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.