On gradient Ricci soliton space-time warped product

Abstract

In this article, we introduce the notion of space-time warped product $\displaystyle \tilde{M}=(B\times I)\times _f F$ with potentially infinite metric $\displaystyle \tilde{g}=(g_{\textrm{B}}+(R+\frac{N}{2t} )dt^2)+f^2g_F.$ First we find connection and Ricci curvatures of space-time manifold, $\displaystyle (B\times I, g_{\textrm{B}}+(R+\frac{N}{2t})dt^2)$. Then, we discuss Ricci curvature approximation upto $O(N^{-1})$ and potentially gradient Ricci soliton identities for space-time warped product. We also investigate the Deturck’s trick and study Variation of metric $\tilde{g}$ on $\displaystyle (B\times I)\times _f F$. Next, we prove Existence conditions for the gradient Ricci soliton space-time warped product. For a compact base and fiber manifold of dimension at least two, we obtain several results for expanding or steady and shrinking gradient Ricci soliton $(\tilde{M}, \tilde{g},\tilde{\nabla }\phi , \lambda )$. Further, we prove the compactness of space-time manifold $B\times I$ when it satisfy some inequality. Finally, we give some examples of generalized black hole solutions whose metrics can be written as a space-time warped product metric.