On the construction of gradient Ricci soliton warped product

TL;DR

This paper investigates conditions under which warped product manifolds form gradient Ricci solitons, providing new criteria for their construction and examples of expanding solitons with Einstein fibers.

Contribution

It establishes that certain warped products with specific warping functions are necessarily Riemannian products and offers a complete characterization for constructing gradient Ricci soliton warped products.

Findings

Warped products with warping functions reaching both maximum and minimum are Riemannian products.

A necessary and sufficient condition for constructing gradient Ricci soliton warped products is provided.

A class of expanding Ricci solitons with Einstein fibers of non-positive scalar curvature is constructed.

Abstract

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and sufficient condition for constructing a gradient Ricci soliton warped product. As an application, we present a class of expanding Ricci soliton warped product having as a fiber an Einstein manifold with non-positive scalar curvature. We also discuss some obstructions to this construction, especially in the case when the base of the warped product is compact.