# Calabi-Yau manifold

From Diffgeom

Template:Kahler manifold property

## 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 is termed a Calabi-Yau -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.