Harmonic Morphisms with Fibers of Dimension One

The harmonic morphisms φ : M → Nn are studied using the methods of the moving frame and exterior differential systems and three main results are achieved. The first result is a local structure theorem for such maps in the case that φ is a submersion, in particular, a normal form is found for all such φ once the metric on the target manifold N is specified. The second result is a finiteness theorem, which says, in a certain sense, that, when n ≥ 3, the set of harmonic morphisms with a given Riemannian domain ( M, g ) is a finite dimensional space. The third result is the explicit classification when n ≥ 3 of all local and global harmonic morphisms with domain ( M, g ) , a space of constant curvature. 0. Introduction A smooth map φ : M → N between Riemannian manifolds is said to be a harmonic morphism if, for any harmonic function f on any open set V ⊂ N , the pullback f ◦ φ is a harmonic function on φ(V ) ⊂ M . By a simple argument (see §1), any non-constant harmonic morphism φ : M → N between connected Riemannian manifolds must be a submersion away from a set of measure zero in M . Thus, a necessary condition for the existence of a non-constant harmonic map φ : M → N is that dimM ≥ dimN . When the dimension of N is 1, so that N can be regarded, at least locally, as R with its standard metric, a map φ : M → N is a harmonic morphism if and only if it is a harmonic function in the usual sense. Thus, at least locally, there are many harmonic morphisms from M to N. However, when the dimension of N is greater than 1, the condition of being a harmonic morphism turns out to be much more restrictive, being essentially equivalent to an overdetermined system of pde for the map φ. Thus, for generic Riemannian metrics onM andN , one does not expect there to be any harmonic morphisms, even 1991 Mathematics Subject Classification. 58E20, 58A15, 53C12.