Differential Geometry of Curves and Surfaces

Contents
Preface
Acknowledgments
Authors
1. Plane Curves: Local Properties
	1.1 Parametrizations
	1.2 Position, Velocity, and Acceleration
	1.3 Curvature
	1.4 Osculating Circles, Evolutes, Involutes
	1.5 Natural Equations
2. Plane Curves: Global Properties
	2.1 Basic Properties
	2.2 Rotation Index
	2.3 Isoperimetric Inequality
	2.4 Curvature, Convexity, and the Four-Vertex Theorem
3. Curves in Space: Local Properties
	3.1 Definitions, Examples, and Differentiation
	3.2 Curvature, Torsion, and the Frenet Frame
	3.3 Osculating Plane and Osculating Sphere
	3.4 Natural Equations
4. Curves in Space: Global Properties
	4.1 Basic Properties
	4.2 Indicatrices and Total Curvature
	4.3 Knots and Links
		4.3.1 Knots
		4.3.2 Links
5. Regular Surfaces
	5.1 Parametrized Surfaces
	5.2 Tangent Planes; The Differential
		5.2.1 Tangent Planes
		5.2.2 the Differential
	5.3 Regular Surfaces
	5.4 Change of Coordinates; Orientability
		5.4.1 Change of Coordinates
		5.4.2 Change of Coordinates and the Tangent Plane
		5.4.3 Orientability
6. First and Second Fundamental Forms
	6.1 The First Fundamental Form
	6.2 Map Projections (Optional)
		6.2.1 Metric Properties of Maps of the Earth
		6.2.2 Azimuthal Projections
		6.2.3 Cylindrical Projections
		6.2.4 Coordinate Changes on the Sphere
	6.3 The Gauss Map
	6.4 The Second Fundamental Form
	6.5 Normal and Principal Curvatures
	6.6 Gaussian and Mean Curvatures
	6.7 Developable Surfaces; Minimal Surfaces
		6.7.1 Developable Surfaces
		6.7.2 Minimal Surfaces
7. Fundamental Equations of Surfaces
	7.1 Gauss’s Equations; Christoffel Symbols
	7.2 Codazzi Equations; Theorema Egregium
	7.3 Fundamental Theorem of Surface Theory
8. Gauss-Bonnet Theorem; Geodesics
	8.1 Curvatures and Torsion
		8.1.1 Natural Frames
		8.1.2 Normal Curvature
		8.1.3 Geodesic Curvature
		8.1.4 Geodesic Torsion
	8.2 Gauss-Bonnet Theorem, Local Form
	8.3 Gauss-Bonnet Theorem, Global Form
	8.4 Geodesics
	8.5 Geodesic Coordinates
		8.5.1 General Geodesic Coordinates
		8.5.2 Geodesic Polar Coordinates
	8.6 Applications to Plane, Spherical, and Elliptic Geometry
		8.6.1 Plane Geometry
		8.6.2 Spherical Geometry
		8.6.3 Elliptic Geometry
	8.7 Hyperbolic Geometry
		8.7.1 Synthetic Hyperbolic Geometry
		8.7.2 The Poincaré Upper Half-plane
		8.7.3 The Poincaré Disk
		8.7.4 The Pseudosphere Revisited
9. Curves and Surfaces in n-Dimensional Space
	9.1 Curves in n-Dimensional Euclidean Space
		9.1.1 Curvatures and the Frenet Frame
		9.1.2 Osculating Planes, Circles, and Spheres
		9.1.3 the Fundamental Theorem of Curves in ℝⁿ
	9.2 Surfaces in Euclidean n-Space
		9.2.1 Regular Surfaces in ℝⁿ
		9.2.2 Intrinsic Geometry for Surfaces
		9.2.3 Orientability
		9.2.4 The Gauss-Bonnet Theorem
A. Tensor Notation
	A.1 Tensor Notation
	A.1.1 Curvilinear Coordinate Systems
	A.1.2 Tensors: Definitions and Notation
	A.1.3 Operations on Tensors
	A.1.4 Examples
	A.1.5 Symmetries
	A.1.6 Numerical Tensors
Bibliography
[13]
[30]
Index
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