𝐺-structures on spheres

Leonard, Peter

doi:10.1090/S0002-9947-1971-0275468-0

$G$-structures on spheres
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by Peter Leonard PDF

Trans. Amer. Math. Soc. 157 (1971), 311-327 Request permission

Abstract:

${G_n}$ denotes one of the classical groups $SO(n),SU(n)$ or $Sp(n)$ and $H$ a closed connected subgroup of ${G_n}$. We ask whether the principal bundle ${G_n} \to {G_{n + 1}} \to {G_{n + 1}}/{G_n}$ admits a reduction of structure group to $H$. If $n$ is even and ${G_n}$ is $SO(n)$ or $SU(n)$ or if $n \not \equiv 11\bmod 12$ and ${G_n}$ is $Sp(n)$, we prove that there are no such reductions unless $n = 6,{G_6} = SO(6)$ and $H = SU(3)$ or $U(3)$. In the remaining cases we consider the problem for $H$ maximal. We divide the maximal subgroups into three main classes: reducible, nonsimple irreducible and simple irreducible. We find a necessary and sufficient condition for reduction to a reducible maximal subgroup and prove that there are no reductions to the nonsimple irreducible maximal subgroups. The remaining case is unanswered.

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