Modern Geometry — Methods and Applications

1 Geometry in Regions of a Space. Basic Concepts.- §1. Co-ordinate systems.- §2. Euclidean space.- §3. Riemannian and pseudo-Riemannian spaces.- §4. The simplest groups of transformations of Euclidean space.- §5. The Serret—Frenet formulae.- §6. Pseudo-Euclidean spaces.- 2 The Theory of Surfaces.- §7. Geometry on a surface in space.- §8. The second fundamental form.- §9. The metric on the sphere.- §10. Space-like surfaces in pseudo-Euclidean space.- §11. The language of complex numbers in geometry.- §12. Analytic functions.- §13. The conformal form of the metric on a surface.- §14. Transformation groups as surfaces in N-dimensional space.- §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.- §18. Tensors of type (0, k).- §19. Tensors in Riemannian and pseudo-Riemannian spaces.- §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.- §22. The behaviour of tensors under mappings.- §23. Vector fields.- §24. Lie algebras.- 4 The Differential Calculus of Tensors.- §25. The differential calculus of skew-symmetric tensors.- §26. Skew-symmetric tensors and the theory of integration.- §27. Differential forms on complex spaces.- §28. Covariant differentiation.- §29. Covariant differentiation and the metric.- §30. The curvature tensor.- 5 The Elements of the Calculus of Variations.- §31. One-dimensional variational problems.- §32. Conservation laws.- §33. Hamiltonian formalism.- §34. The geometrical theory of phase space.- §35. Lagrange surfaces.- §36.The second variation for the equation of the geodesics.- 6 The Calculus of Variations in Several Dimensions. Fields and Their Geometric Invariants.- §37. The simplest higher-dimensional variational problems.- §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.- §41. Covariant differentiation of fields with arbitrary symmetry.- §42. Examples of gauge-invariant functionals. Maxwell’s equations and the Yang—Mills equation. Functionals with identically zero variational derivative (characteristic classes).