Harmonic morphisms with fibers of dimension one
Robert L. Bryant · Communications in Analysis and Geometry · 2000
The harmonic morphisms (j) : M n+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 (f) is a submersion, in particular, a normal form is found for all such (j) 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 n+1 , #) 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 n+1 ,p), a space of constant curvature.