A dual leaf approach to fatness

The motivation for this work stems from the authors' desire to revive the study of ``fat bundles''. Recently, several new works have emerged in this field, including \cite{DV2022, cavenaghilinollohann2, ovando1, ovando2, BOCHENSKI2016131}, while the most notable reference being W. Ziller's unpublished notes on fat bundles (\cite{Ziller_fatnessrevisited}). Given these developments, it is natural to revisit this topic, recalling known facts and furnishing a new perspective to the known results, despite obtaining new ones. One uses recent advances in positive curvatures to obtain a simple and geometric classification of fat homogeneous Riemannian foliations. Although B\'erard-Bergery established a similar classification in \cite{bergeryfat}, we believe one of the novelties of this paper lies in the combination of dual foliations and Ambrose--Singer's Theorem with metric deformations: motivated by the regularization properties of Cheeger deformations and the non-metric dependence of fatness, we show that: the vertical bundle of a fat submersion is obstructed, being wholly generated by the O'Neill tensor (this is a consequence of a dual-leaf argument); we prove that any fat foliation induced by the fibers of a fiber bundle with compact structure group is \emph{twisted} (any Riemannian submersion compatible metric has dual-foliation with only one dual leaf). We also prove that if the total space of a fat Riemannian submersion has non-negative sectional curvature, then their leaves are a symmetric space. We conclude further classification results.
Comment: Major changes: the title was changed to reflect the new perspective of the paper. Results were added, proofs revisited, and exposition improved. Some results contained mistakes or imprecisions in their statements. This was also fixed