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In order to define spinor fields on a space-time M, it is necessary first to endow M with some further structure in addition to its Lorentz metric. This is the spinor structure. The definition and the elementary implications of the existence of a spinor structure are discussed. It is proved that a necessary and sufficient condition for a noncompact space-time M to admit a spinor structure is that M have a global field of orthonormal tetrads.
Robert Geroch (Fri,) studied this question.