This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks.
With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation.
The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.
Author(s): David Xianfeng Gu, Emil Saucan
Publisher: CRC Press
Year: 2022
Language: English
Pages: 588
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
SECTION I: Differential Geometry, Classical and Discrete
CHAPTER 1: Curves
1.1. CURVES
1.2. CURVATURE
1.2.1. The Osculating Circle
1.2.2. Menger Curvature
1.2.2.1. Applications of Menger Curvature
1.2.3. Haantjes Curvature
1.2.4. Applications of Haantjes Curvature
1.3. TORSION
1.3.1. The Serret-Frènet Formulas
1.3.2. Haantjes Curvature Revisited
1.3.3. The Local Canonical Form
1.3.4. Existence and Uniqueness Theorem
1.3.5. Metric Torsion
1.3.5.1. The Metric Existence and Uniqueness Theorem of Curves
1.4. HIGHER DIMENSIONAL CURVES
CHAPTER 2: Surfaces: Gauss Curvature - First Definition
2.1. SURFACES
2.2. GAUSS CURVATURE – FIRST DEFINITION
2.3. THE FUNDAMENTAL FORMS
2.3.1. The First Fundamental Form
2.3.1.1. Examples
2.3.1.2. The Second Fundamental Form
2.3.1.3. Distinguished Curves Revisited
2.4. SOME IMPLEMENTATION ASPECTS
CHAPTER 3: Metrization of Gauss Curvature
3.1. METRIC APPROXIMATION OF SECTIONAL CURVATURES
3.2. WALD CURVATURE
3.2.1. Computation of Wald Curvature I: The Exact Formula
3.2.2. Computation of Wald Curvature II: An Approximation
3.2.3. Applications of Wald Curvature
3.2.4. Wald Curvature Revisited
CHAPTER 4: Gauss Curvature and Theorema Egregium
4.1. THEOREMA EGREGIUM
4.1.1. The Tube Formula and Approximation of Surface Curvatures
4.2. NORMAL CYCLE
CHAPTER 5: The Mean and Gauss Curvature Flows
5.1. CURVE SHORTENING FLOW
5.2. MEAN CURVATURE FLOW
5.3. GAUSS CURVATURE FLOW
CHAPTER 6: Geodesics
6.1. COVARIANT DERIVATIVE
6.2. GEODESICS
6.2.1. The Hopf-Rinow Theorem
6.3. DISCRETIZATION OF GEODESICS
CHAPTER 7: Geodesics and Curvature
7.1. GAUSS CURVATURE AND PARALLEL TRANSPORT
CHAPTER 8: The Equations of Compatibility
8.1. APPLICATIONS AND DISCRETIZATIONS
CHAPTER 9: The Gauss-Bonnet Theorem and the Poincare Index Theorem
9.1. THE GAUSS-BONNET THEOREM
9.1.1. The Local Gauss-Bonnet Theorem
9.1.2. The Global Gauss-Bonnet Theorem
9.2. THE POINCARÉ INDEX THEOREM
9.2.1. Discretizations of the Gauss-Bonnet Theorem
9.2.2. Discretizations of the Poincaré Index Theorem
CHAPTER 10: Higher Dimensional Curvatures
10.1. MOTIVATION AND BASICS
10.1.1. The Curvature Tensor
10.1.2. Sectional Curvature
10.1.3. Ricci Curvature
10.1.4. Scalar Curvature
CHAPTER 11: Higher Dimensional Curvatures 2
11.1. MOTIVATION
11.2. THE LIPSCHITZ-KILLING CURVATURES
11.2.1. Curvatures’ Approximation
11.2.1.1. Thick Triangulations
11.2.1.2. Curvatures’ Approximation Results
11.3. GENERALIZED PRINCIPAL CURVATURES
11.4. OTHER APPROACHES
11.4.1. Banchoff’s Definition Revisited
11.4.2. Stone’s Sectional Curvature
11.4.3. Glickenstein’s Sectional, Ricci and Scalar Curvatures
11.4.4. The Ricci Tensor of Alsing and Miller
11.4.5. The Metric Approach
11.4.5.1. Metrization of the Lipschitz-Killing Curvatures
