Manifolds

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…