EDB — 0Y7

Exercises

1. [0Y7]Through this bijection we transport the Euclidean distance from \(S^ 1\) to \(ℝ/2𝜋\) defining

   \[ d_ e([s],[t])=|Φ([s])-Φ([t])|_{ℝ^ 2} ~ ~ . \]

   With this choice the map \(Φ\) turns out to be an isometry between \((S^ 1,d)\) and \((ℝ/2𝜋,d_ e)\) (see the Definition [0TK]↺↻). So the latter is a complete metric space.

   With some simple calculations it can be deduced that

   \[ d_ e([s],[t])= \sqrt{ |\cos (t)-\cos (s)|^ 2 + |\sin (t)-\sin (s)|^ 2}= \sqrt{ 2 - 2 \cos (t-s)} ~ ~ . \]

   Then we define the function

   \[ d_ a([s],[t]) = \inf \{ |s-t-2𝜋 k| : k∈ℤ\} ~ ~ , \]

   show that it is a distance on \(ℝ/2𝜋\).

   Solution 1

   [0Y8]↺↻