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.