Contents


Volume 1

 Preface to the Russian Edition 
 Preface to the English Edition 
 Lecture 1 
Simple lines in the plane. Giving lines by an equation. Whitney's theorem. Jordan curves. Smooth and regular Curves. Nonparametrized curves. Natural parameter 
 Lecture 2. 
Cuives in the plane. Frenet formulas for a space curve. Projections of a curve onto the coordinate planes of the canonical frame. Frenet formulas for a curve in an ndimensional space. The existence and uniqueness of a curve with given curvature 
 Lecture 3 
Elementary surfaces and their parametrizations. Examples of surfaces. Tangent plane and tangent subspace. Smooth mappings of surfaces and their differentials. Diffeomorphisms of surfaces. The first quadratic form of a surface. Isometrics. Beltrami's first differential parameter. Examples of computation of first quadratic forms. Developable surfaces 
 Lecture 4 
Normal vector. Surface as the graph of a function. Normal sections. The second quadratic form of a surface. The Dupin indicatrix. Principal, total and mean curvatures. The second quadratic form of a graph. Ruled surfaces of zero curvature. Surfaces of revolution 
 Lecture 5 
Weingarten formulas. Coefficients of connection. The Gauss theorem. Explicit formula for Gaussian curvature. The necessary and sufficient conditions of isometry. Surfaces of constant curvature 
 Lecture 6 
Introductory remarks. Open subsets of the space R" and their
diffeomorphisms. Charts and atlases. Maximal atlases. Smooth manifolds. Examples of smooth manifolds 
 Lecture 7 
Topology of a smooth manifold. Open submanifolds. Neighbourhoods and interior points. Homeomorphisms. The first axiom of countability and the property of being locally flat. The second axiom of countability. NonHausdorff manifolds. Smoothnesses of a topological space. Topological manifolds. Zerodimensional manifolds. The category TOP. The category DIFF. Fullback of smoothness 
 Lecture 8 
Topological invariance of the dimension of a manifold. The dimension and coverings. Compact spaces. Lebesgue lemma. The upper estimate of the dimension of compact subsets of a space R". The monotonicity property of the dimension. Closed sets. The monotonicity of the dimension and closed sets. Direct product of topological spaces. The compactness of the direct product of compact spaces 
 Lecture 9 
The drum theorem. The Brouwer fixedpoint theorem. Cube separation theorem. Normal and completely normal spaces. Separation extension. The Lebesgue theorem on coverings of a cube. The lower estimate of the dimension of a cube 
 Lecture 10 
Ordinals. Interval topology in sets of ordinals. Zerodimensional spaces. Tychonoffs example. Tychonoff product ol topological spaces. Filters. Centred sets of sets. Ultrafilters. Compactness criterions. The Tychonoff theorem 
 Lecture 11 
Smoothness on an affine space. The manifold of matrices of a given rank. Stiefel manifolds. Matrix rows. The exponential of a matrix. The logarithm of a matrix. Orthogonal and Jorthogonal matrices. Matrix Lie groups. Groups of ./orthogonal matrices. Unitary and Junitary matrices. Complex matri> Lie groups. Complexanalytic manifolds. Arcwise connected spaces. Connected spaces. The coincidence of connectedness
and arcwise connectedness for manifolds. Smooth and piecewise smooth paths. Connected manifolds failing to satisfy the second axiom of countability 
 Lecture 12 
Vectors tangent to a smooth manifold. Derivatives of holomorphic functhons. Tangent vectors to complex analytic manifolds. The differential of a smooth mapping. The chain rule. The gradient of a smooth function. The etale mapping theorem. The theorem on the change of local coordinates. Locally flat mappings 
 Lecture 13 
Proof of the theorem on locally flat maps. Immersions and submersions. Submanifolds of a smooth manifold. A subspace tangent to a submanifold. Giving locally a submanifold. The uniqueness of the submanifold structure. The case of embedded submanifolds. The theorem on the inverse image of a regular value. Solutions of systems of equations. The group SL(n) as a submanifold 
 Lecture 14 
The embedding theorem. Compact sets revisited. Urysohn functions. Proof of the embedding theorem. Manifolds satisfying the second axiom of countability. Scattered and meager
sets. Null sets 
 Lecture 15 
The Sard theorem. The analytic part of the proof of the Sard theorem. Direct product of manifolds. The manifold of tangent vectors. Proof of the Whitney theorem 
Volume 2

