When we have an Einstein warped manifold where the base is a Riemannian manifold, independently of dimension, and the fiber is a Ricci-flat, we have: $|\nabla f|^2+[\frac{\lambda (m-n)+ R}{m(m-1)}]f^2=0$ (with $n$ and $m$ the dimension of the base and the fiber, respectively and $R$ is the scalar curvature of the base). Then, either $R$ $\leq$ $\lambda (n − m)$ or $f$ is trivial. This means that the Einstein warped manifold will not necessarily be flat (for details see, for example, https://www.researchgate.net/publication/319151263_On_the_structure_of_Einstein_warped_product_semi-Riemannian_manifolds).

When we have an Einstein warped-product manifold where the base is a Riemannian manifold, independently of dimension, and the fiber is a Ricci-flat, we have: $|\nabla f|^2+[\frac{\lambda (m-n)+ R}{m(m-1)}]f^2=0$ (with $n$ and $m$ the dimension of the base and the fiber, respectively and $R$ is the scalar curvature of the base). Then, either $R$ $\leq$ $\lambda (n − m)$ or $f$ is trivial. This means that the Einstein warped-product manifold will not necessarily be flat even though the fiber is Ricci-flat (for details see, for example, https://www.researchgate.net/publication/319151263_On_the_structure_of_Einstein_warped_product_semi-Riemannian_manifolds).