Geometry has been and remains the Cinderella of the curriculum at the Moscow University's Mathematics and Mechanics faculty. Never once during the last fifty years has the curriculum contained a course on the foundations of geometry or algebraic curves or transformation groups or even protective geometry (if we do not count the scraps in the first-semester courses in analytic geometry which may only be given under special circumstances, and nobody cares when the lecturer curtails them up or even drops them altogether). A student might well graduate from the faculty—with honours!—with no idea about Lobachevskian geometry, Caley-Klein ideas in the foundations of geometry, or the properties of algebraic curves or Lie groups.

Some twelve years ago the overflow into the calculus course of geometric material due to the ever increasing implementation of geometric methods led to the creation in the second year of a new course with the ad hoc name "Smooth manifolds and differential geometry". This course was delivered at a rate of a lecture a week and it was hoped that the course wotld free lecturers from presenting the extraneous geometric material. The course, however, was not well thought-out, and the parallel courses in calculus and differential equations were not coordinated with it. As a result the lecturers on calculus did not derive any advantage, and ridiculous as it may seem, integration of the differential forms on manifolds and Stokes' formula were discussed twice in as much detail but from slightly different points of view, in the two concurrent courses.Delivering the geometry course in the third semester did not allow the generalizing and unifying role played by geometric concepts in modern mathematics to be brought out since to do so it is necessary for the main analytic courses to have been covered.

These and various more particular considerations led to the transfer of geometry to the third year (the fifth and sixth semesters). It has immediately become clear that this also had disadvantages. A necessary constituent part of any course in geometry is the theory of curves and surfaces in three-dimensional Euclidean space, which is important both in its own right and as a source of examples and analogues for Riemannian geometry and the geometry of affine connections. By the third year, this material is too elementary (by this time students have already acquired a knack and a taste for more complicated constructions and concepts) and for it to play its propaedeutic role one cannot pass too quickly onto Riemannian geometry.

It is clear that this theory must be presented no later than the third semester (or perhaps earlier, as suggested by me in the first Russian edition of Semester II of these Lectures, even in the second semester). Moreover, a third-year geometry course does not help lecturers presenting second-year analysis (which I am sure will soon lead to the abolition of the course in geometry in the third semester and may be, alas, to its ousting from the schedule of the curriculum).

The radical solution is, of course, to overhaul the traditional system of mathematical courses. However, since there is an acute struggle between the departments for hours and courses such a review, which will have to be carried out sooner or later, is at present not possible, and a temporary solution would be the return of the geometry course in the third or fourth semester with the presentation of integration topics clearly distributed between the courses in calculus and geometry, each passing on the baton to the other as it were.

The following distribution of topics is suggested. After the integral of functions over domains in HI" has been discussed in calculus, the geometry lecturers cover the integration of densities and forms on manifolds. Simultaneously the calculus lecturer illustrates the general theory by particular cases of line and surface integrals of the first (density) and the second (form) kind. During this time, the generalized Stokes theorem is discussed which in the calculus course is immediately rendered concrete in the form of the Green, Gauss-Ostrogradskii and Stokes formulas. This duet, in which the general melody sometimes drifts, sometimes merges, ends in the apotheosis of vector analysis with elements of potential theory where the course in calculus changes freely into the theory of multidimensional improper integrals and the geometry course into cohomology theory. All this, of course, requires close coordination between the lecturers, which is not easy to achieve.

This book has, like its predecessors*, grown out of lectures given at the mathematics and mechanics faculty of Moscow University in different years. It is not, however, a recording of any particular course, but is instead a realization of the proposed geometry syllabus for the third semester. It can, however, certainly be used as the text for the fifth semester course.'

The textbook'is intended as a normal course presented at two lectures a week. The number of lectures (29) arises because although the winter semester formally contains 18 weeks, in practice it is impossible to deliver more than 11 to 15 weeks of lectures. The course can be used, however, even if the curriculum assigns only one or one and a half lectures a week (11 to 15 and respectively 16 to 22 lectures). To be able to estimate the time required for a syllabus, I have tried to make each lecture in the book correspond to a two-hour lecture. Repeating material from other courses and considering examples, in written form, requires much more time. This accounts for difference in the volume of the lectures and the unexpectedly large size of Lectures 3, 11, and 20.

