Classical and Discrete Differential Geometry. Theory, Applications and Algorithms

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