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Supplemental material, sj-pdf-1-jmx-10.1177_00222429221151058 for The Role of Advertising in High-Tech Medical Procedures: Evidence from Robotic Surgeries by Tae Jung Yoon and TI Tongil Kim in Journal of Marketing

In this paper, we consider Ricci curvature of conformal deformation on compact 2-manifolds. And we prove that, by the conformal deformation, the resulting manifold is an Einstein manifold.

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, 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

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations on space-times M = with prescribed scalar curvature functions.

In this paper, when N is a compact Riemannian manifold of class (A), we consider the nonexistence of some warping functions on space-times M = 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.

In this paper, when N is a compact Riemannian manifold, using warped products, we discuss the conformal deformation of a warped product metric on with specific warping functions.

Using Riccati equation techniques and the Raychaudhuri equation from General Relativity, volume comparison results are obtained for compact geodesic wedges in the chronological future of some point in a globally hyperbolic space-time and corresponding wedges in a Lorentzian space-form.