Let ({\mathbf P},F_{\mathbf P}) be a
spin structure on a
Riemannian manifold (M, g),\,that is, an
equivariant lift of the oriented
orthonormal frame bundle \mathrm F_{SO}(M)\to M with respect to the double covering \rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n) of the
special orthogonal group by the
spin group. The
spinor bundle {\mathbf S}\, is defined to be the
complex vector bundle {\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\, associated to the
spin structure {\mathbf P} via the
spin representation \kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\, where {\mathrm U}({\mathbf W})\, denotes the
group of
unitary operators acting on a
Hilbert space {\mathbf W}.\, The spin representation \kappa is a faithful and
unitary representation of the group {\mathrm {Spin}}(n). ==See also==