*(English)*Zbl 0592.53020

[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 $\varphi $ : $M\to N$ 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 $\varphi $ : $M\to ({S}^{2},h)$ be a harmonic submersion to the 2-sphere with any Riemannian metric. Suppose $\varphi $ has minimal fibres and integrable horizontal distribution with leaves homeomorphic to 2-spheres. Then $\varphi $ must be a harmonic morphism.

Let ${G}_{k}\left(TM\right)$ 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 $ : $M\to {G}_{k}\left(TM\right)$. The Gauss section of the associated horizontal distribution H will be denoted by $\gamma $ : $M\to {G}_{2}\left(TM\right)$. 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 $ : $M\to {G}_{2}\left(TM\right)$ is vertically antiholomorphic. (ii) V is conformal if and only if the Gauss section $\gamma $ : $M\to {G}_{2}\left(TM\right)$ is horizontally holomorphic.

Under the same assumption in the above, let ${Z}^{+}$ (resp. ${Z}^{-})$ 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 ${\gamma}_{1}:M\to {Z}^{+}$, ${\gamma}_{2}:M\to {Z}^{-}$. 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 ${\gamma}_{1}$ is holomorphic with respect to the almost complex structure defined by ${\gamma}_{2}$ and the section ${\gamma}_{2}$ is antiholomorphic with respect to the almost complex structure defined by ${\gamma}_{1}\xb7$

Harmonicity properties of the Gauss sections are also investigated.

##### MSC:

53C12 | Foliations (differential geometry) |

53C15 | Differential geometric structures on manifolds |

58E20 | Harmonic maps between infinite-dimensional spaces |