Harmonic morphisms with fibers of dimension one

The harmonic morphisms phi : Mn+1 --> N-n 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 phi is a submersion, in particular, a normal form is found for all such phi 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 greater than or equal to 3, the set of harmonic morphisms with a given Riemannian domain (Mn+1, g) is a finite dimensional space. The third result is the explicit classification when n greater than or equal to 3 of all local and global harmonic morphisms with domain (Mn+1, g), a space of constant curvature.