Infinitesimal harmonic transformations and ricci solitons on complete Riemannian manifolds

Abstract

Ricci solitons were introduced by R. Hamilton as natural generalizations of Einstein metrics. A Ricci soliton on a smooth manifold M is a triple (g0,ξ, λ), where g0 is a complete Riemannian metric, ξ a vector field, and λ a constant such that the Ricci tensor Ric0 of the metric g0 satisfies the equation O2 Ric0 = Lξg0 + 2λgo. The following statement is one of the main results of the paper. Let (g0,ξ, λ) be a Ricci soliton such that M,g0 is a complete noncompact oriented Riemannian manifold, $$ \int\limits_M {\left\| \xi \right\|dv < \infty } $$ , and the scalar curvature s0 of g0 has a constant sign on M, then (M, g0) is an Einstein manifold