Real spin-Clifford bundle and the spinor structure of space-time

Abstract

By analyzing the conditions for the existence on a space-time ℒ of a global algebraic spinor field, we prove the following result, known as Geroch's theorem: A necessary and sufficient condition for ℒ to admit a spinor structure is that the orthonormal frame bundleF ₀(ℒ) have a global section. Our proof, which does not use in any stage the complexification of ℝ_1,3 (the space-time Clifford algebra), is simple, requiring only the explicit construction of the algebraic spinor and the spinorial metric within ℝ_1,3 and elementary facts about associated bundles and the bundle reduction process. This is to be compared with the original proof, which uses the full algebraic topology machinery. We also clarify the relation of the covariant spinor structure and Graf'se-spinor structure.