Constant scalar curvatures on warped product manifolds
The authors study the question when a pseudo-Riemannian metric of constant scalar curvature can be produced on a product manifold by a warped product construction. Specifically, they produce results for two different classes of pseudo-Riemannian manifolds. The first class consists of generalized Robertson-Walker spacetimes, where the base manifold \(B\) is an open interval with negative definite metric and the fiber \(F\) is a Riemannian manifold of dimension greater than 1 and with constant scalar curvature. In this case, the problem reduces to studying an associated differential equation. The second class consists of warped products, where the base \(B\) is a compact Riemannian manifold and the fiber \(F\) is a pseudo-Riemannian manifold of constant scalar curvature. In this case, the method of upper and lower solutions of partial differential equations is employed, as well as variational arguments.
scientific article; zbMATH DE number 1102816