Anatomy of Einstein manifolds

An Einstein manifold in four dimensions has some configuration of SU(2)(+) Yang-Mills instantons and SU(2)_ anti-instantons associated with it. This fact is based on the fundamental theorems that the four-dimensional Lorentz group Spin(4) is a direct product of two groups SU(2)(+/-) and the vector space of 2-forms decomposes into the space of self-dual and anti-self-dual 2-forms. It explains why the four-dimensional spacetime is special for the stability of Einstein manifolds. We now consider whether such a stability of four-dimensional Einstein manifolds can be lifted to a fivedimensional Einstein manifold. The higher-dimensional embedding of four-manifolds from the viewpoint of gauge theory is similar to the grand unification of the Standard Model, since the group SO(4) congruent to Spin(4)/Z (2) = SU(2)(+) circle times SU(2)_/Z(2) must be embedded into the simple group SO(5) = Sp(2)/Z(2). Our group-theoretic approach reveals the anatomy of Riemannian manifolds quite similar to the quark model of hadrons in which two independent Yang-Mills instantons represent a substructure of Einstein manifolds.