Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 2 / 11G (Geometry)

If $(x,y)\sim (x+a,y+b)$ in the torus, thenThis is Klein-equivalent to $(2x,y)$ by taking $(c,d)=(2a,b)$. Hence $\tilde{\pi }$ descends to a well-defined continuous map

The deck group of the Klein bottle has an index-two subgroup consisting of transformations with even $c$; it is generated by translations $(x,y)\mapsto (x+2,y)$ and $(x,y)\mapsto (x,y+1)$. This is exactly the image under $\tilde{\pi }$ of the torus deck lattice. Therefore the orientation double cover of the Klein bottle has two sheets:

For $T^2$, take the unit square $[0,1]\times [0,1]$. Identifyso both pairs of opposite edges have matching directions.

For $K$, take $[0,1]\times [-1/2,1/2]$. Identifywith matching directions, andwith reversed directions. These equations specify the arrows on the two fundamental-domain drawings.

In the Klein fundamental square from part (b), the following straight lines project to the required closed geodesics:this is the one-sided horizontal core, and cutting along it leaves a Möbius strip.

Takethis is a two-sided vertical geodesic, and cutting along it leaves a cylinder.

Finally takeIts endpoints differ by the deck translation $(2,1)$, so it is closed. The parameters $t=1/4$ and $t=3/4$ represent the same Klein-bottle point through an odd-$c$ glide reflection, and no other interior pair does. Thus its image has exactly one transverse self-intersection. This is the flat Klein-bottle geodesic model.

A loop whose Klein deck transformation has odd $c$ has one closed lift to the orientation double cover; one with even $c$ has two. Thus the preimage of ${\gamma }_1$ contains one closed geodesic, while those of ${\gamma }_2$ and ${\gamma }_3$ contain two each.

In the torus unit square, representatives arewhich maps twice around ${\gamma }_1$;andAll coordinates are taken modulo one. Hence the requested numbers areThe two diagonal lifts of ${\gamma }_3$ meet over its self-intersection point but remain distinct closed geodesics.