Yang–Mills connection

A connection whose curvature is a critical point of the Yang–Mills functional, equivalently satisfying the Yang–Mills equation.

Yang–Mills connection

Let $P\to M$ be a principal $G$ -bundle over an oriented Riemannian manifold.

Definition (Yang–Mills connection)

A principal connection $A$ on $P$ is called a Yang–Mills connection if it satisfies the Yang–Mills equation

d_A(*F_A)=0,

where $F_A$ is its curvature .

Equivalently, $A$ is a Yang–Mills connection if it is a critical point of the Yang–Mills functional with respect to compactly supported variations.

Flat connections. Any flat connection is Yang–Mills, since its curvature vanishes.
Anti-self-dual connections. On an oriented 4-manifold, ASD (or SD) connections are Yang–Mills; these are the basic instanton solutions in gauge theory.
Constant central curvature on surfaces. On a closed oriented surface, Yang–Mills connections are precisely those whose curvature is covariantly constant and takes values in the center of the Lie algebra (a two-dimensional special feature).