In this paper, we consider the compact, connected, orientable Riemannian manifold M of dimension n admitting a projective vector field and obtain the series of integral formulas and inequalities without assuming that the scalar curvature of M is constant . Using these integral formulas,it is obtained that the necessary and sufficient conditions for the projective vector field to be the Killing vector field. Further, if the scalar curvature of M is constant on M , then the integral formulas , inequalities and the Propositions of H. Hiramatu¹ will include as special cases from the main Lemmas and Theorems.