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 of a tensor of arbitrary rank …… CHAPTER 4 The Differential Calculus of Tensors CHAPTER 5 The Elements of the Calculus of Variations CHAPTER 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants Bibliography Index
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