In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in a vacuum with vanishing cosmological constant.
In Lorentzian geometry, a number of Ricci-flat metrics are known from works of Karl Schwarzschild, Roy Kerr, and Yvonne Choquet-Bruhat.
In Riemannian geometry, Shing-Tung Yau's resolution of the Calabi conjecture produced a number of Ricci-flat metrics on Kähler manifolds.
[6] However, there are homogeneous (and even symmetric) Lorentzian manifolds which are Ricci-flat but not flat, as follows from an explicit construction and computation of Lie algebras.
[7] Until Shing-Tung Yau's resolution of the Calabi conjecture in the 1970s, it was not known whether every Ricci-flat Riemannian metric on a closed manifold is flat.
[9] Relative to harmonic coordinates, the condition of Ricci-flatness for a Riemannian metric can be interpreted as a system of elliptic partial differential equations.
[10] Analogously, relative to harmonic coordinates, Ricci-flatness of a Lorentzian metric can be interpreted as a system of hyperbolic partial differential equations.
She reached a definitive result in collaboration with Robert Geroch in the 1960s, establishing how a certain class of maximally extended Ricci-flat Lorentzian metrics are prescribed and constructed by certain Riemannian data.
This is already indicated by the fundamental distinction between the geodesically complete metrics which are typical of Riemannian geometry and the maximal globally hyperbolic developments which arise from Choquet-Bruhat and Geroch's work.
A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must satisfy the Hitchin–Thorpe inequality on its topological data.
Every Ricci-flat Riemannian manifold in this class is flat, which is a corollary of Cheeger and Gromoll's splitting theorem.
This condition on a Riemannian manifold may also be characterized (roughly speaking) by the existence of a 2-sphere of complex structures which are all parallel.
[18] Marcel Berger commented that all known examples of irreducible Ricci-flat Riemannian metrics on simply-connected closed manifolds have special holonomy groups, according to the above possibilities.