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In this paper, we prove the existence of warping functions on Riemannian warped product manifolds with some prescribed scalar curvatures according to the fiber manifolds of class (B).

In this paper, we prove the existence and the nonex-istence of warping functions on Riemannian warped product mani-folds with some prescribed scalar curvatures according to the bermanifolds of class (A). 1. IntroductionOne of the basic problems in the di erential geometry is studying theset of curvature functions which a given manifold possesses.The well-known problem in di erential geometry is that of whetherthere exists a warping function of warped metric with some prescribedscalar curvature function. One of the main methods of studying di er-ential geometry is by the existence and the nonexistence of a warpedmetric with prescribed scalar curvature functions on some Riemannianwarped product manifolds. In order to study these kinds of problems,we need some analytic methods in di erential geometry.For Riemannian manifolds, warped products have been useful in pro-ducing examples of spectral behavior, examples of manifolds of negativecurvature (cf. [2, 3, 4, 5, 7, 13, 14]), and also in studying L

In this paper, we prove the existence of warping functions on Riemannian warped product manifolds with fiber manifold of class (B) having some prescribed scalar curvatures.

In this paper, when N is a compact Riemannian manifold of class (C), we consider the nonexistence of some warping functions on Riemannian warped product manifolds with prescribed scalar curvatures.

In this paper, we consider the problem of achieving constant scalar curvature on warped product manifolds according to fiber manifolds with constant scalar curvature.