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 ${\bar{G}}_{AB}=-\bar{\mathrm{\Lambda }}{\bar{g}}_{AB}$, with cosmological constant $\bar{\mathrm{\Lambda }}$, 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 ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$, respectively, where $\bar{\mathrm{\Lambda }}$, ${\mathrm{\Lambda }}_1$ and ${\mathrm{\Lambda }}_2$ are functions of $f_1$, $f_2$ and $n_1$, $n_2$.