Multiply-warped product metrics and reduction of Einstein equations

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 $\bar{G}_{AB}=-\barΛ\bar{g}_{AB}$, with cosmological constant $\barΛ$, which are reducible to the Einstein equations $G_{αβ} = -Λ_1 g_{αβ}$ and $G_{ij} =-Λ_2 h_{ij}$ on the submanifolds $M_1$, $M_2$, with cosmological constants ${Λ_1}$ and ${Λ_2}$, respectively, where $\barΛ$, ${Λ_1}$ and ${Λ_2}$ are functions of ${f_1}$, ${f_2}$ and $n_1$, $n_2$.

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