Spin structure

A spin structure on an orientable Riemannian manifold $(M, g)$ with an oriented vector bundle $E$ is an equivariant lift of the orthonormal frame bundle $P_{SO} (E) \to M$ with respect to the double covering $ρ : Spin (n) \to SO (n)$ . In other words, a pair $(P_{Spin}, ϕ)$ is a spin structure on the $SO (n)$ -principal bundle $π : P_{SO} (E) \to M$ when
a) $π_{P} : P_{Spin} \to M$ is a principal $Spin (n)$ -bundle over $M$ , and
b) $ϕ : P_{Spin} \to P_{SO} (E)$ is an equivariant 2-fold covering map such that

π \circ ϕ = π_{P} and ϕ (p q) = ϕ (p) ρ (q) for all p \in P_{Spin} and q \in Spin (n) .

\begin{array}{ccc} P_{Spin} & \overset{ϕ}{\to} & P_{SO} (E) \\ ↓ π_{P} & ↓ π \\ M & = & M \end{array}

Two spin structures $(P_{1}, ϕ_{1})$ and $(P_{2}, ϕ_{2})$ on the same oriented Riemannian manifold are called "equivalent" if there exists a $Spin (n)$ -equivariant map $f : P_{1} \to P_{2}$ such that

ϕ_{2} \circ f = ϕ_{1} and f (p q) = f (p) q for all p \in P_{1} and q \in Spin (n) .

In this case $ϕ_{1}$ and $ϕ_{2}$ are two equivalent double coverings.