Magnetic maps
The authors introduce magnetic maps as critical points of the Landau-Hall integral and characterize them in terms of prescribed tension fields. Let \((N, h)\) and \((M, g)\) be two Riemannian manifolds and let \(f: (N, h)\longrightarrow (M, g)\) be a smooth map. Let \(\xi\) be a global vector field on \(N\) and \(\omega\) be a 1-form on \(M\). Then they prove that \(f\) is a magnetic map with respect to \(\xi\) and \(\omega\) if and only if it satisfies NEWLINE\[NEWLINE \tau(f) = \phi(f_{*}\xi), NEWLINE\]NEWLINE where \(\tau(f)\) is the tension field of \(f\) and \(\omega\) is defined by \(g(\phi X, Y)= d\omega(X, Y)\), for all \(X\), \(Y\) tangent on \(M\). They provide some examples in Euclidean, contact and Kähler manifolds, and study some classes of magnetic surfaces in Euclidean 3-spaces.
scientific article; zbMATH DE number 6362162
scientific article