7 - Manifolds

Summary

In this chapter we want to generalize the concepts introduced so far to a class of spaces, the differentiable manifolds, which in general cannot be considered as open subsets of ℝⁿ. An example of such a space is the 2-sphere, e.g. the surface of the Earth: The globe as a whole cannot be mapped homeomorphically onto an open subset of the Euclidean plane. Instead, one has to be content with local mappings collected to make an atlas. If it is indicated how points in the overlap of different maps are to be identified, e.g. by means of parallels and meridians, one gets nevertheless a complete description of the surface of the Earth. Note that there is a lot of freedom connected with these local mappings: One can choose among a variety of map projections.

In chapter 2 we already encountered surfaces which could not be described by a single parameter representation. Since we used them only as domains of integration, the details of the gluing procedure were irrelevant. In this chapter, however, we shall be more careful about the transition from one parameter representation to another.

Differentiable manifolds

An n-dimensional manifold is a topological space M which locally looks like an open subset of ℝⁿ.