special case of the Yang–Mills equations, which come from quantum field theory, in contrast to the regular Einstein equations which come from general relativity. In the case where the holomorphic vector bundle is again the holomorphic tangent bundle and the Hermitian metric is the Kähler metric, the Hermite–Einstein equation reduces to the Kähler–Einstein equation. In general however, the geometry of the Kähler manifold is often fixed and only the bundle metric is allowed to vary, and this causes the Hermite–Einstein equation to be easier to study than the Kähler–Einstein equation in general. In particular, a complete algebro-geometric characterisation of the existence of solutions is given by the Kobayashi–Hitchin correspondence. References Notes Differential geometry