Harmonic morphisms and circle actions on 3- and 4-manifolds
Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms \(\phi\) : \(M\to N\) from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on M with possible fixed points. This restricts the topology of M. In all cases, a harmonic morphism \(\phi\) : \(M\to N\) from a closed \((n+1)\)-dimensional manifold to an n- dimensional manifold (n\(\geq 2\), with M,N analytic in the case \(n=2)\) determines a locally smooth circle action on M.
generalization of the analytic functions
scientific article; zbMATH DE number 4107782
scientific article