Fatness and positive curvatures on Riemannian submersions

Abstract

Guided by rather counterintuitive properties of \emph{dual holonomy fields}, we introduce different notions of \emph{fatness} for Riemannian submersions. We introduce \emph{integral fatness} as an intrinsic condition implied by positive sectional curvature; \emph{sub-fatness} as a rigidity constraint that naturally arises in \emph{canonical variation} limits; and then develop the concept of \emph{weakly nonnegative curvature} (WNN), an intrinsic condition depending only on the horizontal distribution and the base metric. Positive sectional curvature implies fatness in this class, verifying Petersen--Wilhelm's Conjecture for homogeneous submersions and beyond. Finally, we introduce the notion of Riemannian submersion which are \emph{$A$-Clifford}: we show that them, like fatness and WNN, is intrinsic to the pair $(\mathcal{H},b)$; that it singles out a canonical vertical gauge; and that in rank three it characterizes $3$-Sasakian geometry, yielding rigidity on ${\mathrm{S}}^3,\mathrm{SO}(3)$-principal bundles.

arXiv commentMajor revision following the anonymous referee's suggestions. The new title aims to reflect that we give more emphasis to submersions with adapted metrics. Many new notions similar to fatness are included, and new connections between these notions and positive sectional curvature are verified. The Petersen--Wilhelm conjecture is verified for many classes