| 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 |
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