Constant scalar curvatures on warped product manifolds

The Completeness of Some Metrics on Lorentzian Warped Product Manifolds With Fiber Manifold of Class (B)

“…Then, by Theorem 3.1, Theorem 3.5 and Theorem 3.7 in [4], we have the following proposition. Proposition 2.1 implies that in Lorentzian warped product there is no obstruction of the existence of metric with positive scalar curvature.…”

“…And also in [4], authors considered the existence of a nonconstant warping function on a Lorentzian warped product manifold such that the resulting warped product metric produces the constant scalar curvature when the fiber manifold has the constant scalar curvature.…”

“…Ironically, even though there exists some obstruction of positive or zero scalar curvature on a Riemannian manifold, results of [4], say, Theorem 3.1, Theorem 3.5 and Theorem 3.7 of [4] show that there exists no obstruction of positive scalar curvature on a Lorentzian warped product manifold, but there may exist some obstruction of negative or zero scalar curvature.…”

Special standard static space–times

“…Thus we should focus on standard static space-times with compact fibers of nonconstant scalar curvatures. In [15], a similar problem was considered on a wider class of warped products (see also [14]). In order to make use of [15], we introduce a linear operator L :…”

“…In [23], it is proven that an Einstein Riemannian warped product with a non-positive scalar curvature and compact base is just a trivial Riemannian product. Constant scalar curvature of warped products was studied in [10,12,14,15] when the base is compact and of generalized Robertson-Walker space-times in [14]. Furthermore, partial results for warped products with non-compact base were obtained in [6] and [9].…”