 Lecture 16 
Tensors. Tensor fields. Vector fields and differentiations. Lie algebra of vector fields 
 Lecture 17 
Integral curves of vector fields. Vector fields and flows.
Transfer of vector fields using diffeomorphisms. The Lie derivative of a tensor field 
 Lecture 18 
Linear differential forms. Differential forms of arbitrary degree. Differential forms as functionals of vector fields. The inner product of a vector field and a differential form. Fullback of a differential form via a smooth mapping 
 Lecture 19 
Exterior differential of a differential form. The Lie derivative of a differential form 
 Lecture 20 
The de Rham complex and cohomology groups of a smooth
manifold. The group H°X. Poincare lemma. The group //'s2. The group HlSl. Computing //'$' using integrals. The group H2S2. Groups //'S" for n>2. Groups HmSn,m< n. Groups HnS" 
 Lecture 21 
Simplicial schemes and their geometric realizations. Cohomology groups of simplicial schemes. The double complex of a covering. Cohomology groups of a double complex. Augmented double complexes. Boundary homomorphisms. Acyclic complexes. Row acyclicity for n=0 
 Lecture 22  Row acyclicity of the double complex of a numerable covering. Row acyclicity of the double complex of a Leray covering.
The de RhamLeray theorem. Generalization. Groups fif9. Groups Ft'M. A group adjoined to a graded group with filtration 
 Lecture 23 
Groups E™. Spectral sequences. Spectral sequence of a double complex. Spectral sequence of a covering 
 Lecture 24 
Compactly exhaustible and paracompact topological spaces. Paracompact manifolds. Integrals in /?". Cubable sets and
Contents densities in arbitrary manifolds. Integration of densities
Lecture 25 Orientable manifolds. Integration of forms. Poincare lemma
for finite forms. The group H'lX. An orientable manifold 
 Lecture 26 
The degree of a smooth proper mapping. The algebraic number of inverse images of a regular value. Invariance of the degree under smooth homotopies. Proof of the drum theorem. Invariance of the degree under any homotopies
Lecture 27 Domains with regular boundaries. Stokes' theorem. GaussOstrogradskii, Green's and NewtonLeibniz formulas. Manifolds with border sets. Interior and boundary points. Embedded d submanifolds. Stokes' theorem for manifolds with border set and for d submanifolds. Stokes' theorem for surface integrals. Stokes' theorem for singular submanifolds. Line integrals of the second kind 
 Lecture 28 
Operators of vector analysis. Consequences of the identity d<>d = 0. Consequences of differentiation formulas for products. The Laplacian and the Beltrami operator. The flow of a vector field. The GaussOstrogradskii formulas for divergence and Green's formulas. Convergence as the density of sources. The Stokes formula for circulation. The GaussOstrogradskii formula for rotation. The generalized GaussOstrogradskii formula 
 Lecture 29 
Periods of differential forms. Singular simplexes, chains, cycles, and boundaries. Stokes' theorem for chain integrals. Singular homology groups. The de Rham theorem. Cohomology groups of a chain complex. Singular cohomology groups 
 Subject index 
Preface to the russian edition


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 firstsemester 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, CaleyKlein 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 thoughtout, 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 threedimensional 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 thirdyear geometry course does not help lecturers presenting secondyear 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, GaussOstrogradskii 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 twohour 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 11lecture 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 16lecture 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 11lecture 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 16lecture 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 uptodate 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.
Preface to the english edition


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 geometryoriented 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