Ricci Almost Solitons on semi-Riemannian Warped Products

TL;DR

This paper characterizes Ricci almost solitons on semi-Riemannian warped products, revealing conditions under which the base and fiber are Einstein, and classifies such manifolds under completeness assumptions.

Contribution

It provides a comprehensive characterization of Ricci almost solitons on warped products, including conditions on the potential function and the geometry of base and fiber.

Findings

Fiber must be an Einstein manifold.

If potential depends on fiber and gradient of warping function is not translational, both base and warped product are Einstein.

Existence of conformal vector fields on base, fiber, and warped product.

Abstract

We characterize Ricci almost solitons on semi-Riemannian warped products, considering the potential function to depend on the fiber or not. We show that the fiber is necessarily an Einstein manifold. As a consequence of our characterization we prove that when the potential function depends on the fiber, if the gradient of the warping function does not act by translations then the base and the warped product are also Einstein manifolds. Moreover, we show the existence of conformal vector fields on the base, the fiber and on the warped product. Assuming completeness of the warped product we provide a classification of such manifolds. When the potential function depends on the fiber and the gradient of the warping functions is an improper vector field, we show that the base is a Brinkmann space and the fiber is Ricci flat. We use the characterization also to prove that the potential function of a complete Ricci soliton depends only on the base.