The Petersen--Wilhelm conjecture on principal bundles

This paper studies Cheeger deformations on ${\mathrm{S}}^3,\mathrm{S}\mathrm{O}(3)$ principal bundles to obtain conditions for positive sectional curvature submersion metrics. We conclude, in particular, a stronger version of the Petersen–Wilhelm fiber dimension conjecture to the class of principal bundles. We prove any $\pi :\mathrm{S}\mathrm{O}(3),{\mathrm{S}}^3\hookrightarrow \mathcal{P}\to \mathcal{B}$ principal bundle over a positively curved base admits a metric of positive sectional curvature if, and only if, the submersion is fat, in particular, $\dim B\ge 4$. The proof combines the concept of ``good triples’’ due to Munteanu and Tapp \cite{tappmunteanu2}, with a Chaves–Derdzisnki–Rigas type condition to nonnegative curvature. Additionally, the conjecture is verified for other classes of submersions.