Global spinor fields in space-time

Abstract

This paper is concerned with space-time manifolds that are space- and time-oriented, causal, and possess spinor structures. Five propositions are proven: (1) If a connected, space- and time-oriented manifold is simply con-nected, then it is non-compact; (2) If such a manifold is simply connected, it admits a spinor structure, which, moreover, is unique; (3) If the space-like section of M is compact, then there exists a global system of orthonormal tetrads on M; (4) The necessary and sufficient condition for every space-time M whose space-like section is compact to admit a spinor structure is that M have a global system of orthonormal tetrads; (5) Every space-time M which can be imbedded in R6 admits a spinor structure. It is further suggested that in view of the fact that the existence of a spinor structure is related to homotopy properties, space-time manifolds may be classified in terms of their homotopy groups πi (M), i=1,2, 3,4. In a concluding section, some avenues for future research are discussed.