A Ricci-flat metric is a type of geometric structure on a manifold where the Ricci curvature is zero. This means that, locally, the manifold behaves like flat space, even if it may be curved globally. Ricci-flat metrics are important in the study of general relativity and string theory, as they can describe spaces that are solutions to certain physical equations. One well-known example of a Ricci-flat metric is the K3 surface, a complex manifold that has rich geometric properties. Another example is the Calabi-Yau manifold, which plays a crucial role in compactifying extra dimensions in string theory. These metrics help mathematicians and physicists understand the underlying geometry of various theories.