On Bach flat warped product Einstein manifolds

TL;DR

This paper proves that compact Bach-flat warped product Einstein manifolds are essentially quotients of warped products with Einstein fibers, with the fiber's curvature properties depending on the dimension.

Contribution

It establishes a classification result for Bach-flat warped product Einstein manifolds, linking their structure to Einstein fibers and curvature conditions.

Findings

Such manifolds are finite quotients of warped products with Einstein fibers.

The fiber has constant curvature when the dimension is four.

The result generalizes understanding of Bach-flat conditions in Einstein geometry.

Abstract

In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension $n \geq 4$ is a finite quotient of a warped product with $(n-1)$-dimensional Einstein fiber. The fiber has constant curvature if $n=4$.