本书是莫斯科大学数学力学系经典教材《现代几何学——方法和应用》三卷本的第2卷。全书力求以直观的和物理的视角阐述，是一本难得的现代几何方面的佳作。本书可用作数学和理论物理专业高年级和研究生的教学用书，对从事几何和拓扑研究的工作者也极具参考价值。

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 Poincaré-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