Ever since the launch of Sputnik, Westerners knew that Russians did science differently. Here is evidence of how they do geometry.

As a quick look at the detailed MathSciNet review MR2264644 (2007i:53001) (or simply the table of contents ) will tell the reader how much and what kind of math it actually contains, I will here choose to focus on the philosophy behind this extraordinary book.

Here are the basic foundational assumptions of this book, (I slightly paraphrase from the authors' own preface):

- Geometry is a bridge between pure mathematics and the natural sciences, especially physics, and it is important and useful to relate to the beginning student how these are intertwined.

- The study of global properties of geometric objects leads to far-reaching developments in topology.

- The geometric theory of Hamiltonian systems leads to the development of symplectic and Poisson geometry.

- Geometry of complex and algebraic manifolds unifies Riemannian geometry with complex analysis, algebra and number theory.

Thus the authors believe there is a need for an introductory text that brings the beginning student of geometry up to speed on quite a few different fronts. There are many beginning texts out there which focus on one of these many aspects, but Novikov and Taimanov are up for the bigger challenges: To write a comprehensive text that will take a beginner all the way to the frontlines, to provide the student a smooth passage through various fundamental concepts and perspectives, and to integrate seamlessly many physical applications throughout the text. This is not a text only in complex geometry, symplectic geometry, or Poisson geometry; it is a text which incorporates all of differential geometry.

A question begs an answer: Who should be the lucky beginner student in this differential geometry course of a lifetime? The language used by the authors is simple, fluent and very understandable, even after translation, and the mathematical prerequisites seem at a first glance to be minimal. However, the calculus background required includes the implicit function theorem, there is substantial use of basic linear algebra, and unless one is willing to skip the chapters on complex geometry, one does need to have some exposure to functions of a complex variable. To me, therefore, it makes perfect sense that this book is published in the *Graduate Studies* series of the AMS. Very motivated undergraduates, especially those who are starting graduate school in mathematics soon may of course look in, but it must be clear to anyone that the book is going to require some mathematical muscle.

To me the ideal audience for this book is a beginning mathematics graduate student who is hoping to learn geometry, but who also is interested in learning some (mathematical) physics. Knowing something about physics at the beginning is a plus; in my opinion, Arnold's *Mathematical Methods of Classical Mechanics* (Springer GTM) would provide a more than sufficient preparation for this book, both in substance and in tone. However, the lack of any previous exposure to physics can easily be compensated by the desire to learn about it. It is important to note here that the integration of physics into the main flow of the text is quite smooth, and this adds so much to the text that reading around this material, even though it would still be possible, would be a real waste and a pity.

Between the two of them, the authors have already written many textbooks in geometry before, some of which have been translated into English (see for instance [DFN1–DFN3] and [NF]), but this may be their ultimate masterpiece. Highly recommended for anyone interested in learning geometry, except possibly those who may have physics anxiety (which must exist if math anxiety does).

**[DFN1]** Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P.; *Modern geometry — methods and applications. Part I. The geometry of surfaces, transformation groups, and fields* ; Graduate Texts in Mathematics, 93. Springer-Verlag, New York, 1992.

**[DFN2]** Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P.; *Modern geometry — methods and applications. Part II. The geometry and topology of manifolds* ; Graduate Texts in Mathematics, 104. Springer-Verlag, New York, 1985.

**[DFN3]** Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P.; *Modern geometry — methods and applications. Part III. Introduction to homology theory* ; Graduate Texts in Mathematics, 124. Springer-Verlag, New York, 1990.

**[NF]** Novikov, S. P.; Fomenko, A. T.; *Basic elements of differential geometry and topology* ; Mathematics and its Applications (Soviet Series), 60. Kluwer Academic Publishers Group, Dordrecht, 1990.

Gizem Karaali is assistant professor of Mathematics at Pomona College. She believes she may have physics infatuation, a disorder which should be on the opposite side of the spectrum from physics anxiety but possibly could be just as harmful.