Let ( P , F P ) {\displaystyle ({\mathbf {P} },F_{\mathbf {P} })} be a spin structure on a Riemannian manifold ( M , g ) , {\displaystyle (M,g),\,} that is, an equivariant lift of the oriented orthonormal frame bundle F S O ( M ) → M {\displaystyle \mathrm {F} _{SO}(M)\to M} with respect to the double covering ρ : S p i n ( n ) → S O ( n ) {\displaystyle \rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)} of the special orthogonal group by the spin group.