Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle M of complex spinormodules over the Clifford bundle Cl(g) of a Riemannian space(M, g) with local model (V, h)is associated with an lpin(Lipschitz) structure on M, this being a reduction of theO(h)-bundle of all orthonormal frames on M to the Lipschitzgroup Lpin(h) of all automorphisms of a suitably defined spinspace. An explicit construction is given of the total space of theLpin(h)-bundle defining such a structure. If the dimension mof M is even, then the Lipschitz group coincides with the complexClifford group and the lpin structure can be reduced to a pin _c structure. If m = 2n – 1, then a spinor module on M is of the Cartan type: its fibres are 2 _n -dimensional anddecomposable at every point of M, but the homomorphism of bundlesof algebras Cl(g) End globally decomposes if, andonly if, M is orientable. Examples of such bundles are given. Thetopological condition for the existence of an lpin structure on anodd-dimensional Riemannian manifold is derived and illustrated by theexample of a manifold admitting such a structure, but no pin _c structure.