Semi-Riemannian Geometry

Part III Smooth Manifolds and Semi‐Riemannian Manifolds

In Part II, we covered a range of topics on curves and surfaces. Despite the sometimes abstract nature of the mathematics, the fact that we were dealing with 1‐ and 2‐dimensional geometric objects residing in 3‐dimensional space made the undertaking relatively concrete. The restrictions on dimensions imposed in Part II reflect the classical nature of the study of curves and surfaces. With relatively little effort, it is possible to generalize results to (m − 1)‐dimensional “surfaces” in m ‐dimensional space. However, this does not resolve an important inherent limitation of this approach—the need for an ambient space. Perhaps the most important application of semi‐Riemannian geometry, and the one that historically motivated this area of mathematics, is Einstein's general theory of relativity. According to this cosmological construct, the universe we inhabit has precisely four dimensions; there is no 5‐dimensional ambient space in which our universe resides. Of course, it is possible to fashion one mathematically, but that would not reflect physical reality. In order to model the general theory of relativity, we need a mathematical description of “surfaces” that dispenses with ambient space altogether.

Let us consider where ℝ³ entered (and did not enter) into our discussion of surfaces in an effort to see whether we can circumvent the role of an ambient space. Recall that in the first instance, a surface is defined ...