Locally conformally flat quasi-Einstein manifolds

TL;DR

This paper proves that complete locally conformally flat quasi-Einstein manifolds of dimension three or higher are locally warped products with fibers of constant curvature, encompassing locally conformally flat gradient Ricci solitons.

Contribution

It establishes a classification result for locally conformally flat quasi-Einstein manifolds, showing they are locally warped products with constant curvature fibers.

Findings

Complete locally conformally flat quasi-Einstein manifolds are locally warped products.

Includes locally conformally flat gradient Ricci solitons as a special case.

Provides a geometric structure classification for these manifolds.

Abstract

In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension $n\geq 3$ is locally a warped product with $(n-1)$-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.