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(Table of content)
18 Elementary Group Theory.- 19 Continuous Groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 21 Group Representations II: General; Rigid Motions; Bessel Functions.- 22 Group Representations and Quantum Mechanics.- 23 Elementary Theory of Manifolds.- 24 Covering Manifolds.- 25 Lie Groups.- 26 Metric and Geodesics on a Manifold.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 28 The Extension of Einstein Manifolds.- 29 Bifurcations in Hydrodynamic Stability Problems.- 30 Invariant Manifolds in the Taylor Problem.- 31 The Early Onset of Turbulence.- References. groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 20.1 Finite-dimensional representations of a group.- 20.2 Vector and tensor transformation laws.- 20.3 Other group representations in physics.- 20.4 Infinite-dimensional representations.- 20.5 A simple case: SO(2).- 20.6 Representations of matrix groups on X?.- 20.7 Homogeneous spaces.- 20.8 Regular representations.- 20.9 Representations of the rotation group SO(3).- 20.10 Tesseral harmonics; Legendre functions.- 20.11 Associated Legendre functions.- 20.12 Matrices of the irreducible representations of SO(3); the Euler angles.- 20.13 The addition theorem for tesseral harmonics.- 20.14 Completeness of the tesseral harmonics.- 21 Group Representations II: General; Rigid Motions; Bessel Functions.- 21.1 Equivalence; unitary representations.- 21.2 The reduction of representations.- 21.3 Schur's Lemma and its corollaries.- 21.4 Compact and noncompact groups.- 21.5 Invariant integration; Haar measure.- 21.6 Complete system of representations of a compact group.- 21.7 Homogeneous spaces as configuration spaces in physics.- 21.8 M2 and related groups.- 21.9 Representations of M2.- 21.10 Some irreducible representations.- 21.11 Bessel functions.- 21.12 Matrices of the representations.- 21.13 Characters.- 22 Group Representations and Quantum Mechanics.- 22.1 Representations in quantum mechanics.- 22.2 Rotations of the axes.- 22.3 Ray representations.- 22.4 A finite-dimensional case.- 22.5 Local representations.- 22.6 Origin of the two-valued representations.- 22.7 Representations of SU(2) and SL(2, ?).- 22.8 Irreducible representations of SU(2).- 22.9 The characters of SU(2).- 22.10 Functions of z and z?.- 22.11 The finite-dimensional representations of SL(2, ?).- 22.12 The irreducible invariant subspaces of X? for SL(2, ?).- 22.13 Spinors.- 23 Elementary Theory of Manifolds.- 23.1 Examples of manifolds; method of identification.- 23.2 Coordinate systems or charts; compatibility; smoothness.-23.3 Induced topology.- 23.4 Definition of manifold; Hausdorff separation axiom.- 23.5 Curves and functions in a manifold.- 23.6 Connectedness; components of a manifold.- 23.7 Global topology; homotopic curves; fundamental group.- 23.8 Mechanical linkages: Cartesian products.- 24 Covering Manifolds.- 24.1 Definition and examples.- 24.2 Principles of lifting.- 24.3 Universal covering manifold.- 24.4 Comments on the construction of mathematical models.- 24.5 Construction of the universal covering.- 24.6 Manifolds covered by a given manifold.- 25 Lie Groups.- 25.1 Definitions and statement of objectives.- 25.2 The expansions of m(·, ·) and l(·, ·).- 25.3 The Lie algebra of a Lie group.- 25.4 Abstract Lie algebras.- 25.5 The Lie algebras of linear groups.- 25.6 The exponential mapping; logarithmic coordinates.- 25.7 An auxiliary lemma on inner automorphisms; the mappings Ad?.- 25.8 Auxiliary lemmas on formal derivatives.- 25.9 An auxiliary lemma on the differentiation of exponentials.- 25.10 The Campbell-Baker-Hausdorf (CBH) formula.- 25.11 Translation of charts; compatibility; G as an analytic manifold.- 25.12 Lie algebra homomorphisms.- 25.13 Lie group homomorphisms.- 25.14 Law of homomorphism for Lie groups.- 25.15 Direct and semidirect sums of Lie algebras.- 25.16 Classification of the simple complex Lie algebras.- 25.17 Models of the simple complex Lie algebras.- 25.18 Note on Lie groups and Lie algebras in physics.- Appendix to Chapter 25-Two nonlinear Lie groups.- 26 Metric and Geodesics on a Manifold.- 26.1 Scalar and vector fields on a manifold.- 26.2 Tensor fields.- 26.3 Metric in Euclidean space.- 26.4 Riemannian and pseudo-Riemannian manifolds.- 26.5 Raising and lowering of indices.- 26.6 Geodesies in a Riemannian manifold.- 26.7 Geodesies in a pseudo-Riamannian manifold M.- 26.8 Geodesies; the initial-value problem; the Lipschitz condition.- 26.9 The integral equation; Picard iterations.- 26.10 Geodesies; the two-point problem.- 26.11 Continuation
Contents
18 Elementary Group Theory.- 19 Continuous Groups.- 20 Group Representations I: Rotations and Spherical Harmonics.- 21 Group Representations II: General; Rigid Motions; Bessel Functions.- 22 Group Representations and Quantum Mechanics.- 23 Elementary Theory of Manifolds.- 24 Covering Manifolds.- 25 Lie Groups.- 26 Metric and Geodesics on a Manifold.- 27 Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds.- 28 The Extension of Einstein Manifolds.- 29 Bifurcations in Hydrodynamic Stability Problems.- 30 Invariant Manifolds in the Taylor Problem.- 31 The Early Onset of Turbulence.- References.
