On the relation between Ricci-Harmonic solitons and Ricci solitons

Abstract

Let (Mm,gij) and (Nn,hβγ) be two Riemannian manifolds, and ϕ:M→N a smooth map. By definition, a gradient Ricci-Harmonic soliton satisfies(0.1){Rij−α∇iϕ∇jϕ+∇i∇jf=λgij;τgϕ=∇iϕ∇if, for some f∈C∞(M) and constants α and λ. Here τgϕ=trg(∇dϕ) is the tension filed of ϕ. We prove that when α>0 and the sectional curvature of N is bounded from above by αm, any shrinking or steady Ricci-Harmonic soliton (i.e., λ>0 or λ=0, respectively) must be a Ricci soliton, namely, ϕ is a constant map. In particular, it implies that the shrinking and steady solitons generated from Bernhard List's flow [9] are exactly the corresponding solitons of the Ricci flow, and hence some recent results regarding the shrinking solitons of List's flow are actually duplications of the previous results for Ricci solitons.