(Note that this corresponds to putting a spin structure on our surface.)

Spinors on a manifold equipped with a spin structure.

A magnon is a collective excitation of the electrons' spin structure in a crystal lattice.

M is orientable and admits a spin structure.

One method for dealing with this problem is to require that M has a spin structure.

This is not always possible since there is potentially a topological obstruction to the existence of spin structures.

When a manifold carries a spin structure at all, the set of spin structures forms an affine space.

To be more precise, one must add additional data to the Lagrangian - a grading and a spin structure.

(which is not an inclusion: it's a 2-fold covering space) is a spin structure.

By colliding spin-polarized protons, the spin structure of the proton is explored.