Infinitesimal harmonic transformations and ricci solitons on complete Riemannian manifolds

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 ò2 Ric₀ = L_ξg₀ + 2λ_go. 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 } $ \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