Roughly speaking, geometry begins with the measurement of distances and angles. We shall see that the geometry of Euclidean space can be derived from the dot product, the natural inner product on Euclidean space.
Much of this chapter is devoted to the geometry of curves in R3. We emphasize this topic not only because of its intrinsic importance, but also because the basic method used to investigate curves has proved effective throughout differential geometry. A curve in R3 is studied by assigning at each point a certain frame-- that is, set of three orthogonal unit vectors. The rate of change of these vectors along the curve is then expressed in terms of the vectors themselves by the celebrated Frenet formulas (Thm.3.2). In a real sense, the theory of curves in R3 is merely a corollary of these fundamental formulas.
Later on we shall use this "method of moving frames" to study a surface in R3. The general idea is to think of a surface as a kind of two-dimensional curve and follow the Frenet approach as closely as possible.To carry out this scheme we need the generalization (Thm.7.2) of the Frenet formulas devised by E. Cartan. It was Cartan who, in the early 1900s, first realized the full power of this method not only in differential geometry, but also in a variety of related fields.