Spin lifts of oriented orthonormal frame bundles

Let $E\to M$ be an oriented Euclidean real vector bundle of rank $n\ge 1$. Its oriented frame bundle Oriented real bundles and oriented frame bundles has structure group $G{L}_n^{+}(\mathbb{R})$, so the supplied metric gives an $SO(n)$ reduction: fiberwise Gram–Schmidt sends every oriented frame to an oriented orthonormal frame, and its formulas are continuous because each normalization divides by the positive norm of a nonzero orthogonal vector. When the base, bundle and metric are smooth, these formulas and the resulting reduction are smooth. Denote this principal right $SO(n)$-bundle by $P_{SO}(E)$.

For the positive Clifford convention $v^2=\mathrm{\mid }v{\mathrm{\mid }}^2$ and the full-even group of Real Clifford group, Pin, and the two Spin subgroups, write $\rho :{\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(n)\to SO(n)$ for the two-sheeted covering of Pin and full-even Spin are topological double covers. A Spin lift of $P_{SO}(E)$ is a right principal ${\mathrm{Spin}}_{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(n)$-bundle $P$ and a continuous map $\lambda :P\to P_{SO}(E)$ over $M$ satisfying $\lambda (ph)=\lambda (p)\rho (h)$. A Spin structure means an isomorphism class of such lifts over the identity of $P_{SO}(E)$. This definition concerns ranks $n\ge 1$; the rank-zero Clifford convention is treated separately in the index theorem and is not assigned a torsor of unstabilized Spin structures.

The right-action convention is that of Principal g bundle and associated fiber bundle.