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Let $(M, g) = (N_1, g_1) \times_f(N_2, g_2)$ be an Einstein warped-product manifold, with metric $g=g_1+f^2g_2$.

What does it mean if the scalar curvature of its base-manifold $(N_1, g_1)$, equal to a multiple of the $f$ warping function?

Does it has any geometrical or physical meaning?

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