Гравитационные инстантоны: четырехмерные римановы многообразия в общей теории относительности.

Four dimensional complete Riemannian manifold satisfying the vacuum Einstein equations
In mathematical physics and differential geometry, a gravitational instanton is a four dimensional complete Riemannian manifold satisfying the vacuum Einstein equations. They are so named because they are analogues in quantum theories of gravity of instantons in Yang–Mills theory. In accordance with this analogy with self dual Yang–Mills instantons, gravitational instantons are usually assumed to look like four dimensional Euclidean space at large distances, and to have a self dual Riemann tensor. Mathematically, this means that they are asymptotically locally Euclidean (or perhaps asymptotically locally flat) hyperkähler 4 manifolds, and in this sense, they are special examples of Einstein manifolds. From a physical point of view, a gravitational instanton is a non singular solution of the vacuum Einstein equations with positive definite, as opposed to Lorentzian, metric. There are many possible generalizations of the original conception of a gravitational instanton: for example one can allow gravitational instantons to have a nonzero cosmological constant or a Riemann tensor which is not self dual. One can also relax the boundary condition that the metric is asymptotically Euclidean. There are many methods for constructing gravitational instantons, including the Gibbons–Hawking Ansatz, twistor theory, and the hyperkähler quotient construction.

Gravitational instantons are interesting, as they offer insights into the quantization of gravity. For example, positive definite asymptotically locally Euclidean metrics are needed as they obey the positive action conjecture; actions that are unbounded below create divergence in the quantum path integral. A four dimensional Kähler–Einstein manifold has a self dual Riemann tensor. Equivalently, a self dual gravitational instanton is a four dimensional complete hyperkähler manifold. Gravitational instantons are analogous to self dual Yang–Mills instantons. Several distinctions can be made with respect to the structure of the Riemann curvature tensor, pertaining to flatness and self duality. These include:
Einstein (non zero cosmological constant)
Ricci flatness (vanishing Ricci tensor)
Conformal flatness (vanishing Weyl tensor)
Self duality
Anti self duality
Conformally self dual
Conformally anti self dual

By specifying the 'boundary conditions', i. e. the asymptotics of the metric 'at infinity' on a noncompact Riemannian manifold, gravitational instantons are divided into a few classes, such as asymptotically locally Euclidean spaces (ALE spaces), asymptotically locally flat spaces (ALF spaces). They can be further characterized by whether the Riemann tensor is self dual, whether the Weyl tensor is self dual, or neither; whether or not they are Kähler manifolds; and various characteristic classes, such as Euler characteristic, the Hirzebruch signature (Pontryagin class), the Rarita–Schwinger index (spin 3/2 index), or generally the Chern class. The ability to support a spin structure (i. e. to allow consistent Dirac spinors) is another appealing feature.

Eguchi et al. list a number of examples of gravitational instantons. These include, among others:
Flat space , the torus and the Euclidean de Sitter space , i. e. the standard metric on the 4 sphere. The product of spheres The Schwarzschild metric and the Kerr metric The Eguchi–Hanson instanton , given below. The Taub–NUT solution, given below. The Fubini–Study metric on the complex projective plane Note that the complex projective plane does not support well defined Dirac spinors. That is, it is not a spin structure. It can be given a spinc structure, however. Page space, a rotating compact metric on the direct sum of two complex projective planes The Gibbons–Hawking multi center metrics, given below. The Taub bolt metric and the rotating Taub bolt metric. The "bolt" metrics have a cylindrical type coordinate singularity at the origin, as compared to the "nut" metrics, which have a sphere coordinate singularity. In both cases, the coordinate singularity can be removed by switching to Euclidean coordinates at the origin. The K3 surfaces. The asymptotically locally Euclidean self dual manifolds, including the lens spaces , the double coverings of the dihedral groups, the tetrahedral group, the octahedral group, and the icosahedral group. Note that corresponds to the Eguchi–Hanson instanton, while for higher k, the corresponds to the Gibbons–Hawking multi center metrics. This is an incomplete list; there are others.

It will be convenient to write the gravitational instanton solutions below using left invariant 1 forms on the three sphere S3 (viewed as the group Sp(1) or SU(2)). These can be defined in terms of Euler angles by

Note that for cyclic.

In 2021 it was found that if one views the curvature parameter of a foliated maximally symmetric space as a continuous function, the gravitational action, as a sum of the Einstein–Hilbert action and the Gibbons–Hawking–York boundary term, becomes that of a point particle. Then the trajectory is the scale factor and the curvature parameter is viewed as the potential. For the solutions restricted like this, general relativity takes the form of a topological Yang–Mills theory.