Differential Geometry

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ISBN: 9780135699638 | 0135699630
Cover: Hardcover
Copyright: 8/1/1997

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Appropriate for introductory undergraduate courses in Differential Geometry with a prerequisite of multivariable calculus and linear algebra courses. This is the only text that introduces differential geometry by combining an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach. Text has running theme of intrinsic vs. extrinsic ways of looking at curves and surfaces. Starting with basic geometric ideas and proceeding to the analytic and algebraic formalisms, this text provides a common and accessible foundation on which all of the various formalisms of differential geometry can be based and from which they can be assessed.

Preface

(4)

Useful Supplements

xvii

(1)

Acknowledgements

xvii

(2)

How to Use This Book

xix

Chapter 1 Surfaces and Straightness

(36)

PROBLEM 1.1. When Do You Call a Line Straight?

(2)

How Do You Construct a Straight Line?

(3)

Local (and Infinitesimal) Straightness

(2)

PROBLEM 1.2. Intrinsic Straight Lines on Cylinders

(4)

PROBLEM 1.3. Geodesics on Cones

(4)

Is "Shortest" Always "Straight"?

(2)

Locally Isometric Surfaces

(1)

Local Coordinates for Cylinders and Cones

(3)

PROBLEM 1.4. Geodesics in Local Coordinates

(2)

PROBLEM 1.5. What Is Straight on a Sphere?

(1)

Intrinsic Curvature on a Sphere

(2)

Local Coordinates on a Sphere

(1)

PROBLEM 1.6. Strakes, Augers, and Helicoids

(2)

PROBLEM 1.7. Surfaces of Revolution

(1)

PROBLEM 1.8. Hyperbolic Plane

(3)

PROBLEM 1.9. Surface as Graph of a Function z=f(x,y)

(3)

Chapter 2 Extrinsic Curves

(24)

Introduction

(1)

PROBLEM 2.1. Give Examples of F.O.V.'s

(2)

Archimedian Property

(1)

Vectors and Affine Linear Space

(1)

PROBLEM 2.2. Smoothness and Tangent Directions

(7)

PROBLEM 2.3. Curvature of a Curve in Space

(2)

Curvature of the Graph of a Function

(2)

PROBLEM 2.4. Osculating Circle

(3)

PROBLEM 2.5. Strakes

(1)

PROBLEM 2.6. When a Curve Does Not Lie in a Plane

(5)

Chapter 3 Extrinsic Descriptions of Intrinsic Curvature

(12)

PROBLEM 3.1. Smooth Surfaces and Tangent Planes

(2)

PROBLEM 3.2. Extrinsic Curvature: Geodesics on Sphere

(1)

PROBLEM 3.3. Intrinsic Curvature: Curves on Sphere

(2)

Intrinsic (Geodesic) Curvature

(2)

PROBLEM 3.4. Geodesics on Surfaces--the Ribbon Test

(1)

Ruled Surfaces and the Converse of the Ribbon Test

(3)

Chapter 4 Tangent Space, Metric, Directional Derivative

(30)

PROBLEM 4.1. The Tangent Space

(3)

PROBLEM 4.2. Mean Value Theorem: Curves, Surfaces

(1)

Natural Parametrizations of Curves

(2)

PROBLEM 4.3. Riemannian Metric

(3)

Riemannian Metric in Local Coordinates on a Sphere

(1)

Riemannian Metric in Local Coordinates on a Strake

(2)

PROBLEM 4.4. Vectors in Extrinsic Local Coordinates

(2)

PROBLEM 4.5. Measuring Using the Riemannian Metric

(2)

Directional Derivatives

(3)

Directional Derivative in Local Coordinates

(1)

PROBLEM 4.6. Differentiating a Metric

(1)

PROBLEM 4.7. Expressing Normal Curvature

(3)

Geodesic Local Coordinates

(1)

PROBLEM 4.8. Differential Operator

(2)

PROBLEM 4.9. Metric in Geodesic Coordinates

100

(3)

Chapter 5 Area, Parallel Transport, Intrinsic Curvature

103

(16)

PROBLEM 5.1. The Area of a Triangle on a Sphere

103

(1)

Introducing Parallel Transport

104

(2)

The Holonomy of a Small Geodesic Triangle

106

(2)

PROBLEM 5.2. Dissection of Polygons into Triangles

108

(1)

PROBLEM 5.3. Gauss-Bonnet for Polygons on a Sphere

109

(2)

PROBLEM 5.4. Parallel Fields and Intrinsic Curvature

111

(4)

PROBLEM 5.5. Holonomy on Surfaces

115

(1)

PROBLEM 5.6. Holonomy Explains Foucault's Pendulum

116

(1)

PROBLEM 5.7. Intrinsic Curvature of a Surface

117

(2)

Chapter 6 Gaussian Curvature Extrinsically Defined

119

(16)

Pep Talk to the Reader

119

(1)

PROBLEM 6.1. Gaussian Curvature, Extrinsic Definition

120

(3)

PROBLEM 6.2. Second Fundamental Form

123

(3)

PROBLEM 6.3. The Gauss Map

126

(2)