11.4.5.2. A Metric Gauss-Bonnet Theorem and PL Curvatures
CHAPTER 12: Discrete Ricci Curvature and Flow
12.1. PL MANIFOLDS – FROM COMBINATORIAL TO METRIC RICCI CURVATURE
12.1.1. Definition and Convergence
12.1.2. The Bonnet-Myers Theorem
12.1.2.1. The 2-Dimensional Case
12.1.2.2. Wald Curvature and Alexandrov Spaces
12.1.3. A Comparison Theorem
12.2. RICCI CURVATURE AND FLOW FOR 2-DIMENSIONAL PL SURFACES
12.2.1. Combinatorial Surface Ricci Flow
12.2.2. The Metric Ricci Flow for Surfaces
12.2.2.1. Smoothings and Metric Curvatures
12.2.2.2. A Metric Ricci Flow
12.2.3. Combinatorial Yamabe Flow
12.3. RICCI CURVATURE AND FLOW FOR NETWORKS
12.3.1. Metric Ricci Curvature of Networks
12.3.2. Ollivier Ricci Curvature
CHAPTER 13: Weighted Manifolds and Ricci Curvature Revisited
13.1. WEIGHTED MANIFOLDS
13.1.1. The Curvature-Dimension Condition of Lott-Villani and Sturm
13.1.2. Corwin et al.
13.1.2.1. Curvature of Curves in Weighted Surfaces
13.1.2.2. The Mean Curvature of Weighted Surfaces
13.1.2.3. Gauss Curvature of Weighted Surfaces
13.2. FORMAN-RICCI CURVATURE
13.2.1. The General Case
13.2.2. Two-Dimensional Complexes
13.2.3. The Forman-Ricci Curvature of Networks
13.2.3.1. From Networks to Simplicial Complexes
SECTION II: Differential Geometry, Computational Aspects
CHAPTER 14: Algebraic Topology
14.1. INTRODUCTION
14.2. SURFACE TOPOLOGY
14.3. FUNDAMENTAL GROUP
14.4. WORD GROUP REPRESENTATION
14.5. FUNDAMENTAL GROUP CANONICAL REPRESENTATION
14.6. COVERING SPACE
14.7. COMPUTATIONAL ALGORITHMS
CHAPTER 15: Homology and Cohomology Group
15.1. SIMPLICIAL HOMOLOGY
15.2. HOMOLOGY VS. HOMOTOPY
15.3. SIMPLICIAL COHOMOLOGY
15.4. SIMPLICIAL MAPPING
15.5. FIXED POINT
15.6. COMPUTATIONAL ALGORITHMS
CHAPTER 16: Exterior Calculus and Hodge Decomposition
16.1. EXTERIOR DIFFERENTIALS
16.2. DE RHAM COHOMOLOGY
16.3. HODGE STAR OPERATOR
16.4. HODGE DECOMPOSITION
16.5. DISCRETE HODGE THEORY
CHAPTER 17: Harmonic Map
17.1. PLANAR HARMONIC MAPS
17.2. SURFACE HARMONIC MAPS
17.3. DISCRETE HARMONIC MAP
17.4. COMPUTATIONAL ALGORITHM
CHAPTER 18: Riemann Surface
18.1. RIEMANN SURFACE
18.2. MEROMORPHIC DIFFERENTIAL
18.3. RIEMANN-ROCH THEOREM
18.4. ABEL-JACOBIAN THEOREM
CHAPTER 19: Conformal Mapping
19.1. TOPOLOGICAL QUADRILATERAL
19.2. TOPOLOGICAL ANNULUS
19.3. RIEMANN MAPPING FOR TOPOLOGICAL DISK
19.4. TOPOLOGICAL POLY-ANNULUS SLIT MAP
19.5. KOEBE’S ITERATION FOR POLY ANNULUS
19.6. TOPOLOGICAL TORUS
CHAPTER 20: Discrete Surface Curvature Flows
20.1. YAMABE EQUATION
20.2. SURFACE RICCI FLOW
20.3. DISCRETE SURFACE
20.4. DISCRETE SURFACE YAMABE FLOW
20.5. TOPOLOGICAL QUADRILATERAL
20.6. TOPOLOGICAL ANNULUS
20.7. TOPOLOGICAL POLY-ANNULUS
20.8. TOPOLOGICAL TORUS
CHAPTER 21: Mesh Generation Based on Abel-Jacobi Theorem
21.1. QUAD-MESHES AND MEROMORPHIC QUARTIC FORMS
21.2. METRICS WITH SPECIAL HOLONOMIES
21.3. MESH GENERATION
SECTION III: Appendices
APPENDIX A: Alexandrov Curvature
A.1. ALEXANDROV CURVATURE
A.2. ALEXANDROV CURVATURE VS. WALD CURVATURE
A.3. RINOW CURVATURE
APPENDIX B: Thick Triangulations Revisited
APPENDIX C: The Gromov-Hausdorff Distance
Bibliography
Index
