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

2.4 Exercises and Some Further Results

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 Geometry

5.1 Local Riemannian Geometry

5.2 The Tangent Bundle and the Exponential Map

5.3 Geodesic Polar Coordinates

5.4 Jacobi Fields

5.5 Manifolds

5.6 Differential Forms

5.7 Exercises and Some Further Results

Chapter 6 The Global Geometry of Surfaces

6.1 Surfaces in Euclidean Space

6.2 Ovaloids

6.3 The Gauss-Bonnet Theorem

6.4 Completeness

6.5 Conjugate Points and Curvature

6.6 Curvature and the Global Geometry of a Surface

6.7 Closed Geodesics and the Fundamental Group

6.8 Exercises and Some Further Results

References

Index

Index of Symbols