Conformal Diffeomorphisms of Gradient Ricci Solitons and Generalized Quasi-Einstein Manifolds

Jeffrey L. Jauregui, William Wylie

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preserving, non-homothetic conformal diffeomorphism, then the metric is conformal to the sphere, and isometric to the sphere in the case of a gradient Ricci soliton. In the complete case, the only structure-preserving non-homothetic conformal diffeomorphisms from a shrinking or steady gradient Ricci soliton to another soliton are the conformal transformations of spheres and inverse stereographic projection.

Original languageEnglish (US)
Pages (from-to)668-708
Number of pages41
JournalJournal of Geometric Analysis
Volume25
Issue number1
DOIs
StatePublished - Jan 1 2013

Keywords

  • Conformal Killing field
  • Conformal diffeomorphism
  • Generalized quasi Einstein space
  • Gradient Ricci soliton
  • Warped product

ASJC Scopus subject areas

  • Geometry and Topology

Fingerprint Dive into the research topics of 'Conformal Diffeomorphisms of Gradient Ricci Solitons and Generalized Quasi-Einstein Manifolds'. Together they form a unique fingerprint.

  • Cite this