Semi-Riemannian Geometry

Chapter 12 Curves and Regular Surfaces in

Chapter 11 was devoted to a discussion of curves and regular surfaces in ℝ³ . A regular surface was defined to be a subset of ℝ³ with certain properties specified in terms of the subspace topology, smooth maps, immersions, homeomorphisms, and so on. The fact that ℝ³ has an inner product (which gives rise to a norm, which in turn gives rise to a distance function, which in turn gives rise to a topology) was relegated to the background—present but largely unacknowledged. The topological and metric aspects of ℝ³ were central to our discussion of what it means for a regular surface to be “smooth”, and in that way the inner product (through the distance function) was involved.

In this chapter, we continue our discussion of regular surfaces, but this time endow each tangent plane with additional linear structure induced by the linear structure on ℝ³ . Specifically, we view ℝ³ as either Euclidean 3‐space, that is, or Minkowski 3‐space, that is, and give each tangent plane the corresponding inner product or Lorentz scalar product obtained by restriction. It must be stressed that this additional linear structure changes nothing regarding ...