Multiply-warped product metrics and reduction of Einstein equations

It is shown that for every multidimensional metric in the multiply-warped product form (M) over bar = K x f(1) M-1 x 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 (G) over bar (AB) = -(A) over bar(g) over bar (AB), with cosmological constant (A) over bar, which are reducible to the Einstein equations G(alpha beta) = -Delta(1g alpha beta) and G(ij) = -Delta(2)h(ij) on the submanifolds M-1, M-2, with cosmological constants Lambda(1) and Lambda(2), respectively, where Lambda, Lambda(1) and Lambda(2) are functions of f(1), f(2) and n(1), n(2).