Euclidean Geometry

Abstract

This chapter begins with a standard elementary introduction to the theory of surfaces immersed in Euclidean space R ³, whose Riemannian metric is the standard dot product. Section 4.2 will be review for readers who have studied basic differential geometry of curves and surfaces in Euclidean space. Geometric intuition is used to construct Euclidean frames on a surface. Section 4.3 repeats the exposition, but this time following the frame reduction procedure outlined in Chapter 3 The classical existence and congruence theorems of Bonnet are stated and proved as consequences of the Cartan–Darboux Theorems. A section on tangent and curvature spheres provides needed background for Lie sphere geometry. The Gauss map helps tie together the formalism of Gauss and that of moving frames. We discuss special examples, such as surfaces of revolution, tubes about a space curve, inversions in a sphere, and parallel transforms of a given immersion. These constructions provide many valuable examples throughout the book. The latter two constructions introduce for the first time Möbius, respectively Lie sphere, transformations that are not Euclidean motions. The section on elasticae contains material needed in our introduction of the Willmore problems.