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If you also assume that $(N,\ddot g)$ is complete, then your assumptions imply that $(N,\ddot g)$ is Ricci flat. I proved this in my article The nonexistence of quasi-{E}instein metrics.
If you also assume that $(N,\ddot g)$ is complete, then your assumptions imply that $(N,\ddot g)$ is Ricci flat. I proved this in my article The nonexistence of quasi-Einstein metrics.
If you also assume that $(N,\ddot g)$ is complete, then your assumptions imply that $(N,\ddot g)$ is Ricci flat. I proved this in my article The nonexistence of quasi-{E}instein metrics.
