Modern Geometry -- Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Paperback, 2, 1992. Softcover)

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. Exer