The Weyl Tensor of Gradient Ricci Solitons

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.

42 pages

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Existence of Gradient Kahler-Ricci Solitons
• A Myers-type theorem and compact Ricci solitons

Cited by in corpus (6)

• Isotropic quasi-Einstein manifolds
• On Closed Manifolds with Harmonic Weyl Curvature
• Einstein four-manifolds of pinched sectional curvature
• Half conformally flat gradient Ricci almost solitons
• Four-dimensional complete gradient shrinking Ricci solitons
• Sharp upper diameter bounds for compact shrinking Ricci solitons