EDB — 1P1

[1P1] Prerequisites:[1P0]↺↻.Difficulty:*.

Fix a curve \(𝛾:I→ℝ^ n\). We define in the following \(\hat I=\{ t∈ℝ:-t∈ I\} \) and \(\hat𝛾:\hat I→ℝ^ n\) via \(\hat𝛾(t)=𝛾(-t)\).

We want to show that, in certain hypotheses, two curves have the same support if and only if they are equivalent.

• Let \(𝛾,𝛿:[0,1]→ℝ^ n\) be simple curves, but not closed, and with the same support. Show that if \(𝛾(0)=𝛿(t)\) then \(t=0\) or \(t=1\). In case \(𝛾(0)=𝛿(0)\), show that \(𝛾∼𝛿\). If instead \(𝛾(0)=𝛿(1)\) then \(\hat𝛾∼𝛿\).

• Let \(𝛾,𝛿:[0,1]→ℝ^ n\) be simple immersed curves, but not closed, and with the same support, and let \(𝛾(0)=𝛿(0)\): show that \(𝛾≈𝛿\). If instead \(𝛾(0)=𝛿(1)\) then \(\hat𝛾≈𝛿\).

(For the case of closed curves see [1PT]↺↻)