Ancient Solutions on Bundles with Non-abelian Structural Group

TL;DR

This paper extends the construction of ancient Ricci flow solutions to new non-abelian bundle settings, specifically $RP^3$ bundles over quaternionic K"ahler manifolds, revealing new geometric behaviors.

Contribution

It generalizes previous solutions to include $RP^3$ bundles over products of quaternionic K"ahler manifolds and constructs analogous solutions on non-K"ahlerian bundles.

Findings

Solutions are of Type I and $ppa$-noncollapsed

Solutions have positive Ricci curvature

New geometric structures on non-abelian bundles

Abstract

We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic K\"ahler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic K\"ahler manifolds. The ancient solutions are of Type I, $\kappa$-noncollapsed, and have positive Ricci curvature. Analogous solutions on even-dimensional non-K\"ahlerian bundle spaces are also constructed.