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Quasi-Einstein manifolds with Harmonic Weyl curvature

Huai-Dong Cao, Fengjiang Li, James Siene

TL;DR

This work extends the classification of quasi-Einstein manifolds with harmonic Weyl curvature from four dimensions to all n≥5, offering a comprehensive local-to-global analysis. By establishing a local multiply warped product structure and constraining Ricci eigenvalue configurations, the authors derive a finite set of local models and assemble them into global classifications. The results yield new examples of quasi-Einstein manifolds that are neither locally conformally flat nor D-flat, enriching the landscape of Einstein-warped product geometries. The framework builds on and refines prior work on warped products, Ricci solitons, and harmonic Weyl curvature, providing explicit geometric structures and completeness criteria.

Abstract

In this paper, we classify $n$-dimensional ($n\geq 5$) quasi-Einstein manifolds with harmonic Weyl curvature, thus extending the work of Shin \cite{Shin} in dimension four for quasi-Einstein manifolds and refining the work of He-Petersen-Wylie \cite{HPW}. As a consequence, we provide new examples of quasi-Einstein manifolds which are neither locally conformally flat nor D-flat in the sense of \cite{CC12}.

Quasi-Einstein manifolds with Harmonic Weyl curvature

TL;DR

This work extends the classification of quasi-Einstein manifolds with harmonic Weyl curvature from four dimensions to all n≥5, offering a comprehensive local-to-global analysis. By establishing a local multiply warped product structure and constraining Ricci eigenvalue configurations, the authors derive a finite set of local models and assemble them into global classifications. The results yield new examples of quasi-Einstein manifolds that are neither locally conformally flat nor D-flat, enriching the landscape of Einstein-warped product geometries. The framework builds on and refines prior work on warped products, Ricci solitons, and harmonic Weyl curvature, providing explicit geometric structures and completeness criteria.

Abstract

In this paper, we classify -dimensional () quasi-Einstein manifolds with harmonic Weyl curvature, thus extending the work of Shin \cite{Shin} in dimension four for quasi-Einstein manifolds and refining the work of He-Petersen-Wylie \cite{HPW}. As a consequence, we provide new examples of quasi-Einstein manifolds which are neither locally conformally flat nor D-flat in the sense of \cite{CC12}.

Paper Structure

This paper contains 12 sections, 21 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and basics of Riemannian geometry
  4. Basic facts and identities for quasi-Einstein manifolds
  5. Local multiply warped product structure of quasi-Einstein manifolds with harmonic Weyl curvature
  6. The local structure of the case with more than two distinct Ricci-eigenfunctions
  7. The local structure of the case with two distinct Ricci-eigenfunctions
  8. Case I: Either $X$ or $Y$ vanishes
  9. Case II: Neither $X$ nor $Y$ vanishes
  10. The local structure theorem and the global classifcations
  11. The local structure theorem
  12. The global classifications

Key Result

Theorem 1.1

Let $(M^n, g, f)$, $n\geq 5$, be an $n$-dimensional, non-trivial, complete quasi-Einstein manifold satisfying qe with harmonic Weyl curvature and $m>1$ ($m\neq \infty$). Then, it is one of the following types:

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3: Chen-He CH
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 30 more