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1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。整套书内容包括张量分析、曲线和曲面几何、一维和高维变分法（第1卷），微分流形的拓扑和几何（2卷），以及同调与上同调理论（第3卷）。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。

Preface to the First Edition
CHAPTER 1 Geometry in Regions of a Space.Basic Concepts
1.Co-ordinate systems
1.1 Cartesian co-ordinates in a space
1.2 Co-ordinate changes
2.Euclidean space
2.1 Curves in Euclidean space
2.2 Quadratic forms and vectors
3.Riemannian and pseudo-Riemannian spaces
3.1 Riemannian metrics
3.2 The Minkowski metric
4.The simplest groups of transformations of Euclidean space
4.1 Groups of transformations of a region
4.2 Transformations of the plane
4.3 The isometries of 3-dimensional Euclidean space
4.4 Further examples of transformation groups
4.5 Exercises
5.The Serret-Frenet formulae
5.1 Curvature of curves in the Euclidean plane
5.2 Curves in Euclidean 3-space.Curvature and torsion
5.3 Orthogonal transformations depending on a parameter
5.4 Exercises
6.Pseudo-Euclidean spaces
6.1 The simplest concepts of the special theory of relativity
6.2 Lorentz transformations
6.3 Exercises
CHAPTER 2 The Theory of Surfaces
7.Geometry on a surface in space
7.1 Co-ordinates on a surface
7.2 Tangent planes
7.3 The metric on a surface in Euclidean space
7.4 Surface area
7.5 Exercises
8.The second fundamental form
8.1 Curvature of curves on a surface in Euclidean space
8.2 Invariants of a pair of quadratic forms
8.3 Properties of the second fundamental form
8.4 Exercises
9.The metric on the sphere
10.Space-like surfaces in pseudo-Euclidean space
10.1 The pseudo-sphere
10.2 Curvature of space-like curves in R
11.The language of complex numbers in geometry
11.1 Complex and real co-ordinates
11.2 The Hermitian scalar product
11.3 Examples of complex transformation groups
12.Analytic functions
12.1 Complex notation for the element of length, and for the differential of a function
12.2 Complex co-ordinate changes
12.3 Surfaces in complex space
13.The conformal form of the metric on a surface
13.1 Isothermal co-ordinates.Gaussian curvature in terms of conformal co-ordinates
13.2 Conformal form of the metrics on the sphere and the Lobachevskian plane
13.3 Surfaces of constant curvature
13.4 Exercises
14.Transformation groups as surfaces in N-dimensional space
14.1 Co-ordinates in a neighbourhood of the identity
14.2 The exponential function with matrix argument
14.3 The quaternions
14.4 Exercises
15.Conformal transformations of Euclidean and pseudo-Euclidean spaces of several dimensions
CHAPTER 3 Tensors: The Algebraic Theory
16.Examples of tensors
17.The general definition of a tensor
17.1 The transformation rule for the components of a tensor of arbitrary rank
17.2 Algebraic operations on tensors
17.3 Exercises
18.Tensors of type (0,k)
18.1 Differential notation for tensors with lower indices only
18.2 Skew-symmetric tensors of type (0, k)
18.3 The exterior product of differential forms.The exterior algebra
18.4 Skew-symmetric tensors of type (k, 0)(polyvectors).Integrals with respect to anti-commuting variables
18.5 Exercises
19.Tensors in Riemannian and pseudo-Riemannian spaces
19.1 Raising and lowering indices
19.2 The eigenvalues of a quadratic form
19.3 The operator
19.4 Tensors in Euclidean space
19.5 Exercises
20.The crystallographic groups and the finite subgroups of the rotation group of Euclidean 3-space.Examples of invariant tensors
21.Rank 2 tensors in pseudo-Euclidean space, and their eigenvalues
21.1 Skew-symmetric tensors.The invariants of an electromagnetic field
21.2 Symmetric tensors and their eigenvalues.The energy-momentum tensor of an electromagnetic feltauvs. v
22.The behaviour of tensors under mappings
22.1 The general operation of restriction of tensors with lower indices
22.2