Abstract
It is shown that every bundle Σ → 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 Σ 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 Σ 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.
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Friedrich, T., Trautman, A. Spin Spaces, Lipschitz Groups, and Spinor Bundles. Annals of Global Analysis and Geometry 18, 221–240 (2000). https://doi.org/10.1023/A:1006713405277
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DOI: https://doi.org/10.1023/A:1006713405277