Einstein Manifold - Applications

Applications

Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in quantum theories of gravity. The term "gravitational instanton" is usually used restricted to Einstein 4-manifolds whose Weyl tensor is self-dual, and it is usually assumed that metric is asymptotic to the standard metric of Euclidean 4-space (and are therefore complete but non-compact). In differential geometry, self-dual Einstein 4-manifolds are also known as (4-dimensional) hyperkähler manifolds in the Ricci-flat case, and quaternion Kähler manifolds otherwise.

Higher dimensional Lorentzian Einstein manifolds are used in modern theories of gravity, such as string theory, M-theory and supergravity. Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for nonlinear σ-models with supersymmetry.

Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging. Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example.