Differential Geometry: A Geometric Introduction

Abstract

Preface. How to Use This Book. 1. Surfaces and Straightness. When Do You Call a Straight Line? How Do You Construct a Straight Line? Local (and Infinitesimal) Straightness. Intrinsic Straight Lines on Cylinders. Geodesics on Cones. Is Always Straight? Locally Isometric Surfaces. Local Coordinates for Cylinders and Cones. Geodesics in Local Coordinates. What Is Straight on a Sphere? Intrinsic Curvature on a Sphere. Local Coordinates on a Sphere. Strakes, Augers, and Helicoids. Surfaces of Revolution. Hyperbolic Plane. Surface as Graph of a Function z=f(x,y) 2. Extrinsic Curves. Introduction. Give Examples of F.O.V.'s. Archimedian Property. Vectors and Affine Linear Space. Smoothness and Tangent Directions. Curvature of a Curve in Space. Curvature of the Graph of a Function. Osculating Circle. Strakes. When a Curve Does Not Lie in a Plane. 3. Extrinsic Descriptions of Intrinsic Curvature. Smooth Surfaces and Tangent Planes. Extrinsic Curvature - Geodesics on Sphere. Intrinsic Curvature - Curves on Sphere. Intrinsic (Geodesic) Curvature. Geodesics on Surfaces--the Ribbon Test. Ruled Surfaces and the Converse of the Ribbon Test. 4. Tangent Space, Metric, Directional Derivative. The Tangent Space. Mean Value Theorem: Curves, Surfaces. Natural Parametrizations of Curves. Riemannian Metric. Riemannian Metric in Local Coordinates on a Sphere. Riemannian Metric in Local Coordinates on a Strake. Vectors in Extrinsic Local Coordinates. Measuring Using the Riemannian Metric. Directional Derivatives. Directional Derivative in Local Coordinates. Differentiating a Metric. Expressing Normal Curvature. Geodesic Local Coordinates. Differential Operator. Metric in Geodesic Coordinates. 5. Area, Parallel Transport, Intrinsic Curvature. The Area of a Triangle on a Sphere. Introducing Parallel Transport. The Holonomy of a Small Geodesic Triangle. Dissection of Polygons into Triangles. Gauss-Bonnet for Polygons on a Sphere. Parallel Fields and Intrinsic Curvature. Holonomy on Surfaces. Holonomy Explains Foucault's Pendulum. Intrinsic Curvature of a Surface. 6. Gaussian Curvature Extrinsically Defined. Pep Talk to the Reader. Gaussian Curvature, Extrinsic Definition. Second Fundamental Form. The Gauss Map. Gauss-Bonnet and Intrinsic Curvature. Matrix of the Second Fundamental. Mean Curvature and Minimal Surfaces. Celebration of Our Hard Work. 7. Applications of Gaussian Curvature. Gaussian Curvature in Local Coordinates. Curvature on Sphere, Strake, Catenoid. Circles, Polar Coordinates, and Curvature. Exponential Map and Shortest Is Straight. Ruled Surfaces and Ribbons. Surfaces with Constant Curvature. Curvature of the Hyperbolic Plane. 8. Intrinsic Local Descriptions and Manifolds. Covariant Derivative and Connections. Manifolds--Intrinsic and Extrinsic. Christoffel Symbols, Intrinsic Descriptions. Intrinsic Curvature and Geodesics. Lie Brackets, Coordinate Vector Fields. Riemann Curvature Tensors. Calculation of Curvature Tensors in Local Coordinates. Intrinsic Calculations in Examples. Appendix A. Linear Algebra--a Geometric Point of View. Where Do We Start? Geometric Affine Spaces. Vector Spaces. Inner Product--Lengths and Angles. Linear Transformations and Operators. Areas, Cross Products, and Triple Products. Volumes, Orientation, and Determinants. Eigenvalues and Eigenvectors. Introduction to Tensors. Appendix B. Analysis from a Geometric Point of View. Smooth Functions. Invariance of Domain. Inverse Function Theorem. Implicit Function Theorem. Appendix C. Computer Scripts. Standard Functions. Strake. Surfaces of Revolution. Surfaces as Graph of a Function. Tangent Vectors to Curves. Curvature and Tangent Vectors. Osculating Planes. Osculating Circles. Frenet Frame. Tangent Planes to Surfaces. Curves on a Surface. Extrinisic Curvature Vectors. The Three Curvature Vectors. Ruled Surfaces. Non-dissectable Polyhedron. Sign of (Gaussian) Curvature. Mulitple Principle Directions. Gauss Map. Helicoid to Catenoid. Bibliography. Notation Index. Subject Index.