Ricci solitons on low-dimensional generalized symmetric spaces

TL;DR

This paper classifies Ricci solitons on low-dimensional pseudo-Riemannian generalized symmetric spaces, revealing differences between Riemannian and pseudo-Riemannian cases and providing explicit solutions.

Contribution

It demonstrates that most four-dimensional type B pseudo-Riemannian generalized symmetric spaces are Ricci solitons, contrasting with Riemannian cases, and fully characterizes three-dimensional cases.

Findings

Four-dimensional type A, C, D spaces are algebraic Ricci solitons.

Type B four-dimensional spaces are mostly Ricci solitons, unlike in Riemannian case.

All three-dimensional generalized symmetric spaces are Ricci solitons.

Abstract

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are algebraic Ricci solitons, the ones of type B are not so. The Ricci soliton equation for their metrics yields a system of partial differential equations. Solving such system, we prove that almost all the four-dimensional pseudo-Riemannian generalized symmetric spaces of type B are Ricci solitons. These examples show some deep differences arising for the Ricci soliton equation between the Riemannian and the pseudo-Riemannian cases, as any homogeneous Riemannian Ricci soliton is algebraic [21]. We also investigate three-dimensional generalized symmetric spaces of any signature and prove that they are Ricci solitons.