Solutions to the affine quasi-Einstein equation for homogeneous surfaces

TL;DR

This paper characterizes the solution spaces of the affine quasi-Einstein equation on homogeneous surfaces, enabling the construction of various special geometric structures like Yamabe solitons and Einstein manifolds.

Contribution

It provides explicit descriptions of solution spaces for the affine quasi-Einstein equation on homogeneous surfaces, linking them to geometric structures via affine Killing vector fields.

Findings

Explicit descriptions of solution spaces for homogeneous surfaces

Connections between solutions and gradient Yamabe solitons

Framework based on affine Killing vector fields

Abstract

We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.