Differential Geometry and Lie Groups - A Computational Perspective

Preface
Contents
1 Introduction
Part I Introduction to Differential Manifolds and Lie Groups
	2 The Matrix Exponential: Some Matrix Lie Groups
		2.1 The Exponential Map
		2.2 The Lie Groups GL(n, R), SL(n, R), O(n), SO(n), the Lie Algebras gl(n, R), sl(n, R), o(n), so(n), and the Exponential Map
		2.3 Symmetric Matrices, Symmetric Positive Definite Matrices, and the Exponential Map
		2.4 The Lie Groups GL(n, C), SL(n, C), U(n), SU(n), the Lie Algebras gl(n, C), sl(n, C), u(n), su(n), and the Exponential Map
		2.5 Hermitian Matrices, Hermitian Positive Definite Matrices, and the Exponential Map
		2.6 The Lie Group SE(n) and the Lie Algebra se(n)
		2.7 Problems
	3 Adjoint Representations and the Derivative of exp
		3.1 The Adjoint Representations Ad and ad
		3.2 The Derivative of exp
		3.3 Problems
	4 Introduction to Manifolds and Lie Groups
		4.1 Introduction to Embedded Manifolds
		4.2 Linear Lie Groups
		4.3 Homomorphisms of Linear Lie Groups and Lie Algebras
		4.4 Problems
	5 Groups and Group Actions
		5.1 Basic Concepts of Groups
		5.2 Group Actions: Part I, Definition and Examples
		5.3 Group Actions: Part II, Stabilizers and Homogeneous Spaces
		5.4 The Grassmann and Stiefel Manifolds
		5.5 Topological Groups
		5.6 Problems
	6 The Lorentz Groups
		6.1 The Lorentz Groups O(n, 1), SO(n, 1), and SO0(n, 1)
		6.2 The Lie Algebra of the Lorentz Group SO0(n, 1)
		6.3 The Surjectivity of exp2mu-:6muplus1muso(1, 3)→SO0(1, 3)
		6.4 Problems
	7 The Structure of O(p, q) and SO(p, q)
		7.1 Polar Forms for Matrices in O(p, q)
		7.2 Pseudo-Algebraic Groups
		7.3 More on the Topology of O(p, q) and SO(p, q)
		7.4 Problems
	8 Manifolds, Tangent Spaces, Cotangent Spaces, and Submanifolds
		8.1 Charts and Manifolds
		8.2 Tangent Vectors and Tangent Spaces
		8.3 Tangent Vectors as Derivations
		8.4 Tangent and Cotangent Spaces Revisited
		8.5 Tangent Maps
		8.6 Submanifolds, Immersions, and Embeddings
		8.7 Problems
	9 Construction of Manifolds from Gluing Data
		9.1 Sets of Gluing Data for Manifolds
		9.2 Parametric Pseudo-Manifolds
	10 Vector Fields, Lie Derivatives, Integral Curves, and Flows
		10.1 Tangent and Cotangent Bundles
		10.2 Vector Fields and Lie Derivative
		10.3 Integral Curves, Flow of a Vector Field, and One-Parameter Groups of Diffeomorphisms
		10.4 Log-Euclidean Polyaffine Transformations
		10.5 Fast Polyaffine Transforms
		10.6 Problems
	11 Partitions of Unity and Covering Maps
		11.1 Partitions of Unity
		11.2 Covering Maps and Universal Covering Manifolds
		11.3 Problems
	12 Basic Analysis: Review of Series and Derivatives
		12.1 Series and Power Series of Matrices
		12.2 The Derivative of a Function Between Normed Vector Spaces
		12.3 Linear Vector Fields and the Exponential
		12.4 Problems
	13 A Review of Point Set Topology
		13.1 Topological Spaces
		13.2 Continuous Functions and Limits
		13.3 Connected Sets
		13.4 Compact Sets
		13.5 Quotient Spaces
		13.6 Problems
Part II Riemannian Geometry, Lie Groups, and Homogeneous Spaces
