Length of Closed Lines on a Twisted Cylinder

Question

Show that there exists a closed line $L$ of minimum length in the twisted cylinder, and that all other closed lines have twice the length of $L$.

It is intuitively clear that closed lines exist in the twisted cylinder, and so such a line of minimum length exists. I do not know how to show that this minimum length line is unique, and that all other closed lines have twice the length of $L$. I was thinking about trying to cut the twisted cylinder along $L$, but I am not sure if this is helpful or not. We have not defined any concrete measure of distance on the twisted cylinder.

Answer 1

I have since figured this problem out. We know that vertical and diagonal lines in the plane will be infinite when mapped to the twisted cylinder. Thus we restrict ourselves to horizontal lines only. Consider the horizontal line of reflection for the twisted cylinder in the plane. For simplicity, we will assume that this line is the x-axis, but the results will hold for any horizontal line. Similarly, we assume that we are translating right one unit, but the results will hold for any translation of the form $t_{(x,0)}$. Let $P=(x,y)$ be a point in the fundamental region. If $P$ lies on the x-axis, then $t_{(1,0)}sP$ = $(x+1,y)$. The closed line connecting these two points will be of minimum length, $L$. If $P$ does not lie on the x-axis, then $t_{(1,0)}sP$ $=$ $(x+1, -y)$, and there is no closed line in the twisted cylinder connecting this point with $P$. So we move to the next region, performing another glide reflection: $t_{(1,0)}st_{(1,0)}sP$ $=$ $(x+2, y)$. the line between $P$ and this point will be closed in the twisted cylinder, but it clearly has length $2L$.