**Leonardo Francisco Cavenaghi, Universidade de São Paulo (USP)**

**Title:** 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 S^{3}. In particular, we present the
ideas to show that: a SO(3), S^{3} 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),
S^{3}-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.