	14 Riemannian Metrics and Riemannian Manifolds
		14.1 Frames
		14.2 Riemannian Metrics
		14.3 Problems
	15 Connections on Manifolds
		15.1 Connections on Manifolds
		15.2 Parallel Transport
		15.3 Connections Compatible with a Metric: Levi-Civita Connections
		15.4 Problems
	16 Geodesics on Riemannian Manifolds
		16.1 Geodesics, Local Existence, and Uniqueness
		16.2 The Exponential Map
		16.3 Complete Riemannian Manifolds, the Hopf-Rinow Theorem, and the Cut Locus
		16.4 Convexity and Convexity Radius
		16.5 Hessian of a Function on a Riemannian Manifold
		16.6 The Calculus of Variations Applied to Geodesics: the First Variation Formula
		16.7 Problems
	17 Curvature in Riemannian Manifolds
		17.1 The Curvature Tensor
		17.2 Sectional Curvature
		17.3 Ricci Curvature
		17.4 The Second Variation Formula and the Index Form
		17.5 Jacobi Fields and Conjugate Points
		17.6 Jacobi Fields and Geodesic Variations
		17.7 Topology and Curvature
		17.8 Cut Locus and Injectivity Radius: Some Properties
		17.9 Problems
	18 Isometries, Local Isometries, Riemannian Coverings and Submersions, and Killing Vector Fields
		18.1 Isometries and Local Isometries
		18.2 Riemannian Covering Maps
		18.3 Riemannian Submersions
		18.4 Isometries and Killing Vector Fields
		18.5 Problems
	19 Lie Groups, Lie Algebras, and the Exponential Map
		19.1 Lie Groups and Lie Algebras
		19.2 Left- and Right-Invariant Vector Fields, the Exponential Map
		19.3 Homomorphisms of Lie Groups and Lie Algebras, Lie Subgroups
		19.4 The Correspondence Lie Groups–Lie Algebras
		19.5 Semidirect Products of Lie Algebras and Lie Groups
		19.6 Universal Covering Groups
		19.7 The Lie Algebra of Killing Fields
		19.8 Problems
	20 The Derivative of exp and Dynkin\'s Formula
		20.1 The Derivative of the Exponential Map
		20.2 The Product in Logarithmic Coordinates
		20.3 Dynkin\'s Formula
		20.4 Problems
	21 Metrics, Connections, and Curvature on Lie Groups
		21.1 Left (Resp. Right) Invariant Metrics
		21.2 Bi-Invariant Metrics
		21.3 Connections and Curvature of Left-Invariant Metrics on Lie Groups
		21.4 Connections and Curvature of Bi-Invariant Metrics on Lie Groups
		21.5 Simple and Semisimple Lie Algebras and Lie Groups
		21.6 The Killing Form
		21.7 Left-Invariant Connections and Cartan Connections
		21.8 Problems
	22 The Log-Euclidean Framework Applied to SPD Matrices
		22.1 Introduction
		22.2 A Lie Group Structure on SPD(n)
		22.3 Log-Euclidean Metrics on SPD(n)
		22.4 A Vector Space Structure on SPD(n)
		22.5 Log-Euclidean Means
		22.6 Problems
	23 Manifolds Arising from Group Actions
		23.1 Proper Maps
		23.2 Proper and Free Actions
		23.3 Riemannian Submersions and Coverings Induced by Group Actions
		23.4 Reductive Homogeneous Spaces
		23.5 Examples of Reductive Homogeneous Spaces
		23.6 Naturally Reductive Homogeneous Spaces
		23.7 Examples of Naturally Reductive Homogeneous Spaces
		23.8 A Glimpse at Symmetric Spaces
		23.9 Examples of Symmetric Spaces
		23.10 Types of Symmetric Spaces
		23.11 Problems
Bibliography
Symbol Index
Index