Yang-Mills theory and conjugate connections.
Let \(E\) be a vector bundle with a bundle metric \(h\) over a Riemannian manifold \((M,g)\). The authors define the notion of a Yang-Mills connection \(D\) in the bundle \(E\) in the standard way (as a critical point of the Yang-Mills functional), dropping the standard assumption that the connection \(D\) preserves the metric \(h\). In some cases, they express the corresponding Yang-Mills equation in terms of the connection \(D^*\), which is metrically conjugate to the connection \(D\). For example, let \((D,g)\) be a Weyl structure on a 4-manifold \(M\) that is a Riemannian metric \(g\) and a torsion-free linear connection \(D\) which preserves the conformal structure \([g]\). Then \(D\) is a minimum of the Yang-Mills functional if and only if the curvatures \(R^D\) and \(R^{D^*}\) of the connections \(D, D^*\) satisfy the following generalization of the self-duality equation: \[ \ast R^D = R^{D^*}. \] As another application, they prove the following theorem: Let \(f: M \rightarrow \mathbb R^{n+1}\) be a non-degenerate affine immersion of an \(n\)-manifold, \(D\) the induced connection on \(M\), \(h\) the affine second fundamental form and \(S\) the affine shape operator. Then \(D\) is a Yang-Mills connection on the manifold \((M,h)\) if and only if the dual connection \(D^*\) satisfies the equation \[ D^*_X(SY) = S(D_XY) \] for any vector fields \(X,Y\) on \(M\). In particular, if \(f(M)\) is an affine hypersphere (i.e. \(S = c \text{ Id}\) for a nonzero constant \(c\)), then \(D\) is a Yang-Mills connection if and only if \(f(M)\) is a quadratic affine hypersurface.
scientific article; zbMATH DE number 1970799