Generalized quasi-Einstein manifolds with harmonic Weyl tensor

TL;DR

This paper introduces generalized quasi-Einstein manifolds, extending existing concepts, and characterizes their local structure under harmonic Weyl tensor conditions, revealing they are locally warped products with Einstein fibers.

Contribution

It defines generalized quasi-Einstein manifolds and proves their local warped product structure under harmonic Weyl tensor and zero radial Weyl curvature conditions.

Findings

Complete generalized quasi-Einstein manifolds with harmonic Weyl tensor are locally warped products.

Characterization of locally conformally flat gradient Ricci almost solitons.

Extension of structure results from Ricci solitons to generalized quasi-Einstein manifolds.

Abstract

In this paper we introduce the notion of generalized quasi--Einstein manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi--Einstein manifolds. We prove that a complete generalized quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature, is locally a warped product with $(n-1)$--dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.