Undergrad Constructing an Oriented Atlas for S1 (Self Study)

[Amateur659]

Hello,

I am trying to construct an oriented atlas for S1 by considering the 8 transitions from the basic coordinate projection atlas. However, it does not appear to be possible to modify this atlas without destroying existing progress.

Am I on the right track towards constructing the oriented atlas? Is there some other direction I should choose? Is there something obvious I am missing?

I've attached my work below.

Thank you for your time.

 

[andrewkirk]

I think using Cartesian coordinates makes it unnecessarily complex. Instead use a one-coordinate system where the coordinate is angle from a given reference ray through the origin. Then your atlas needs only two charts. Make each one cover say 3/4 of the ring, and make the reference ray for the second one 180 degrees from the first reference ray. Then you have two overlapping part-rings covering the circle, each with a bicontinuous mapping to [0, 3 pi / 2], and quarter-circle overlaps at each end. You only have two overlap regions to look at, and in both the Jacobian is 1 x 1, ie a scalar.

 

[Orodruin]

Make each one cover say 3/4 of the ring

Am I just obsessed if I always make my circle charts cover the full circle minus a point?