Abstract
Spinor fields can only be defined on a space‐time which has been given a spinor structure. A number of conditions (some sufficient, others necessary and sufficient) for the existence of a spinor structure are derived. By applying one or another of these conditions, it is shown that many well‐known solutions of Einstein's equations do have spinor structure. The question of the existence of spinor structure depends only on the topology of the underlying manifold, not on the (time‐ and space‐oriented) metric. It is shown that, nonetheless, a certain ``threshold'' of curvature must be exceeded before there can be even the possibility of a space‐time's having no spinor structure.

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