1999/01/29 by Thomas Friedrich, Friedrich, Thomas, Andrzej Trautman +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Primary 15A66 and 53A50 #Secondary 81R25 and 83C60 #math.DG #msc:15A66 #msc:53A50 #msc:81R25 #msc:83C60
paper · pdf · doi:10.48550/arxiv.math/9901137
Latex2.09, 23 pages
arxiv created 1999/01/29 · arxiv updated 2009/11/30
It is shown that every bundle \varSigma→ M of complex spinor modules 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 the \Ort(h)-bundle of all orthonormal frames on M to the Lipschitz group \Lpin(h) of all automorphisms of a suitably defined spin space. An explicit construction is given of the total space of the \Lpin(h)-bundle defining such a structure. If the dimension m of M is even, then the Lipschitz group coincides with the complex Clifford group and the lpin structure can be reduced to a pinc structure. If m=2n-1, then a spinor module \varSigma on M is of the Cartan type: its fibres are 2n-dimensional and decomposable at every point of M, but the homomorphism of bundles of algebras \Cl(g)→\End\varSigma 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 pinc structure.