Spin Structure

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

a) π_P: P → M is a principal Spin(n)-bundle over M,

b) F_P: P → F_SO(M) is an equivariant 2-fold covering map such that

and F_P(p q) = F_P(p)ρ(q) for all p ∈ P and q ∈ Spin(n).

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

Two spin structures (P₁, F_P1) and (P₂, F_P2) 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 F_SO(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 F_SO(M) → M is due to André Haefliger (1956).