Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed in the literature since the primary interest of mathematicians in surfaces is in the study of one complex variable, number theory, algebraic geometry etc. where all of the surfaces are oriented."
(I won't give context, other than a page number: 446.) This got me thinking, perhaps non-orientability is purely an invention of topologists. Surely non-orientable manifolds play an important role in other areas of mathematics? Are there "real world" examples of non-orientability phenomenon in the natural sciences?
