We continue from Part One of this journey our attempt to illustrate how one can start with calculus and arrive at the definition of a 1-dimensional manifold.

In the previous segment, we concluded with the fact that a curve in  may be viewed as an interval I together with the following set of data:

•  A real-valued function , which will help measure the length,
• A real-valued function , which will help measure the curvature, and,
• A real-valued function , which will help measure the torsion.

In this segment we will see some examples and discuss possible constructions that this allows us.

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Why (yet) another article?

There are competing theories online about possible interpretations of John von Neumann’s quote, but manifolds are definitely some mathematics that “you don’t understand … you just get used to them,” — at least for a while.

In a series of posts reflecting on my own experience, I will try to motivate the conceptualization of manifolds, and the implications such an abstraction has/had on our understanding of, basically, shapes. I hope to point to some beautiful geometry in low dimensions that you may have passed by too quickly to take notice of.

I must underline the subjective nature of my articles, and that by no means are they meant to narrate the history of the subject, or depict a current fashion in the community. This is simply “another article.”

The first three articles will be dedicated to converting the conventional calculus of curves into manifold language. We will see that a curve can be replaced by an interval endowed with some structure. This will pave the way for an exposition of the theory of surfaces in subsequent articles. The reason for such an extended sequence is to include as much detail and as many examples as possible.

The Question

Diagram by Behnam Esmayli

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