In recent years a rather strange view of smooth manifolds has become widespread, a view surprisingly shared by some respected and competent mathematicians. Since a smooth manifold can be regarded as the result of the natural attempt to generalize axiomatically the simple idea of a manifold as a subset of a Euclidean space denned by a system of functionally independent equations, it is argued that the generalization does not actually lead to new objects because of the Whitney embedding theorem and so manifolds should be denned as subsets of that kind and that the general concept of a manifold is just an example of an axiomatic construction which inevitably arises in following a concept to its conclusion but one which it is then better to forget. I cannot share this opinion because in pr/actice—for example, in mechanics— manifolds tend to appear in an abstract form, unembedded in a Euclidean space, and their forced embedding (with great arbitrariness!) introduces an additional structure that is sometimes useful but often having no relevance to the crux of the matter. The adherents of the former opinion appeal to Poincare, who has shared it. In fact Poincare clearly understood the necessity of having a general concept/ of manifold and dwelt on pasting together the charts of an atlas. Referring to the extremes of axiomatization is also wrong, since in reality manifolds were not introduced as the result of "natural attempt to generalize the simple notion of a manifold given by equations" but as an answer to the need to clearly explicate the notion arising in mathematical investigation. A consistent execution of the same principles would throw mathematics a hundred years back, since from this point of View, for example, all linear algebra in its present form has no right to exist, being based as it is, on the concept of a vector space which could be said to "have arisen as a result of a natural attempt to generalize the simple idea of the space 5ln"i (which is as false as it is for manifolds), whereas the isomorphism theorem shows that "the generalization does not actually lead to new objects" (which, though true, does not deprive the concept of a vector space of its value). In this book, therefore, manifolds are defined in the usual way, on the basis of an atlas, while subsets of Euclidean spaces only appear as examples.

The problems in this book are mainly quite trivial and intended exclusively for a reader to test himself. Some more difficult problems are given in small print. Auxiliary material on algebra or calculus is also given in small print.

The first five lectures are only indirectly related to the theory of smooth manifolds, mainly being devoted to elementary differential geometry. The theory of curves (Frenet formulas) is followed by the first and second quadratic forms of a surface, the Weingarten formulas are derived, and Gauss's theorem on the invariance of total curvature is proved. Everything not directly related to the Gauss theorem has been omitted (the Meusnier theorem and the Euler theorem, geodesies, asymptotic curves, lines of curvature, and the like). When delivering lectures in the second year this material had sometimes to be postponed until the middle of the semester (so as to satisfy the needs of the course on differential equations, by introducing the general theory of smooth manifolds as soon as possible). Although this did remove some repetitions (for example, it was then unnecessary to define the differential of a smooth mapping twice, first for surfaces and then in the general case), it was barely justified methodologically (as it links elementary differential geometry, which is local in nature, to the theory of manifolds).

The theory of manifolds begins in Lecture 6. The first ten lectures (from the sixth to the fifteenth) are devoted to basic geometric notions and theorems of the theory of manifolds. In the shorter 11-lecture course one may omit seven of these lectures, reducing the first five lectures to four and sacrificing Lectures 8, 9, and 10 (which treat in the main the topological theory of dimension and the Tychonoff theorems), and Lecture 10 in a 16-lecture course. The remaining lectures in this group (particularly those on the Sard and the Whitney theorems) must, in my opinion, be kept in the course under all circumstances.

An 11-lecture course actually stops here. It turns out to be, however, possible to save, by slightly reducing and condensing the presentation, about an hour and a half of lecture time in presenting some of the material in Lectures 16 and 17. As to the theory of differential forms (Lectures 18, 19, and 20), it must be put off in the shortened course until the next semester (or left to the calculus lectures).

In a 16-lecture course it is possible to finish the course with Lecture 20, which demonstrates various ways of computing de Rham cohomology groups using the example of a sphere. This means that the "integration" Lectures 24 to 29 are excluded from the course and their material is thus left to the calculus course. In Lectures 21 to 23 an attempt is made to expound the theory of homologies and cohomologies (up to spectral sequences!) in a form suitable for the compulsory course. This is made possible by changing the generally accepted point of view and giving up the treatment of simplicial homology theory, which is alleged to be geometrically obvious. I am pleased to note that a similar approach, at a more advanced level, is accepted in Differential Forms in Algebraic Topology by R. Bott and L. W. Tu, which must be read by anyone who wants to become acquainted with the basic ideas and constructions of the classical homology theory in a bright and up-to-date presentation. When time is lacking it is possible to omit the second half of Lecture 22 and all of Lecture 23.

Finally, the concluding Lectures 24 to 29, which, if desired, can be partially interchanged with Lectures 21 to 23, deal with integration. Here the presentation is deliberately incomplete (for example, nothing is said about additive functions of a set), since these lectures reflect only part of the general picture, and omit what relates to calculus. Lecture 28 can be left entirely to the calculus lecturer. It is also possible to confine oneself to just one lecture, Lecture 29, which is virtually independent of the previous four lectures.

This book is actually Semester 3 of my Lectures in Geometry. It starts a new subject, however, and is therefore independent of the previous two semesters.

The book has two major features that distinguish it from other textbooks on elementary smooth manifold theory. Firstly, a lot of space is allotted to topological dimensional theory, the most geometry-oriented branch of general topology, and an acquaintance with it will bring joy to lovers of elegant mathematical constructions which provide deep insights. Secondly, I've ventured, in this elementary course, to present the basic notions of the theory of spectral sequences, a tool whose power and significance is becoming increasingly clear in current studies. This is done without first expounding general (co)homology theory. It is hoped that the two topics will appeal to the interested English reader. The symbol D signifies the end of a proof of a theorem. December 31. 1988 M, Postnikov