Rigid properties of quasi-Einstein metrics

Abstract

Abstract. In this paper we get some rigid results for m−dimensional quasiEinstein metrics on complete Riemannian manifolds.

On noncompactτ-quasi-Einstein metrics

“…It was also proved in [Ivey 1993] that any expanding or steady gradient Ricci solitons on closed manifolds should be trivial. The same rigid properties for the τ -quasiEinstein metrics on closed manifolds were proved in [Kim and Kim 2003;Wang 2011]. But for the τ -quasi-Einstein metrics on closed manifolds with λ > 0, the rigid properties rely on the constant µ which appears in the following identity:…”

“…This identity was proved in [Kim and Kim 2003]. See also [Wang 2011], where the author proved that the quasi-Einstein metrics with λ > 0 should be trivial when µ ≤ 0. In fact, the authors of [Lü et al 2004] constructed nontrivial τ -quasi-Einstein metrics with λ > 0 and τ > 1, which also satisfy µ > 0.…”

“…There has been an active interest in the study of the weighted measure under some conditions about the τ -BakryÉmery Ricci curvature tensor; see [Li 2005;Wang 2010] and the references therein. According to [Kim and Kim 2003;Case 2010;Case et al 2011;Wang 2011], we call a metric g τ -quasi-Einstein with potential function f , if for some constant λ,…”

“…Later, Wang [2011] studied the lower bound estimate for scalar curvature R on complete noncompact τ -quasi-Einstein metrics with λ ≤ 0. We state this result as follows.…”

On Ricci-harmonic metrics

“…The works on the quasi Einstein metric can be referred to [5,6,7,15,36,37,38,39,40] and the references therein. Naturally, we will study the τ -quasi Ricci-harmonic metric, which is defined as follows.…”