Bach-flat noncompact steady quasi-Einstein manifolds

TL;DR

This paper investigates the geometric structure of Bach-flat noncompact steady quasi-Einstein manifolds, revealing conditions under which they form warped products with Einstein fibers, especially in four dimensions.

Contribution

It establishes that such manifolds with positive Ricci curvature and a critical point in the potential function are warped products with Einstein fibers, with special results in four dimensions.

Findings

Manifolds are warped products with Einstein fibers.

Fiber has constant curvature when dimension is four.

Conditions relate Bach-flatness, steady quasi-Einstein, and positive Ricci curvature.

Abstract

The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold $(M^{n},\,g)$ with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstein fiber. In addition, the fiber has constant curvature if $n = 4.$