The Weyl tensor of gradient Ricci solitons

Cao, Xiaodong; Tran, Hung

doi:10.2140/gt.2016.20.389

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The Weyl tensor of gradient Ricci solitons

Xiaodong Cao and Hung Tran

Geometry & Topology 20 (2016) 389–436

DOI: 10.2140/gt.2016.20.389

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

Volume 20, issue 1 (2016)