For an example, let $N$ be a compact complex hypersurface of degree $m+1$ of the complex projective space $\mathbb{CP}^m$ with complex dimension $m\ge 3$ (for $m=3$ this is a complex $K3$ surface, as mentioned above). The first Chern class of $N$ vanishes, and hence $N$ admits a Ricci-flat but nonflat Riemannian metric, by a theorem of Yau.

For an example, let $N$ be a compact complex hypersurface of degree $m+1$ of the complex projective space $\mathbb{CP}^m$ with complex dimension $m\ge 3$ (for $m=3$ this is a complex $K3$ surface). The first Chern class of $N$ vanishes, and hence $N$ admits a Ricci-flat but nonflat Riemannian metric, by a theorem of Yau.