Contents

Contentsix8 Continuous groups of transformations; isometryand homothety groups 918.1 Lie groups and Lie algebras 918.2 Enumeration of distinct group structures 958.3 Transformation groups 978.4 Groups of motions 988.5 Spaces of constant curvature 1018.6 Orbits of isometry groups 1048.6.1 Simply-transitive groups 1058.6.2 Multiply-transitive groups 1068.7 Homothety groups 1109Invariants and the characterization of geometries 1129.1 Scalar invariants and covariants 1139.2 The Cartan equivalence method for space-times 1169.3 Calculating the Cartan scalars 1209.3.1 Determination of the Petrov and Segre types 1209.3.2 The remaining steps 1249.4 Extensions and applications of the Cartan method 1259.5 Limits of families of space-times 12610 Generation techniques 12910.1 Introduction 12910.2 Lie symmetries of Einstein’s equations 12910.2.1 Point transformations and their generators 12910.2.2 How to find the Lie point symmetries of a givendifferential equation 13110.2.3 How to use Lie point symmetries: similarityreduction 13210.3 Symmetries more general than Lie symmetries 13410.3.1 Contact and Lie–Bäcklund symmetries 13410.3.2 Generalized and potential symmetries 13410.4 Prolongation 13710.4.1 Integral manifolds of differential forms 13710.4.2 Isovectors, similarity solutions and conservation laws 14010.4.3 Prolongation structures 14110.5 Solutions of the linearized equations 14510.6 Bäcklund transformations 14610.7 Riemann–Hilbert problems 14810.8 Harmonic maps 14810.9 Variational Bäcklund transformations 15110.10 Hirota’s method 152