Understanding the Möbius Bundle on a Circle

[seydunas]

Hi,

i can not understand how circle has a nontrivial bundle, Möbius bundle. Can you say me what is its transition function on it.

 

[lavinia]

Hi,

i can not understand how circle has a nontrivial bundle, Möbius bundle. Can you say me what is its transition function on it.

multiplication by -1.

Similarly the Klein bottle is reflection of a circle.

A way to visualize the Klein bottle is to look at the transformation (x,y) -> ( x+ 1/2,-y) on the unit square viewed as the fundamental domain of the torus.