Semi-Riemannian Geometry

Chapter 8 Orientation

Orientation is concerned with “sidedness”, a concept that is intuitively obvious but surprisingly difficult to formulate in a mathematically rigorous fashion. In Section 8.1, we make a few observations based on concrete examples, and then present a preliminary definition of orientation for ℝ ^m . In Section 8.2, these ideas are developed into a computational framework for arbitrary vector spaces using our recently acquired knowledge of multicovectors.

8.1 Orientation of ℝ ^m

Let be a basis for ℝ ^m . Corresponding to each point in ℝ ^m is an m‐tuple of real numbers consisting of the components of the point with respect to . For m = 1, 2, 3, we can plot this m‐tuple on a rectangular coordinate system, with axes labeled h ₁,…,h _m . We use the geometry of this approach to motivate a definition of orientation.

Consider the bases ℰ = (e ₁, e ₂), (e ₂, − e ₁), and (e ₂, e ₁) for ℝ² , where ε is the standard basis. In Figures 8.1.1(a)–(c), these bases are depicted as the axes of rectangular coordinate systems. The configuration in Figure 8.1.1(a), which we call the standard configuration for ℝ² , is said to have a counterclockwise orientation because the 90° rotation taking e ₁ to e ₂ is in a counterclockwise direction. The configuration in Figure  ...