Complex differential geometry and nonlinear differential equations, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 49, 145-184 (1986).
[For the entire collection see Zbl 0575.00016.]
Some holomorphic objects are used to study harmonic morphisms. Firstly, the horizontal quadratic differential analogous to the quadratic differential for harmonic maps of a surface is introduced for a smooth map : with rank 2 somewhere, where M and N are Riemannian manifolds with dimensions m and 2 respectively. Using the horizontal quadratic differential, the author proves: Let : be a harmonic submersion to the 2-sphere with any Riemannian metric. Suppose has minimal fibres and integrable horizontal distribution with leaves homeomorphic to 2-spheres. Then must be a harmonic morphism.
Let denote the Grassmann bundle over a smooth Riemannian manifold M. Associated to a smooth distribution V of dimension k on M, we have the Gauss section : . The Gauss section of the associated horizontal distribution H will be denoted by : . By defining the holomorphicity properties of the Gauss sections of 2-dimensional distributions, the author proves: Let V be a 2- dimensional distribution in a 4-dimensional Riemannian manifold M. Then (i) V is integrable and minimal if and only if its Gauss section : is vertically antiholomorphic. (ii) V is conformal if and only if the Gauss section : is horizontally holomorphic.
Under the same assumption in the above, let (resp. be the fibre bundle over M whose fibre at p consists of all metric almost complex structures on the tangent space at p which are orientation preserving (resp. reversing). The distribution V defines sections , . The above result is translated as follows: Let V be a 2-dimensional distribution on a 4- dimensional Riemannian manifold. Then V is integrable minimal and conformal if and only if the section is holomorphic with respect to the almost complex structure defined by and the section is antiholomorphic with respect to the almost complex structure defined by
Harmonicity properties of the Gauss sections are also investigated.