is a cover of the full orthogonal group . The spin group consists of those elements of that are products of an even number of unit vectors. Thus by the Cartan–Dieudonné theorem Spin is a cover of the group of proper rotations . Let be the automorphism which is given by the mapping acting on pure vectors. Then in particular, is the subgroup of whose elements are fixed by . Let (These are precisely the elements of even degree in .) Then the spin group lies within . The irreducible representations of restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of . To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above) and realize a spin representation in signature as a pin representation in either signature or . Applications Differential geometry One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more important is the link to a spin manifold, its associated spinor bundle and manifolds. Physics Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices called Dirac matrices which have the property that where is the matrix of a quadratic form of signature (or corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra , whose complexification is which, by the classification of Clifford algebras, is isomorphic to the algebra of complex matrices . However, it is best to retain the notation , since any transformation that takes the bilinear form to the canonical form is not a Lorentz transformation of the