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Jan 11, 2018 at 13:13 comment added Tobias Fritz In the pseudo-Riemannian case, the FLRW metric for $k=0$ (spatially flat cosmology) is a nice example of the same flavour.
Jan 11, 2018 at 13:00 comment added MathDG I'm not necessarily interested in an example, I'm interested in knowing if $(M, g)$ must be necessarily Ricci flat or not.
Jan 11, 2018 at 12:54 comment added MathDG thank you very much for your precious answer. And if $(N, \ddot{g})$ is a surface and $(B, \bar{g})$ is a fiber-surface Ricci-flat, $(M, g)$ must be Ricci flat or not necessarily?
Jan 11, 2018 at 12:44 history answered Thomas Richard CC BY-SA 3.0