Exploring the Relationship between Spinors and Mobius Strips in Rotations

[jfy4]

Hi,

I was pondering a bit about the mobius strips and I was wondering if there is a relationship between spinors and there transformation under rotations and that, in a manner of speaking, one must go around a mobius strip twice to return to the original position. To me it seems there would be some underlying relationship here.

I was thinking, abstractly, some sort of closed path, $\gamma$, with non-zero torsion; such that after traversing the loop once, one has been spun around half way, and then after a subsequent traversal, another half to return to the original orientation.

Are there such closed paths such that, say, after one traversal the torsion of the path returns one to their original orientation (a full rotation) and other paths like the one mentioned above.

Thanks in advance,

 

[lavinia]

Not sure about your question but the Spin groups are two fold covers of the rotation groups of Euclidean space. I don't know much about these groups but SO(3) is covered by the 3 sphere and is homeomorphic to the 3 dimensional real projective space. Projective space has fundamental group Z/2Z that arises from a cross cap around its equator. A cross cap is a Moebius band with a disk attached to its bounding circle.