One immediate benefit of considering coordinate-free descriptions of geometric objects is that we may talk about “curves” that are not a priori embedded in . In other words, we don’t have to start with a subset of to be able to study 1-dimensional objects. There is already quite a nontrivial question we can ask: what curves can be embedded in a plane? The answer will be provided as a condition on and , and this description has the advantage of having nothing to do with or any other non-intrinsic data. Later we will talk about surfaces (2-dimensional manifolds) that do not live in 3-space, but rather in 4-space. Having an intrinsic way of seeing objects is liberating and opens up new possibilities.
A 1-dimensional manifold
It may also happen that the data are equivalent only on subsets of the intervals. For example, it could be the case that the data restricted to the first half of interval is equivalent to the data on the last third of interval . Then in this situation, we may “glue” together these overlapping compatible parts and get a longer curve that extends the one on . There will be no ambiguity in measurements over our new curve due to the equivalence of two sets of data in the intersecting parts. This possibility of patching together pieces while maintaining the structures is a fundamental part of the concept of a manifold.
Our definition will be of a Riemannian 1-manifold (because of the metric structures we have decided to keep). A Riemannian 1-manifold is a Hausdorff topological space such that each point in has an open neighborhood homeomorphic to an interval in , along with the set of data as above. If two open neighborhoods have a nonempty intersection, then we require that on the intersection the two localizations be equivalent in the sense of the previous section. There is also the condition of second countability: We like for our manifold to be covered by countably many of such neighborhoods.
Later in this series on manifolds, I will turn to surfaces which are 2-dimensional objects and re-interpret much of their calculus in the intrinsic language of differential geometry.
Let us know if you have any questions on manifolds in the comments below!