The Nonexistence of Conformal Deformations on Riemannian Warped Product Manifolds

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations onRiemannian warped product manifold M =(a, ∞ )× f N with prescribed scalar curvature functions.Key words : Warped Product, Scalar Curvature, Partial Differential Equation 1. Introduction In a recent study [7-9] , M.C. Leung has studied theproblem of scalar curvature functions on Riemannianwarped product manifolds and obtained partial resultsabout the existence and nonexistence of Riemannianwarped metric with some prescribed scalar curvaturefunction. He has studied the uniqueness of positive solu-tion to equation(1.1)where is the Laplacian operator for an n-dimen-sional Riemannian manifold (N, g 0 ) and d n = n−2/4(n−1). Equation (1.1) is derived from the conformal defor-mation of Riemannian metric [1,4-6,8,9] .Similarly, let (N, g 0 ) be a compact Riemanniandimensional manifold. We consider the (n+1)−dimen-sional Riemannian warped product manifold M =(a, ∞)× f N with the metric g = dt