Spin Structure

A spin structure on an orientable Riemannian manifold (M,g) is an equivariant lift of the oriented orthonormal frame bundle FSO(M) → M with respect to the double covering ρ: Spin(n) → SO(n). In other words, a pair (P,FP) is a spin structure on the principal bundle π: FSO(M) → M when

a) πP: P → M is a principal Spin(n)-bundle over M,
b) FP: P → FSO(M) is an equivariant 2-fold covering map such that
and FP(p q) = FP(p)ρ(q) for all p ∈ P and q ∈ Spin(n).

The principal bundle πP: P → M is also called the bundle of spin frames over M.

Two spin structures (P1, FP1) and (P2, FP2) on the same oriented Riemannian manifold (M,g) are called equivalent if there exists a Spin(n)-equivariant map f: P1 → P2 such that

and f(p q) = f(p)q for all and q ∈ Spin(n).

Of course, in this case and are two equivalent double coverings of the oriented orthonormal frame SO(n)-bundle FSO(M) → M of the given Riemannian manifold (M,g).

This definition of spin structure on (M,g) as a spin structure on the principal bundle FSO(M) → M is due to André Haefliger (1956).

Read more about Spin Structure:  Spin Structures On Vector Bundles, Spinc Structures, Vector Structures

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