On gradient Ricci soliton space-time warped product with potentially infinite metric

Abstract

The idea of potentially infinite metrics on space-time was given by Bennett Chow and Sun-Chin Chu in 1995. After that Perelman introduced two new ideas about such metric in 2002. In this article, we use this idea in space-time warped product manifold and study gradient Ricci soliton space-time warped product. First, we introduce the notion of space-time warped product, $P=(B\times I)\times _f F$ with potentially infinite metric, ${\mathcal {G}}=(g_B+(R+\frac{N}{2t} )\textrm{d}t^2)+f^2g_F.$ Then, we investigate the behavior of Ricci curvature tensor up to $O(N^{-1})$ and discuss potentially gradient Ricci soliton for space-time warped product. Next, we prove existence conditions for gradient Ricci soliton space-time warped product. Then, we discuss several results for shrinking, steady, or expanding gradient Ricci soliton $(P,{\mathcal {G}},\nabla \phi , \lambda ),$ with compact base and fiber with at least dimension two. After that, we emphasize the construction of class of expanding Ricci soliton. Furthermore, we discuss the compactness of space-time manifold when space-time warped product satisfies some inequality. We also give examples of generalized black holes, which metric can be written as space-time warped product with potentially infinite metric.