内容

目次

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.- 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 $$ \mathbb{R}_1^3 $$.- §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.- 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 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.- 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. Kählerian 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.- 5 The Elements of the Calculus of Variations.- §31. One-dimensional variational problems.- 31.1. The Euler—Lagrange equations.- 31.2. Basic examples of functional.- §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. Hamiltonians 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.- 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. The spinor representations of the groups SO(3) and O(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).