In studying spaces, we considered the locally Euclidean structure of topological manifolds as defining a subset of spaces that were “nicer,” meeting the minimum requirements of our idea of a geometrical “shape” such as integral dimension. By slightly narrowing our consideration to differentiable manifolds, we can essentially graft calculus onto our “rubber sheet.” The constructions of coordinates and tangent vectors enable us to define a family of derivatives associated with the concept of how vector fields change on the manifold. The challenge is in defining all these objects without an ambient space, which our intuitive picture normally depends upon.
| Δ Note that a differentiable manifold includes no concept of length or distance (a metric), and no structure that allows tangent vectors at different points to be compared or related to each other (a connection). It is important to remember that nothing in this chapter depends upon these two extra structures. |
When dealing with manifolds, there are two main approaches one can take: express everything in terms of coordinates, or strive to express everything in a coordinate-free fashion. In keeping with this book’s attempt to focus on concepts rather than calculations, we will take the latter approach, but will take pains to carefully express fundamental concepts in terms of coordinates in order to derive a picture of what these coordinate-free tools do. Facility in moving back and forth between these two views is a worthwhile goal, best accomplished by combining the material here with a standard text on differential geometry or general relativity.