Applications
Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
Ricci curvature also appears in the Ricci flow equation, where a time-dependent Riemannian metric is deformed in the direction of minus its Ricci curvature. This system of partial differential equations is a non-linear analog of the heat equation, and was first introduced by Richard Hamilton in the early 1980s. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, Ricci flow may be hoped to produce an equilibrium geometry for a manifold for which the Ricci curvature is constant. Recent contributions to the subject due to Grigori Perelman now show that this program works well enough in dimension three to lead to a complete classification of compact 3-manifolds, along lines first conjectured by William Thurston in the 1970s.
On a Kähler manifold, the Ricci curvature determines the first Chern class of the manifold (mod torsion). However, the Ricci curvature has no analogous topological interpretation on a generic Riemannian manifold.
Read more about this topic: Ricci Curvature