Obstruction Class for the Existence of a Conformal Spin Structure in a Strict Sense

Abstract

Let V be a pseudo-Riemannian n-dimensional manifold or, more generally, let $$(\xi ,Q)$$ be a real fibre bundle whose base space is a paracompact space endowed with a non-degenerate quadratic form Q, (that is, with a structure group $${\mathrm {O}}(p,q),$$ $$n=p+q$$.) Let $$K_{p,q}$$ denote the obstruction class for the existence of a $$\mathrm {Pin}(p,q)$$-spin structure on V or over $$\xi .$$ Let $$K_{\mathrm {Conf}}(p,q)$$ denote the obstruction class for the existence of a conformal spin structure in a strict sense on V or over $$\xi ,$$ (simply: a $$C_{n}^{s}(p,q)$$-spin structure), if $$n=2r,$$ or of a conformal special spin structure, if $$n=2r+1.$$ This short self-contained paper will recall the determination of the obstruction class $$K_{p,q}$$ on V, or over $$\xi ,$$ for n even or odd. Then, the obstruction class $$K_{p+1,q+1}$$ for the existence of a $$\mathrm {Pin}(p+1,q+1)$$-spin structure over $$\xi _{j}$$, (Greub’s j-extension of $$\xi ,$$ where j denotes the identity mapping from $${\mathrm {O}}(p,q)$$ into $${\mathrm {O}}(p+1,q+1)$$), will be determined in order to express $$K_{\mathrm {Conf}}(p,q),$$ for $$n=2r$$ or $$n=2r+1,$$ in terms of the Stiefel–Whitney classes $$w_{i}(p,q),$$ $$i=1,2,$$ of $$\xi $$, decomposed as the Whitney sum $$\xi =\xi ^{+}\oplus \xi ^{-},$$ where the restriction of Q to $$\xi ^{+}$$ is positive definite and the restriction of Q to $$\xi ^{-}$$ is negative. If $$n=2r,$$ we find again results obtained in previous publications [4, 5, 7], by different methods.