Some geometric consequences from the Ricci-flow evolution of the geodesic distance function

Let (M,g) be a compact Riemannian manifold and {gt}tisanelementof[0,μ[ a Ricci deformation of the metric g. Denote by θg and θt the volume density functions of g and gt respectively and put θend:=limt→μθt. In this note some geometric consequences and controlling results on the quotient (θend)/θg are deduced from the evolution under the Ricci flow of the density volume θg and the distance function. (author)