This textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.
Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.
Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.
Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.
Author(s): Jean Gallier, Jocelyn Quaintance
Series: Geometry and Computing 12
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 777
Tags: Riemannian Manifolds, Differential Geometry, Lie Groups
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
