2022/07/21 by Cavenaghi, Leonardo F., Grama, Lino, Sperança, Llohann D.
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.10749
This paper studies Cheeger deformations on S3, 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 π: SO(3), S3 \hookrightarrow \cal P → 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 ≥ 4. The proof combines the concept of ``good triples'' due to Munteanu and Tapp \citetappmunteanu2, with a Chaves--Derdzisnki--Rigas type condition to nonnegative curvature. Additionally, the conjecture is verified for other classes of submersions.