Abstract Nonsense

Crushing one theorem at a time

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 Tperiodic 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.

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September 23, 2011 - Posted by | Differential Geometry, Topological Groups | , , , , , ,

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