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
