Multiply-warped product metrics and reduction of Einstein equations

Abstract

It is shown that for every multidimensional metric in the multiply warped product form $\bar{M}=K{\times }_{f_1}{M}_1{\times }_{f_2}{M}_2$ with warp functions $f_1$, $f_2$, associated to the submanifolds $M_1$, $M_2$ of dimensions $n_1$, $n_2$ respectively, one can find the corresponding Einstein equations ${\bar{G}}_{AB}=-\bar{\Lambda }{\bar{g}}_{AB}$, with cosmological constant $\bar{\Lambda }$, which are reducible to the Einstein equations $G_{\alpha \beta }=-{\Lambda }_1{g}_{\alpha \beta }$ and $G_{ij}=-{\Lambda }_2{h}_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${\Lambda }_1$ and ${\Lambda }_2$, respectively, where $\bar{\Lambda }$, ${\Lambda }_1$ and ${\Lambda }_2$ are functions of $f_1$, $f_2$ and $n_1$, $n_2$.

12 pages, Reference updated

Cited by in corpus (2)

• Classification of Einstein equations with cosmological constant in warped product space-time
• On Einstein equations with cosmological constant in braneworld models