{"id":31187,"date":"2016-10-07T07:17:26","date_gmt":"2016-10-07T12:17:26","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=31187"},"modified":"2016-10-11T12:56:09","modified_gmt":"2016-10-11T17:56:09","slug":"manifold-article-16","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/07\/manifold-article-16\/","title":{"rendered":"What is a manifold? Yet another article! (1\/6)"},"content":{"rendered":"

Why (yet) another article?<\/strong><\/p>\n

There are competing theories\u00a0online about possible interpretations of John von Neumann’s quote, but\u00a0manifolds\u00a0<\/em>are definitely some\u00a0mathematics that “you don’t understand … you just get used to them,” — at least for a while.<\/p>\n

In a series of posts reflecting on my own experience, I will try to motivate the conceptualization of manifolds, and the\u00a0implications such an abstraction has\/had on our understanding of, basically, shapes. I hope to point to some beautiful geometry\u00a0in low dimensions that you may have passed by too quickly to take notice of.<\/p>\n

I must underline<\/span> the subjective nature of my articles, and that by no means are they meant\u00a0to narrate the history of the subject, or depict a current fashion in the community. This\u00a0is simply\u00a0“another article.”<\/p>\n

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.<\/p>\n

 <\/p>\n

The Question<\/strong><\/p>\n

\"curve-through-space\"

Diagram by Behnam Esmayli<\/p><\/div>\n

<\/p>\n

What We Know from Calculus<\/strong><\/p>\n

From calculus\u00a0we fix a Cartesian coordinate system for three-space and then parameterize our curve\u00a0by a map from a subset of \"\mathbb{R}\",<\/p>\n

\"\gamma:<\/p>\n

If \"\gamma is a smooth parameterization, then\u00a0the length of the object between two of its points\u00a0\"P=\gamma \u00a0and\u00a0\"Q=\gamma is given by<\/p>\n

\"\int<\/p>\n

The curvature is given by<\/p>\n

\"k(t)=\dfrac{|\gamma<\/p>\n

and the torsion is given by<\/p>\n

\"\tau<\/p>\n

where differentiation is taken coordinate-wise.<\/p>\n

Notice that the integral above involves\u00a0only the<\/em> length <\/em>of \"\gamma. Thus, even if we\u00a0don’t have \"\gamma\"\u00a0itself , but only some \"\ell, we will again\u00a0be able to measure the length of any segment of \u00a0our curve.<\/p>\n

This observation invites\u00a0a search for a representation of our curves independent of the three-space.<\/p>\n

 <\/p>\n

The Answer to the Question<\/strong><\/p>\n

In a search for a 1-dimensional embodiment of a curve, the following\u00a0theorem is the best we could hope for.<\/p>\n

Theorem:<\/strong> Given differentiable real-valued functions \"\ell, \"k(t)>0\" and \"\tau, with \"t\" ranging in an interval \"I\", there is a curve \"\gamma with \"k\" being its curvature and \"\tau\" being its torsion and \"\int giving its length between points. Moreover<\/strong>, any other curve with this description is obtained by changing our origin and rotating the axes rigidly. (See Chapter 1 of\u00a0do Carmo’s book\u00a0Differential Geometry of Curves and Surfaces.<\/em>)<\/p>\n

Thanks to this theorem, an interval \"I\" along with the following set of data\u00a0can be thought of as a curve in \"\mathbb{R}^3\":<\/p>\n