Gradient Ricci almost soliton warped product

TL;DR

This paper characterizes conditions for constructing gradient Ricci almost solitons as warped products, providing existence results, examples, and rigidity theorems using geometric analysis and PDE techniques.

Contribution

It offers new necessary and sufficient conditions for warped product gradient Ricci almost solitons, along with existence proofs and rigidity results under geometric assumptions.

Findings

Derived conditions for warped product gradient Ricci almost solitons.

Provided explicit examples of solutions to the PDEs involved.

Proved a rigidity theorem for gradient Ricci soliton Riemannian products.

Abstract

We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an example of particular solutions of the PDEs that arise from our construction. We also prove a rigidity result for a gradient Ricci soliton Riemannian product in the class of gradient Ricci almost soliton warped products under some natural geometric assumptions on the warping function.