21.1 Closed curves
We add other definitions to those already seen in 415.
[1PB]Let \((X,d)\) be a metric space. Let \(I=[a,b]⊆ {\mathbb {R}}\) be a closed and bounded interval. Let \(𝛾:I\to X\) be a parametric curve.
we also say that the curve is simple and closed if \(𝛾(a)=𝛾(b)\) and \(𝛾\) is injective when restricted to \([a,b)\). 1
If \(X={\mathbb {R}}^ n\) and \(𝛾\) is class \(C^ 1\) and is closed, it is further assumed that \(𝛾'(a)=𝛾'(b)\).
- E418
[1PC] Consider the subsets of the plane described in the following figures 5: which can be the support of a simple curve? or a simple closed curve? or union of supports of two simple curves (possibly closed)? (Prove your claims.)
![\includegraphics[width=\textwidth ]{UUID/1/P/D/blob_zxx}](images/img-0009.png)
Figure 5 Sets for exercise 1 - E418
[1PF] Let \(𝛾:[0,1]\to X\) be a closed curve, show that it admits an extension \(\tilde𝛾:{\mathbb {R}}\to X\) continuous and periodic with period \(1\).
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[1PG] Let \(𝛾:[0,1]\to {\mathbb {R}}^ n\) be a closed \(C^ 1\) curve, show that it admits an extension \(\tilde𝛾:{\mathbb {R}}\to {\mathbb {R}}^ n\) periodic with period \(1\) and of class \(C^ 1\).
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[1PH] We will use the definitions and results of the Section 10.15, in particular 5.
Fix \(\tilde𝛾:ℝ→ X\) continuous and periodic (of period \(1\)); we can define the map \(\hat𝛾:S^ 1→ X\) through the relation
\[ \hat𝛾\Big( (\cos (t),\sin (t))\Big)=\tilde𝛾(t)~ ~ . \]Show that this is a good definition, and that \(\hat𝛾\) is continuous.
Use the exercise 3 to show that every closed simple arc, when viewed equivalently as a map \(\hat𝛾:S^ 1→ X\), is a homeomorphism with its image.
In the following we will use periodic maps to represent the closed curves.
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[1PK]Adapt the notion of equivalence 416 to the case of simple and closed arcs, but considering them as maps \(𝛾:{\mathbb {R}}→ X\) continuous and periodic (of period \(1\)); what hypotheses do we require from the maps \(𝜑:{\mathbb {R}}→{\mathbb {R}}\)?
Hidden solution: [UNACCESSIBLE UUID ’1PM’]
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[1PN]Prerequisites:416,2.Let \(𝛾,𝛿\) be closed curves, but seen as maps defined on \(ℝ\), continuous and periodic of period \(1\).
Let’s discuss a new relation: we write \(𝛾∼_ f𝛿\) if there is an increasing homeomorphism \(𝜑:ℝ→ℝ\) such that \(𝜑(t+1)=𝜑(t)+1\) for every \(t∈ℝ\), and for which \(𝛾=𝛿 ◦𝜑\)
Show that this is an equivalence relation.
Compare it with the relation \(∼\).
Hidden solution: [UNACCESSIBLE UUID ’1PP’]
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[1PQ]Prerequisites:417,3.Let \(𝛾,𝛿\) curves be closed and immersed, but seen as maps defined on \(ℝ\) and \(C^ 1\) and periodic. with periods \(1\).
Let’s see a new relation: you have \(𝛾≈_ f𝛿\) if there is an increasing diffeomorphism \(𝜑:ℝ→ℝ\) such that \(𝜑(t+1)=𝜑(t)+1\) for every \(t∈ℝ\) and for which \(𝛾=𝛿 ◦𝜑\)
Show that this is an equivalence relation.
Compare it with the relation \(≈\).
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[1PR]Prerequisites:417,3,3.Give a simple example of closed curves immersed for which you have \(𝛾≈_ f𝛿\) but not \(𝛾≈𝛿\).
Hidden solution: [UNACCESSIBLE UUID ’1PS’]
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[1PT] Prerequisites:3.Difficulty:*.
Let \(𝛾,𝛿: S^ 1→ℝ^ n\) be simple and immersed closed curves with the same support; Define \(\hat𝛾(t)=𝛾(-t)\): show that either \(𝛾≈_ f𝛿\) or \(\hat𝛾≈_ f𝛿\).
Hidden solution: [UNACCESSIBLE UUID ’1PV’]
Other exercises regarding curves are 2, 8, 4 and 4; see also Section 23.4.