Upon David Hoffman fell the difficult task of transforming the tightly constructed German text into one which would mesh well with the more relaxed format of the Graduate Texts in Mathematics series. There are some elaborations and several new figures have been added. I trust that the merits of the German edition have survived whereas at the same time the efforts of David helped to elucidate the general conception of the Course where we tried to put Geometry before Formalism without giving up mathematical rigour.

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