Semi-Riemannian Geometry

Chapter 13 Examples of Regular Surfaces

This chapter provides worked examples of graphs of functions in and , and surfaces of revolution in . The details of computations, which are not included, are based on formulas appearing in Theorems 12.10.1–12.10.4. In this chapter, we identify ℝ² with the xy ‐plane in ℝ³ . See Example 12.8.6 for a summary of Gauss curvatures as well as related comments.

Each of the regular surfaces to be considered can be parametrized as either the graph of a function or a surface of revolution. The former approach has the advantage that the regular surface can be depicted literally as a graph in ℝ³ . On the other hand, when symmetries are present, the surface of revolution parametrization can be quite revealing and computationally convenient. The choice of parametrization made here is somewhat arbitrary. There is a small issue that differentiates the two computational methods. Parameterizing a regular surface as a surface of revolution leaves out certain points compared with the corresponding parametrization as the graph of a function; more specifically, with the former approach, part of a longitude curve is “missing”. Since we are interested exclusively ...