Harmonic morphisms and circle actions on 3- and 4-manifolds

Baird, Paul

Annales de l'Institut Fourier, Tome 40 (1990), p. 177-212 / Harvested from Numdam

Résumé

On considère les morphismes harmoniques comme généralisation naturelle des fonctions analaytiques qu’on rencontre dans la théorie des surfaces de Riemann. On montre que chaque variété fermée et analytique à 3 dimensions qui supporte un morphisme harmonique à valeurs dans une surface de Riemann est un espace fibré de Seifert. On étudie les morphismes harmoniques $\phi :M\rightarrow N$ définies sur une variété fermée à 4 dimensions et à valeurs dans une variété à 3 dimensions. Ceux-ci déterminent une action du cercle sur $M$ qui est localement différentiable, peut-être avec des points fixes. Par conséquent la topologie de $M$ est limitée. Dans chaque cas, un morphisme harmonique $\phi : M \rightarrow N$ défini sur une variété fermée à $n+1$ dimensions et à valeurs dans une variété à $n$ dimensions ($n\ge 2$, avec $M$, $N$ analytiques dans le cas où $n=2$) détermine une action du cercle sur $M$ qui est localement différentiable.

Abstract

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 $\phi : M\rightarrow N$ 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 $\phi : M\rightarrow N$ from a closed $(n+1)$-dimensional manifold to an $n$-dimensional manifold (n$\ge 2$, with $M$, $N$ analytic in the case $n=2)$ determines a locally smooth circle action on $M$.

Détail

Publié le : 1990-01-01

MR 91e:57025 | Zbl 0676.58023

DOI : https://doi.org/10.5802/aif.1210

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