A course in differential geometry

Chapter 0 Calculus in Euclidean Space 0.1 Euclidean Space 0.2 The Topology of Euclidean Space 0.3 Differentiation in Rn 0.4 Tangent Space 0.5 Local Behavior of Differentiable Functions (Injective and Surjective Functions) Chapter 1 Curves 1.1 Definitions 1.2 The Frenet Frame 1.3 The Frenet Equations 1.4 Plane Curves; Local Theory 1.5 Space Curves 1.6 Exercises Chapter 2 Plane Curves: Global Theory 2.1 The Rotation Number 2.2 The Umlaufsatz 2.3 Convex Curves Chapter 3 Surfaces: Local Theory 3.1 Definitions 3.2 The First Fundamental Form 3.3 The Second Fundamental Form 3.4 Curves on Surfaces 3.5 Principal Curvature, Gauss Curvature, and Mean Curvature 3.6 Normal Form for a Surface, Special Coordinates 3.7 Special Surfaces, Developable Surfaces 3.8 The Gauss and Codazzi-Mainardi Equations 3.9 Exercises and Some Further Results Chapter 4 Intrinsic Geometry of Surfaces: Local Theory 4.1 Vector Fields and Covariant Differentiation 4.2 Parallel Translation 4.3 Geodesics 4.4 Surfaces of Constant Curvature 4.5 Examples and Exercises Chapter 5 Two-dimensional Riemannian Genometry Chapter 6 The Global Geometry of Surfaces References Index Index of Symbols