Nowhere-vanishing spinors and triality rotations in 8-manifolds

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Abstract
Using methods from differential geometry and index theorems, the authors give simple derivations of the criteria for the existence of nowhere-vanishing spinor fields on compact Riemannian manifolds. These had been derived previously using obstruction theory. The possibility of the non-existence of nowhere-vanishing spinor fields arises only in two, four or eight dimensions, and of these, eight is the dimension that presents the most interesting features. This is the same as the dimension of the transverse space for the superstring and they make some observations that follow from this fact.