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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 ϕ : MN 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 ϕ : M(S 2 ,h) be a harmonic submersion to the 2-sphere with any Riemannian metric. Suppose ϕ has minimal fibres and integrable horizontal distribution with leaves homeomorphic to 2-spheres. Then ϕ must be a harmonic morphism.

Let G 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 γ : MG k (TM). The Gauss section of the associated horizontal distribution H will be denoted by γ : MG 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 γ : MG 2 (TM) is vertically antiholomorphic. (ii) V is conformal if and only if the Gauss section γ : MG 2 (TM) 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 γ 1 :MZ + , γ 2 :MZ - . 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 γ 1 is holomorphic with respect to the almost complex structure defined by γ 2 and the section γ 2 is antiholomorphic with respect to the almost complex structure defined by γ 1 ·

Harmonicity properties of the Gauss sections are also investigated.

Reviewer: T.Ishihara

MSC:
53C12Foliations (differential geometry)
53C15Differential geometric structures on manifolds
58E20Harmonic maps between infinite-dimensional spaces