现代几何学方法和应用（第1卷）

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 R3
~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 ora 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 field
~22. The behaviour of tensors under mappings
22.1. The general operation of restriction of tensors with lower indices
22.2. Mappings of tangent spaces
~23. Vector fields
23.1. One-parameter groups of diffeomorphisms
23.2. The exponential function of a vector field
23.3. The Lie derivative
23.4. Exercises
~24. Lie algebras
24.1. Lie algebras and vector fields
24.2. The fundamental matrix Lie algebras
24.3. Linear vector fields
24.4. Left-invariant fields defined on transformation groups
24.5. Invariant metrics on a transformation group
24.6. The classification of the 3-dimensional Lie algebras
24.7. The Lie algebras of the conformal groups
24.8. Exercises
CHAPTER 4
The Differential Calculus of Tensors
~25. The differential calculus of skew-symmetric tensors
25.1. The gradient of a skew-symmetric tensor
25.2. The exterior derivative of a form
25.3. Exercises
~26. Skew-symmetric tensors and the theory of integration
26.1. Integration of differential forms
26.2. Examples of integrals of differential forms
26.3. The general Stokes formula. Examples
26.4. Proof of the general Stokes formula for the cube
26.5. Exercises
~27. Differential forms on complex spaces
27.1. The operators d'' and d
27.2. K/ihlerian metrics. The curvature form
~28. Covariant differentiation
28.1. Euclidean connexions
28.2. Covariant differentiation of tensors of arbitrary rank
~29. Covariant differentiation and the metric
29.1. Parallel transport of vector fields
29.2. Geodesics
29.3. Connexions compatible with the metric
29.4. Connexions compatible with a complex structure Hermitian metric
29.5. Exercises
~30. The curvature tensor
30.1. The general curvature tensor
30.2. The symmetries of the curvature tensor. The curvature tensor
defined by the metric
30.3. Examples: The curvature tensor in spaces of dimensions 2 and 3:
the curvature tensor of transformation groups
30.4. The Peterson-Codazzi equations. Surfaces of constant negative
curvature, and the sine-Gordon equation
30.5. Exercises
CHAPTER 5
The Elements of the Calculus of Variations
~31. One-dimensional variational problems
31.1. The Euler-Lagrange equations
31.2. Basic examples of functionals
~32. Conservation laws
32.1. Groups of transformations preserving a given variational problem
32.2. Examples. Applications of the conservation laws
~33. Hamiltonian formalism
33.1. Legendre''s transformation
33.2. Moving co-ordinate frames
33.3. The principles of Maupertuis and Fermat
33.4. Exercises
~34. The geometrical theory of phase space
34.1. Gradient systems
34.2. The Poisson bracket
34.3. Canonical transformations
34.4. Exercises
~35. Lagrange surfaces
35.1. Bundles of trajectories and the Hamilton-Jacobi equation
35.2. Hamiitonians which are first-order homogeneous with
respect to the momentum
~36. The second variation for the equation of the geodesics
36.1. The formula for the second variation
36.2. Conjugate points and the minimality condition
CHAPTER 6
The Calculus of Variations in Several Dimensions.
Fields and Their Geometric Invariants
~37. The simplest higher-dimensional variational problems
.37.1. The Euler-Lagrange equations
37.2. The energy-momentum tensor
37.3. The equations of an electromagnetic field
37.4. The equations of a gravitational field
37.5. Soap films
37.6. Equilibrium equation for a thin plate
37.7. Exercises
~38. Examples of Lagrangians
~39. The simplest concepts of the general theory of relativity
~40. T-he spinor representations of the groups SO 3 and 0 3, 1 .
Dirac''s equation and its properties
40.1. Automorphisms of matrix algebras
40.2. The spinor representation of the group SO 3
40.3. The spinor representation of the Lorentz group
40.4. Dirac''s equation
40.5. Dirac''s equation in an electromagnetic field. The operation
of charge conjugation
~41. Covariant differentiation of fields with arbitrary symmetry
41.1. Gauge transformations. Gauge-invariant Lagrangians
41.2. The curvature form
41.3. Basic examples
~42. Examples of gauge-invariant functionals. Maxwell''s equations and
the Yang-Mills equation. Functionals with identically zero
variational derivative characteristic classes
Bibliography
Index