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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The Weyl tensor of gradient Ricci solitons

Xiaodong Cao and Hung Tran

Geometry & Topology 20 (2016) 389–436
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.

Keywords
Weyl tensor, Ricci soliton, Bochner–Weitzenböck formula
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 53C21, 53C25
References
Publication
Received: 28 May 2014
Revised: 3 May 2015
Accepted: 27 May 2015
Published: 29 February 2016
Proposed: Gang Tian
Seconded: Tobias H Colding, John Lott
Authors
Xiaodong Cao
Department of Mathematics
Cornell University
507 Malott Hall
Ithaca, NY 14853
USA
Hung Tran
Department of Mathematics
University of California, Irvine
440G Rowland Hall
UCI
Irvine, CA 92697
USA