CHAPTER 1 Examples of Manifolds §1.The concept of a manifold 1.1.Definition of a manifold 1.2.Mappings of manifolds; tensors on manifolds 1.3.Embeddings and immersions of manifolds.Manifolds with boundary §2.The simplest examples of manifolds 2.1.Surfaces in Euclidean space.Transformation groups as manifolds 2.2.Projective spaces 2.3.Exercises §3.Essential facts from the theory of Lie groups 3.1.The structure of a neighbourhood of the identity of a Lie group. The Lie algebra of a Lie group.Semisimplicity 3.2.The concept of a linear representation.An example of a non-matrix Lie group §4.Complex manifolds 4.1.Definitions and examples 4.2.Riemann surfaces as manifolds §5.The simplest homogeneous spaces 5.1.Action of a group on a manifold 5.2.Examples of homogeneous spaces 5.3.Exercises §6.Spaces of constant curvature （symmetric spaces） 6.1.The concept of a symmetric space 6.2.The isometry group of a manifold.Properties of its Lie algebra 6.3.Symmetric spaces of the first and second types 6.4.Lie groups as symmetric spaces 6.5.Constructing symmetric spaces.Examples 6.6.Exercises §7.Vector bundles on a manifold 7.1.Constructions involving tangent vectors. 7.2.The normal vector bundle on a submanifold CHAPTER 2 Foundational Questions.Essential Facts Concerning Functions on a Manifold.Typical Smooth Mappings §8.Partitions of unity and their applications 8.1.Partitions of unity 8.2.The simplest applications of partitions of unity.Integrals over a manifold and the general Stokes formula 8.3.Invariant metrics §9.The realization of compact manifolds as surfaces in RN §10.Various properties of smooth maps of manifolds 10.1.Approximation of continuous mappings by smooth ones 10.2.Sard's theorem 10.3.Transversal regularity 10.4.Morse functions §11.Applications of Sard's theorem 11.1.The existence of embeddings and immersions 11.2.The construction of Morse functions as height functions 11.3.Focal points CHAPTER 3 The Degree of a Mapping.The Intersection Index of Submanifolds. Applications §12.The concept of homotopy 12.1.Definition of homotopy.Approximation of continuous maps
and homotopies by smooth ones 12.2.Relative homotopies §13.The degree of a map 13.1.Definition of degree 13.2.Generalizations of the concept of degree 13.3.Classification of homotopy classes of maps from an arbitrary manifold to a sphere 13.4.The simplest examples §14.Applications of the degree of a mapping. 14.1.The relationship between degree and integral 14.2.The degree of a vector field on a hypersurface 14.3.The Whitney number.The Gauss-Bonnet formula 14.4.The index of a singular point of a vector field 14.5.Transverse surfaces of a vector field.The Poincar?-Bendixson theorem §15.The intersection index and applications 15.1.Definition of the intersection index 15.2.The total index of a vector field 15.3.The signed number of fixed points of a self-map （the Lefschetz,number）.The Brouwer fixed-point theorem 15.4.The linking coefficient CHAPTER 4 Orientability of Manifolds.The Fundamental Group. Covering Spaces （Fibre Bundles with Discrete Fibre） §16.Orientability and homotopies of closed paths 16.1.Transporting an orientation along a path 16.2.Examples of non-orientable manifolds §17.The fundamental group 17.1.Definition of the fundamental group 17.2.The dependence on the base point 17.3.Free homotopy classes of maps of the circle 17.4.Homotopic equivalence 17.5.Examples 17.6.The fundamental group and orientability §18.Covering maps and covering homotopies 18.1.The definition and basic properties of covering spaces 18.2.The simplest examples.The universal covering 18.3.Branched coverings.Riemann surfaces 18.4.Covering maps and discrete groups of transformations §19.Covering maps and the fundamental group.Computation of the fundamental group of certain manifolds 19.1.Monodromy 19.2.Covering maps as an aid in the calculation of fundamental groups 19.3.The simplest of the homology groups 19.4.Exercises §20.The discrete groups of motions of the Lobachevskian plane CHAPTER 5 Homotopy Groups §21.Definition of the absolute and relative homotopy groups.Examples 21.1.Basic definitions 21.2.Relative homotopy groups.The exact sequence of a pair §22.Covering homotopies.The homotopy groups of covering spaces and loop spaces
22.1.The concept of a fibre space 22.2.The homotopy exact sequence of a fibre space 22.3.The dependence of the homotopy groups on the base point 22.4.The case of Lie groups 22.5.Whitehead multiplication §23.Facts concerning the homotopy groups of spheres.Framed normal bundles.The Hopf invariant 23.1.Framed normal bundles and the homotopy groups of spheres 23.2.The suspension map 23.3.Calculation of the groups xn+1（S"） 23.4.The groups 7n+2（S"） CHAPTER 6 Smooth Fibre Bundles §24.The homotopy theory of fibre bundles 24.1.The concept of a smooth fibre bundle 24.2.Connexions 24.3.Computation of homotopy groups by means of fibre bundles 24.4.The classification of fibre bundles 24.5.Vector bundles and operations on them 24.6.Meromorphic functions 24.7.The Picard–Lefschetz formula §25.The differential geometry of fibre bundles 25.1.G-connexions on principal fibre bundles 25.2.G-connexions on associated fibre bundles.Examples 25.3.Curvature 25.4.Characteristic classes: Constructions 25.5.Characteristic classes: Enumeration §26.Knots and links.Braids 26.1.The group of a knot 26.2.The Alexander polynomial of a knot 26.3.The fibre bundle associated with a knot 26.4.Links 26.5.Braids CHAPTER 7 Some Examples of Dynamical Systems and Foliations on Manifolds §27.The simplest concepts of the qualitative theory of dynamical systems. Two-dimensional manifolds 27.1.Basic definitions 27.2.Dynamical systems on the torus §28.Hamiltonian systems on manifolds.Liouville's theorem.Examples 28.1.Hamiltonian systems on cotangent bundles 28.2.Hamiltonian systems on symplectic manifolds.Examples 28.3.Geodesic flows 28.4.Liouville's theorem 28.5.Examples §29.Foliations 29.1.Basic definitions 29.2.Examples of foliations of codimension 1 §30.Variational problems involving higher derivatives 30.1.Hamiltonian formalism 30.2.Examples 30.3.Integration of the commutativity equations.The connexion with
the Kovalevskaja problem.Finite-zoned periodic potentials 30.4.The Korteweg-de Vries equation.Its interpretation as an infinite-dimensional Hamiltonian system 30.5 Hamiltonian formalism of field systems CHAPTER 8 The Global Structure of Solutions of Higher-Dimensional Variational Problems §31.Some manifolds arising in the general theory of relativity （GTR） 31.1.Statement of the problem 31.2.Spherically symmetric solutions 31.3.Axially symmetric solutions 31.4.Cosmological models 31.5.Friedman's models 31.6.Anisotropic vacuum models 31.7.More general models §32.Some examples of global solutions of the Yang-Mills equations. Chiral fields 32.1.General remarks.Solutions of monopole type 32.2.The duality equation 32.3.Chiral fields.The Dirichlet integral §33.The minimality of complex submanifolds Bibliography Index
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