Point of Post: In this post we discuss the notion of closed curves, proving such facts such as the existence of a minimal period.
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 . It’s not hard to see that such plane curves are described parametrically by periodic functions of the input variable . 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 with the property that for all then clearly the number has the same property. In particular, for every 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.
For a smooth curve we say that is –periodic if for every . We say that is a period of and denote the set of all periods of by . This is suggestive notation for a reason. Namely:
Theorem: Let be a smooth curve. Then, is a subgroup of .
Proof: Suppose that then for all and so .
What we claim though is that is a discrete subspace of if is non-constant. Indeed:
Theorem: Let be a smooth curve. Then, if is non-constant then is a discrete subgroup of .
Proof: Since is a subgroup we can transfer the problem back to the origin. To be more specific, it suffices to show that for some . Indeed, suppose that for every there exists some element with . Fix any point . We note then that, by definition, we have that
But, we claim that that the limit on the right hand side is zero. Indeed, since the limit exists we know that if is any sequence of non-zero real numbers converging to zero that
Now, since contains infinitely many points of for every we may, in particular, choose a sequence with . We have then that
But, this implies that for all and so by a common consequence of the mean value theorem that is constant.
But, consider the following theorem:
Theorem: Let be a non-trivial proper disccrete subgroup of . Then, for some .
Proof: We first note that is closed (this is obvious). So, now choose . We may clearly assume without loss of generality that is positive and so is non-empty and bounded below. We then have that exists and since is bounded away from we have that . Now, since is closed we must have that . We claim now that . Indeed, let be arbitrary. We may evidently write where and . Now, and so but since we have by construction that and so . Since we may conclude that .
Integral to this proof was that there is a unique smallest member of . In particular, if is smooth and non-constant we have that has a unique smallest positive member . We call this the fundamental period of .
Truthfully there was an easier way to prove that a fundamental period existed. Indeed, there was no need to prove that is, in fact, a subgroup. Indeed, it’s fairly easy to see that the map given by is a continuous map when equipping with the infinity norm and since we’d have that is a closed subset of and so the same analysis as above would show that it has a unique smallest positive member. That said, figuring out that is a discrete subgroup of of the form suggests that we could perhaps consider the obvious map eliminating the periodicity. But, it’s easy to see using the first topological group isomorphism theorem that is isomorphic as a topological group to thought of as a multiplicative group. Thus, closed curves could really be thought of as maps so that all ‘curves’ representable as a diffeomorphic embedding which perhaps justifies the seeming lack of imagination when a teacher draws a closed curve that looks like a deformed circle.
As a last point about closed curves, it’s clear geometrically that if is a closed curve with fundamental period then the ‘length’ of the entire curve should be the arc length from to for any since this just measure the arc length of one ‘go around’. So, accordingly we define the length of to be equal to
and so it’s clear that if is unit speed then . 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.
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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