Fat bundles and positively curved submersions.
Abstract: A well known problem in Riemannian geometry consists of finding new obstructions to the existence of metrics of positive sectional curvature on closed simply connected manifolds. In the realm of Riemannian submersions, the Petersen--Wilhelm's Fiber Dimension Conjecture states that if the total space of a Riemannian submersion has positive sectional curvature, then the dimension of the base manifold is greater than the dimension of the fiber. In this talk we shall approach this conjecture in the principal bundle case, namely, we study Riemannian submersions on principal bundles with a structure group either SO(3) or S3. In particular, we present the ideas to show that: a SO(3), S3 principal bundle admits a Riemannian submersion metric of positive sectional curvature if, and only if, such a submersion is fat. The fat condition can be interpreted as a non-metric condition: [X,H]V = V for any non-zero horizontal vector X. This result implies that on any SO(3), S3-poisitvely curved Riemannian principal bundle, the base manifold has dimension at least four, verifying Petersen--Wilhelm's Conjecture in this case. We shall focus on the totally geodesic case due to the regularization properties of Cheeger deformations, due to Searle--Solorzano--Wilhelm: invariant metrics with totally geodesic fibers are ``basins of attraction'' to invariant metrics via Cheeger deformations. The ideas to be presented come from a joint work with Llohann D. Sperança and Lino Grama.