PROBLEM 6.4. Gauss-Bonnet and Intrinsic Curvature

128

(1)

PROBLEM 6.5. Matrix of the Second Fundamental Form

129

(2)

PROBLEM 6.6. Mean Curvature and Minimal Surfaces

131

(2)

Celebration of Our Hard Work

133

(2)

Chapter 7 Applications of Gaussian Curvature

135

(16)

PROBLEM 7.1. Gaussian Curvature in Local Coordinates

135

(5)

PROBLEM 7.2. Curvature on Sphere, Strake, Catenoid

140

(1)

PROBLEM 7.3. Circles, Polar Coordinates, and Curvature

141

(1)

PROBLEM 7.4. Exponential Map and Shortest Is Straight

142

(5)

PROBLEM 7.5. Ruled Surfaces and Ribbons

147

(1)

PROBLEM 7.6. Surfaces with Constant Curvature

148

(1)

PROBLEM 7.7. Curvature of the Hyperbolic Plane

149

(2)

Chapter 8 Intrinsic Local Descriptions and Manifolds

151

(24)

PROBLEM 8.1. Covariant Derivative and Connection

151

(3)

PROBLEM 8.2. Manifolds--Intrinsic and Extrinsic

154

(7)

PROBLEM 8.3. Christoffel Symbols, Intrinsic Descriptions

161

(3)

PROBLEM 8.4. Intrinsic Curvature and Geodesics

164

(1)

PROBLEM 8.5. Lie Brackets, Coordinate Vector Fields

165

(2)

PROBLEM 8.6. Riemann Curvature Tensors

167

(4)

Calculation of Curvature Tensors in Local Coordinates

171

(2)

PROBLEM 8.7. Intrinsic Calculations in Examples

173

(2)

Appendix A Linear Algebra--a Geometric Point of View

175

(26)

A.0. Where Do We Start?

175

(1)

A.1. Geometric Affine Spaces

176

(4)

A.2. Vector Spaces

180

(2)

A.3. Inner Product--Lengths and Angles

182

(3)

A.4. Linear Transformations and Operators

185

(7)

A.5. Areas, Cross Products, and Triple Products

192

(2)

A.6. Volumes, Orientation, and Determinants

194

(3)

A.7. Eigenvalues and Eigenvectors

197

(1)

A.8. Introduction to Tensors

198

(3)

Appendix B Analysis from a Geometric Point of View

201

(4)

B.1. Smooth Functions

201

(1)

B.2. Invariance of Domain

202

(1)

B.3. Inverse Function Theorem

202

(1)

B.4. Implicit Function Theorem

203

(2)

Appendix C Computer Scripts

205

(22)

Standard Functions

205

(3)

Computer Exercise 1.6: Strake

208

(1)

Computer Exercise 1.7: Surfaces of Revolution

209

(1)

Computer Exercise 1.9: Surfaces as Graph of a Function

210

(1)

Computer Exercise 2.2: Tangent Vectors to Curves

210

(1)

Computer Exercise 2.3: Curvature and Tangent Vectors

211

(1)

Computer Exercise 2.4a: Osculating Planes

212

(1)

Computer Exercise 2.4b: Osculating Circles

213

(1)

Computer Exercise 2.6: Frenet Frame

214

(1)

Computer Exercise 3.1: Tangent Planes to Surfaces

215

(1)

Computer Exercise 3.2a: Curves on a Surface

216

(1)

Computer Exercise 3.2b: Extrinsic Curvature Vectors

217

(2)

Computer Exercise 3.3: The Three Curvature Vectors

219

(2)

Computer Exercise 3.4: Ruled Surfaces

221

(1)

Computer Exercise 5.2: Non-dissectable Polyhedron

222

(1)

Computer Exercise 5.5: Sign of (Gaussian) Curvature

223

(1)

Computer Exercise 6.1: Multiple Principle Directions

223

(1)

Computer Exercise 6.3: Gauss Map

224

(2)

Computer Exercise 6.6: Helicoid to Catenoid

226

(1)

Bibliography

227

(16)

A. Ancient Texts

227

(1)

AD. Art and Design

227

(1)

An. Analysis

228

(1)

DG. Differential Geometry

228

(2)

DS. Dimensions and Scale

230

(1)

GC. Geometry in Different Cultures

231

(1)

Hi. History

232

(1)

LA. Linear Algebra and Geometry

233

(1)

Mi. Minimal Surfaces

234

(1)

MP. Models, Polyhedra

234

(1)

Na. Nature

235

(1)

NE. Non-Euclidean Geometries (Mostly Hyperbolic)

235

(1)

Ph. Philosophy

236

(1)

RN. Real Numbers

236

(1)

SP. Spherical and Projective Geometry

237

(1)

SG. Symmetry and Groups

238

(1)

SE. Surveys and General Expositions

238

(2)

Tp. Topology

240

(1)

Tx. Geometry Texts

240

(1)

Z. Miscellaneous

241

(2)

Notation Index

243

(2)

Subject Index

245

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Differential Geometry

Summary

Table of Contents

Supplemental Materials