I'm trying to think about what information I need to add to a manifold that it describes a spin structure?
I know you can have spin-structure on a 2d plane, a 2-sphere.
I also know you can define a Dirac equation on a 2-sphere.
I would think if you have two Dirac matrices $\gamma_x$ and $\gamma_y$ you would need to have some idea of an $x$-direction and a $y$-direction at each point. (which is odd because if you use latitude and longitude this would be undefined at the poles).
It seems like spin is fundamentally linked to some underlying Euclidean space which is strange.
Is there a more intuitive way to understand this?