On gradient Ricci soliton Riemannian submersions

Abstract

In this thesis we show how to construct gradient Ricci solitons that are realized as Riemannian submersions with total space having totally umbilical fibers and integrable horizontal distribution. This construction is based on a generalization of warped products to bundles as well as a construction of gradient Ricci soliton warped products, from which we know that the base spaces of such warped products are necessarily Ricci-Hessian type manifolds. By studying this latter class of Riemannian manifolds we also obtain triviality and nonexistence results for gradient Ricci soliton warped products. These results stem from a Liouville type theorem and the validity of a weak maximum principle at infinity for a specific diffusion operator on a Ricci-Hessian type manifold.