Up until recently， Riemannian geometry and basic topology were not included， even by departments or faculties of mathematics， as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead （and still are given in some places） gradually came to be viewed as anachronisms. However， there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date， that is to say，which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education， and what might be the appropriate level of abstractness of their exposition The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that， in addition to the geometry，of curves and surfaces， the following topics are certainly useful in the various areas of application of mathematics （especially in elasticity and relativity，to name but two）， and are therefore essential： the theory of tensors （including eovariant differentiation of them）; Riemannian curvature;
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暫缺《現(xiàn)代幾何學(xué)方法和應(yīng)用（第1卷）》作者簡(jiǎn)介
圖書(shū)目錄
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
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