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 (g(0), xi, lambda), where g(0) is a complete Riemannian metric, xi a vector field, and lambda a constant such that the Ricci tensor Ric(0) of the metric g(0) satisfies the equation -2Ric(0) = L xi g(0) + 2 lambda g(0). The following statement is one of the main results of the paper. Let (g(0), xi, lambda) be a Ricci soliton such that (M, g(0)) is a complete noncompact oriented Riemannian manifold, integral(M) parallel to xi parallel to dv < infinity, and the scalar curvature s(0) of g(0) has a constant sign on M, then (M, g(0)) is an Einstein manifold.