Calabi-Yau manifold

Definition

A Kahler manifold is said to be a Calabi-Yau manifold if its first Chern class vanishes.

A Calabi-Yau manifold of complex dimension $n$ is termed a Calabi-Yau $n$-fold.

Facts

Ricci-flatness

Yau's theorem tells us that every Calabi-Yau manifold admits a Ricci-flat metric, viz a compatible metric suchthat the Ricci curvature tensor is zero at all points.