The Petersen--Wilhelm conjecture on principal bundles

TL;DR

This paper advances the understanding of positive sectional curvature on principal bundles by establishing conditions under which such metrics exist, confirming a stronger version of the Petersen--Wilhelm fiber dimension conjecture.

Contribution

It proves that principal bundles over positively curved bases admit positive curvature metrics if and only if the submersion is fat, extending the conjecture to a broader class of bundles.

Findings

Principal bundles over positively curved bases admit positive curvature if submersion is fat.

Verification of the conjecture for other classes of submersions.

Integration of 'good triples' and curvature conditions in the proof.

Abstract

This paper studies Cheeger deformations on $\mathrm{S}^3, \mathrm{SO}(3)$ 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 $\pi: \mathrm{SO}(3), \mathrm{S}^3 \hookrightarrow \cal P \rightarrow B$ principal bundle over a positively curved base admits a metric of positive sectional curvature if, and only if, the submersion is fat, in particular, $\dim B \geq 4$. 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.