biographical references for results and have given only references to the sources we ourselves have used. If we have overlooked someone's work or state a result without reference that someone knows to be his, we apologize in advance. However, we would be less than honest if we did not admit the great influence of the ideas of J Brezin, G W Mackey and A Weil on our work. Indeed, after so many years of talking with Brezin many of the ideas or germs of ideas in these notes may be his . In addition, we should also mention the work of Cartier which stands somewhere in the middle ground between the work of Weil and that presented in these notes.

9.2. Review by: Leonard F Richardson.
Mathematical Reviews, MR0414785 (54 #2877).

... the book under review contains much interesting material about (general) function theory on the Heisenberg manifold.

Note. These expository lectures were given at the Conference Board of the Mathematical Sciences regional conference held at Cleveland State University, 5-10 June 1977.

10.1. From the Publisher.

This monograph consists of three chapters covering the following topics: Foundations, (1) Bilinear forms and presentations of certain 2-step nilpotent Lie groups, (2) Discrete subgroups of the Heisenberg group, (3) The automorphism group of the Heisenberg group, (4) Fundamental unitary representations of the Heisenberg group, (5) The Fourier transform and the Weil-Brezin map, (6) Distinguished subspaces and left action: Jacobi theta functions and the finite Fourier transform, (1) Nil-theta functions and Jacobi-theta functions, (2) The algebra of the finite Fourier transform; Abelian varieties, nil-theta and theta functions, (1) A general construction and algebraic foundations, (2) Nil-theta functions associated with a positive definite H-morphism of an Abelian variety, (3) The relation between nil-theta and classical theta functions. In presenting the material, the author has attempted to lay a careful foundation and has stressed low-dimensional examples and special computations when proving general results by general techniques. This an expository piece of work, although some of the results are new.

10.2. Review by: Leonard F Richardson.
Mathematical Reviews, MR0466409 (57 #6289).

The theme of this book is the isolation of useful and interesting classes of nilpotent groups. The first part of the book is largely preparatory to the introduction and application of nil-theta functions. ... In the second half of the book the author introduces nil-theta functions.