Semi-Riemannian Geometry

Chapter 11 Curves and Regular Surfaces in ℝ³

In earlier discussions, the set ℝ ^m appeared in a variety of contexts: as a vector space (also denoted by ℝ ^m ), an inner product space (ℝ ^m , ), a normed vector space (ℝ ^m , ||·||), a metric space (ℝ ^m , ), and a topological space (ℝ ^m , ). Section 9.4 outlines the logical connections between these spaces. Looking back at Chapter 10, it would have been more precise, although cumbersome, to use the notation (ℝ ^m , , ||·||, , ), or at least (ℝ ^m , ), instead of simply ℝ ^m when discussing Euclidean derivatives and integrals. In this chapter, we are concerned with many of the same concepts considered in Chapter 10, but this time exclusively for M = 3. We use the notation ℝ³ in the preceding generic manner, allowing the structures ...