Differential Geometry and Lie Groups - A Second Course

Preface
Contents
1 Introduction
2 Tensor Algebras and Symmetric Algebras
	2.1 Linear Algebra Preliminaries: Dual Spaces and Pairings
	2.2 Tensor Products
	2.3 Bases of Tensor Products
	2.4 Some Useful Isomorphisms for Tensor Products
	2.5 Duality for Tensor Products
	2.6 Tensor Algebras
	2.7 Symmetric Tensor Powers
	2.8 Bases of Symmetric Powers
	2.9 Duality for Symmetric Powers
	2.10 Symmetric Algebras
	2.11 Tensor Products of Modules Over a Commutative Ring
	2.12 Problems
3 Exterior Tensor Powers and Exterior Algebras
	3.1 Exterior Tensor Powers
	3.2 Bases of Exterior Powers
	3.3 Duality for Exterior Powers
	3.4 Exterior Algebras
	3.5 The Hodge *-Operator
	3.6 Left and Right Hooks
	3.7 Testing Decomposability
	3.8 The Grassmann-Plücker's Equations and Grassmannian Manifolds
	3.9 Vector-Valued Alternating Forms
	3.10 Problems
4 Differential Forms
	4.1 DifferentialFormsonSubsetsofRnanddeRhamCohomology
	4.2 Pull-Back of Differential Forms
	4.3 Differential Forms on Manifolds
	4.4 Lie Derivatives
	4.5 Vector-Valued Differential Forms
	4.6 Differential Forms on Lie Groups and Maurer-Cartan Forms
	4.7 Problems
5 Distributions and the Frobenius Theorem
	5.1 Tangential Distributions and Involutive Distributions
	5.2 Frobenius Theorem
	5.3 Differential Ideals and Frobenius Theorem
	5.4 A Glimpse at Foliations and a Global Version of Frobenius Theorem
	5.5 Problems
6 Integration on Manifolds
	6.1 Orientation of Manifolds
	6.2 Volume Forms on Riemannian Manifolds and Lie Groups
	6.3 Integration in Rn
	6.4 Integration on Manifolds
	6.5 Densities
	6.6 Manifolds with Boundary
	6.7 Integration on Regular Domains and Stokes' Theorem
	6.8 Integration on Riemannian Manifolds and Lie Groups
	6.9 Problems
7 Spherical Harmonics and Linear Representations of Lie Groups
	7.1 Hilbert Spaces and Hilbert Sums
	7.2 Spherical Harmonics on the Circle
	7.3 Spherical Harmonics on the 2-Sphere
	7.4 The Laplace-Beltrami Operator
	7.5 Harmonic Polynomials, Spherical Harmonics, and L2(Sn)
	7.6 Zonal Spherical Functions and Gegenbauer Polynomials
	7.7 More on the Gegenbauer Polynomials
	7.8 The Funk–Hecke Formula
	7.9 Linear Representations of Compact Lie Groups: A Glimpse
	7.10 Consequences of the Peter–Weyl Theorem
	7.11 Gelfand Pairs, Spherical Functions, and Fourier Transform
	7.12 Problems
8 Operators on Riemannian Manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, the Bochner Laplacian, and Weitzenböck Formulae
	8.1 The Gradient and Hessian Operators on RiemannianManifolds
	8.2 The Hodge * Operator on Riemannian Manifolds
	8.3 The Hodge Laplacian and the Hodge Divergence Operators on Riemannian Manifolds
	8.4 The Hodge and Laplace–Beltrami Laplacians of Functions
	8.5 Divergence and Lie Derivative of the Volume Form
	8.6 Harmonic Forms, the Hodge Theorem, and Poincaré Duality
	8.7 The Bochner Laplacian, Weitzenböck Formula, and the Bochner Technique
	8.8 Problems
9 Bundles, Metrics on Bundles, and Homogeneous Spaces
	9.1 Fiber Bundles
	9.2 Bundle Morphisms, Equivalent Bundles, and Isomorphic Bundles
	9.3 Bundle Constructions via the Cocycle Condition
	9.4 Vector Bundles
	9.5 Operations on Vector Bundles
	9.6 Properties of Vector Bundle Sections
	9.7 Duality Between Vector Fields and Differential Forms and Covariant Derivatives of Tensor Fields
	9.8 Metrics on Vector Bundles, Reduction of Structure Groups, and Orientation
	9.9 Principal Fiber Bundles
	9.10 Proper and Free Actions and Homogeneous Spaces
	9.11 Problems
10 Connections and Curvature in Vector Bundles
	10.1 Introduction to Connections in Vector Bundles
	10.2 Connections and Connection Forms in Vector Bundles
	10.3 Connection Matrices
	10.4 Parallel Transport
	10.5 Curvature, Curvature Form, and Curvature Matrix
	10.6 Structure Equations
	10.7 A Formula for dd
	10.8 Connections Compatible with a Metric: Levi-CivitaConnections
	10.9 Connections on the Dual Bundle
	10.10 The Levi-Civita Connection on TM Revisited
	10.11 Pontrjagin Classes and Chern Classes: a Glimpse
	10.12 The Pfaffian Polynomial
	10.13 Euler Classes and the Generalized Gauss-Bonnet Theorem
	10.14 Problems
11 Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n)
	11.1 Introduction: Rotations as Group Actions
	11.2 Preliminaries
	11.3 Clifford Algebras
	11.4 Clifford Groups
	11.5 The Groups Pin(n) and Spin(n)
	11.6 The Groups Pin(p, q) and Spin(p, q)
	11.7 The Groups Pin(p, q) and Spin(p, q) as Double Covers of O(p, q) and SO(p, q)
	11.8 Periodicity of the Clifford Algebras Clp, q
	11.9 The Complex Clifford Algebras Cl(n, C)
	11.10 Clifford Groups Over a Field K
	11.11 Problems
Bibliography
Symbol Index
Index