Harmonic morphisms, foliations and Gauss maps. (English) Zbl 0592.53020
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 $\phi$ : $M\to N$ 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 $\phi$ : $M\to (S\sp 2,h)$ be a harmonic submersion to the 2-sphere with any Riemannian metric. Suppose $\phi$ has minimal fibres and integrable horizontal distribution with leaves homeomorphic to 2-spheres. Then $\phi$ must be a harmonic morphism.
Let $G\sb k(TM)$ 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 $\gamma$ : $M\to G\sb k(TM)$. The Gauss section of the associated horizontal distribution H will be denoted by $\gamma$ : $M\to G\sb 2(TM)$. 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 $\gamma$ : $M\to G\sb 2(TM)$ is vertically antiholomorphic. (ii) V is conformal if and only if the Gauss section $\gamma$ : $M\to G\sb 2(TM)$ is horizontally holomorphic.
Under the same assumption in the above, let $Z\sp+$ (resp. $Z\sp-)$ 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 $\gamma\sb 1: M\to Z\sp+$, $\gamma\sb 2: M\to Z\sp-$. 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 $\gamma\sb 1$ is holomorphic with respect to the almost complex structure defined by $\gamma\sb 2$ and the section $\gamma\sb 2$ is antiholomorphic with respect to the almost complex structure defined by $\gamma\sb 1.$
Harmonicity properties of the Gauss sections are also investigated.
|53C12||Foliations (differential geometry)|
|53C15||Differential geometric structures on manifolds|
|58E20||Harmonic maps between infinite-dimensional spaces|