Rigidity results on gradient Schouten solitons

TL;DR

This paper classifies gradient Schouten solitons on certain product manifolds, showing that complete Riemannian cases have a specific geometric structure involving spheres and compact Einstein manifolds.

Contribution

It provides a complete classification of gradient Schouten solitons on product manifolds with conformal and Einstein components, including explicit solutions and geometric characterizations.

Findings

All solutions for gradient Schouten solitons are characterized.

Complete Riemannian gradient Schouten solitons have a specific geometric structure.

The base manifold is isometric to a product of a sphere and a line.

Abstract

In this paper we consider $\rho$-Einstein solitons of type $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$, where $\left(B^n,g^{*}\right)$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group, and $\left(F^m,g_{F}\right)$ is an Einstein manifold. We provide all the solutions for the gradient Schouten soliton case. Moreover, in the Riemannian case, we prove that if $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$ is a complete gradient Schouten soliton then $\left(B^{n},g^{*}\right)$ is isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ and $F^m$ is a compact Einstein manifold.