# Abstract Nonsense

## Closed Curves

Point of Post: In this post we discuss the notion of closed curves, proving such facts such as the existence of a minimal period.

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Motivation

Most times when people think of honest to god curves they often think of curves that ‘closed up’ in the sense that eventually the curve curves back over itself. A prime example of this is the classic unit circle $\mathbb{S}^1$. It’s not hard to see that such plane curves are described parametrically by periodic functions of the input variable $t$. One may then begin to ask, is there a ‘fundamental period’ for such a parametrized curve? By this, I mean that for a given curve with one number $T$ with the property that $\gamma(t+T)=\gamma(t)$ for all $t\in I$ then clearly the number $2T$ has the same property. In particular, $nT$ for every $n\in\mathbb{Z}$ also has this property. One may then begin to ask whether there is a ‘smallest’ positive such period. The answer turns out to be yes for a very cool reason.

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Closed Curves

For a smooth curve $\gamma:\mathbb{R}\to\mathbb{R}^n$ we say that $\gamma$ is $T$periodic if $\gamma(t+T)=\gamma(t)$ for every $t\in T$. We say that $T$ is a period of $\gamma$ and denote the set of all periods of $\gamma$ by $G_\gamma$. This is suggestive notation for a reason. Namely:

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Theorem: Let $\gamma:\mathbb{R}\to\mathbb{R}^n$ be a smooth curve. Then, $G_\gamma$ is a subgroup of $\left(\mathbb{R},+\right)$.

Proof: Suppose that $T,S\in G_\gamma$ then $\gamma(t+(T-S))=\gamma((t+(T-S))+S)=\gamma(t+T)=\gamma(t)$ for all $t\in \mathbb{R}$ and so $T-S\in G_\gamma$. $\blacksquare$

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What we claim though is that $G_\gamma$ is a discrete subspace of $\mathbb{R}$ if $\gamma$ is non-constant. Indeed:

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Theorem: Let $\gamma:\mathbb{R}\to\mathbb{R}^n$ be a smooth curve. Then, if $\gamma$ is non-constant then $G_\gamma$ is a discrete subgroup of $\mathbb{R}$.

Proof: Since $G_\gamma$ is a subgroup we can transfer the problem back to the origin. To be more specific, it suffices to show that $B_\varepsilon( 0 )\cap G_\gamma=\{0\}$ for some $\varepsilon>0$. Indeed, suppose that for every $\varepsilon>0$ there exists some element $T\in G_\gamma$ with $|T|<\varepsilon$. Fix any point $x\in\mathbb{R}$. We note then that, by definition, we have that

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$\displaystyle D_\gamma(x)=\lim_{t\to 0}\frac{f(x+t)-f(x)}{t}$

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But, we claim that that the limit on the right hand side is zero. Indeed, since the limit exists we know that if $\{x_n\}_{n\in\mathbb{N}}$ is any sequence of non-zero real numbers converging to zero that

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$\displaystyle \lim_{t\to0}\frac{f(x+t)-f(x)}{t}=\lim_{n\to\infty}\frac{f(x+x_n)-f(x)}{x_n}$

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Now, since $B_\varepsilon(0)\cap G_\gamma$ contains infinitely many points of $G_\gamma$ for every $\varepsilon>0$ we may, in particular, choose a sequence $\{g_n\}\in G_\gamma$ with $g_n\to 0$. We have then that

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$\displaystyle D_\gamma(x)=\lim_{t\to0}\frac{f(x+t)-f(x)}{t}=\lim_{n\to\infty}\frac{f(x+g_n)-f(x)}{g_n}=0$

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But, this implies that $D_\gamma(x)=0$ for all $x\in\mathbb{R}$ and so by a common consequence of the mean value theorem that $\gamma$ is constant. $\blacksquare$

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But, consider the following theorem:

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Theorem: Let $G$ be a non-trivial proper disccrete subgroup of $\mathbb{R}$. Then, $G=x\mathbb{Z}$ for some $x\in\mathbb{R}$.

Proof: We first note that $G$ is closed (this is obvious). So, now choose $x\in G-\{0\}$. We may clearly assume without loss of generality that $x$ is positive and so $G\cap (0,\infty)$ is non-empty and bounded below. We then have that $x=\inf G\cap(0,\infty)$ exists and since $G$ is bounded away from $0$ we have that $x>0$. Now, since $G$ is closed we must have that $x\in G$. We claim now that $G=x\mathbb{Z}$. Indeed, let $y\in G$ be arbitrary. We may evidently write $y=mx+r$ where $m\in\mathbb{Z}$ and $r\in[0,m)$.  Now, $mx,y\in G$ and so $r\in G$ but since $r\in[0,m)\cap G$ we have by construction that $r=0$ and so $y=mx$. Since $y$ we may conclude that $G=x\mathbb{Z}$. $\blacksquare$

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Integral to this proof was that there is a unique smallest member of $G\cap (0,\infty)$. In particular, if $\gamma:\mathbb{R}\to\mathbb{R}^n$ is smooth and non-constant we have that $G_\gamma$ has a unique smallest positive member $T$. We call this the fundamental period of $\gamma$.

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Truthfully there was an easier way to prove that a fundamental period existed. Indeed, there was no need to prove that $G_\gamma$ is, in fact, a subgroup. Indeed, it’s fairly easy to see that the map $f:\mathbb{R}\to C(\mathbb{R})$ given by $T\mapsto \gamma(t+T)-\gamma(t)$ is a continuous map when equipping $C(\mathbb{R})$ with the infinity norm and since $f^{-1}(\{0\})=G_\gamma$ we’d have that $G_\gamma$ is a closed subset of $\mathbb{R}$ and so the same analysis as above would show that it has a unique smallest positive member. That said, figuring out that $G_\gamma$ is a discrete subgroup of $\mathbb{R}$ of the form $x\mathbb{Z}$ suggests that we could perhaps consider the obvious map $\gamma:\mathbb{R}/G_\gamma\to\mathbb{R}^n$ eliminating the periodicity. But, it’s easy to see using the first topological group isomorphism theorem that $G/x\mathbb{Z}$ is isomorphic as a topological group to $\mathbb{S}^1$ thought of as a multiplicative group. Thus, closed curves could really be thought of as maps $\gamma:\mathbb{S}^1\to\mathbb{R}^n$ so that all ‘curves’ representable as a diffeomorphic embedding $\mathbb{S}_1\hookrightarrow\mathbb{R}^n$ which perhaps justifies the seeming lack of imagination when a teacher draws a closed curve that looks like a deformed circle.

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As a last point about closed curves, it’s clear geometrically that if $\gamma:\mathbb{R}\to\mathbb{R}^n$ is a closed curve with fundamental period $T$ then the ‘length’ of the entire curve should be the arc length from $t$ to $t+T$ for any $t\in\mathbb{R}$ since this just measure the arc length of one ‘go around’. So, accordingly we define the length $\ell(\gamma)$ of $\gamma$ to be equal to

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$\displaystyle \int_0^T \|\gamma'(t)\|\; dt$

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and so it’s clear that if $\gamma$ is unit speed then $\ell(\gamma)=T$. Moreover, it’s clear that any unit speed reparamaterization of a closed regular curve is closed, and so since we may, up to reparamaterization, always assume a regular curve is unit speed we can with these observations assume any closed regular curve is closed unit speed with fundamental period equal to its length.

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References:

1. Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

2.  Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.