The Weyl tensor of gradient Ricci solitons

2013/11/04 by Xiaodong Cao, Hung Tran
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Conformal map #Curvature #Curvature of Riemannian manifolds #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hessian matrix #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Ricci curvature #Ricci decomposition #Ricci flow #Riemann curvature tensor #Rigidity (electromagnetism) #Scalar curvature #Sectional curvature #Topology (electrical circuits) #Weyl tensor #Weyl transformation #math.DG #msc:53C21 #msc:53C44

paper · pdf · doi:10.2140/gt.2016.20.389

published as Geom. Topol. 20 (2016) 389-436 · 42 pages

arxiv created 2013/11/04 · openalex publication_date 2016/02/29 · arxiv updated 2016/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner–Weitzenböck-type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.

Citations

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