Editing Spinor bundle (section)

==Formal definition==

Let <math>({\mathbf P},F_{\mathbf P})</math> be a [[spin structure]] on a [[Riemannian manifold]] <math>(M, g),\,</math>that is, an [[equivariant]] lift of the oriented [[orthonormal frame bundle]] <math>\mathrm F_{SO}(M)\to M</math> with respect to the double covering <math>\rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n)</math> of the [[special orthogonal group]] by the [[spin group]].

The '''spinor bundle''' <math>{\mathbf S}\,</math> is defined <ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}} page 53
</ref> to be the [[complex vector bundle]]
<math display=block>{\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\,</math>
associated to the [[spin structure]] <math>{\mathbf P}</math> via the [[spin representation]] <math>\kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\,</math> where <math>{\mathrm U}({\mathbf W})\,</math> denotes the [[Group (mathematics)|group]] of [[unitary operator]]s acting on a [[Hilbert space]] <math>{\mathbf W}.\,</math> The spin representation <math>\kappa</math> is a faithful and [[unitary representation]] of the group <math>{\mathrm {Spin}}(n).</math><ref>{{citation | last1=Friedrich|first1=Thomas| title = Dirac Operators in Riemannian Geometry| publisher=[[American Mathematical Society]] | year=2000|isbn=978-0-8218-2055-1}} pages 20 and 24</ref>