Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle ! M of complex spinor modules over the Cliiord 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 the O(h)-bundle of all orthonormal frames on M to the Lips-chitz group Lpin(h) of all automorphisms of a suitably deened spin space. An explicit construction is given of the total space of the Lpin(h)-bundle deening such a structure. If the dimension m of M is even, then the Lipschitz group coincides with the complex Cliiord 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 bres are 2 n-dimensional and decomposable at every point of M, but the homomorphism of bundles of algebras Cl(g) ! End globally decomposes if, and only if, M is orientable. Examples of such bundles are given. The topological condition for the existence of an lpin structure on an odd-dimensional Riemannian manifold is derived and illustrated by the example of a manifold admitting such a structure, but no pin c structure.