Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0592.53020
Wood, J.C.
Harmonic morphisms, foliations and Gauss maps. (English)
[A] 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.] \par 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. \par 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. \par 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 $\gamma\sb 1.$ \par Harmonicity properties of the Gauss sections are also investigated.
[T.Ishihara]

MSC 2000:: *53C12 Foliations (differential geometry)
53C15 Geometric structures on manifolds
58E20 Harmonic maps between infinite-dimensional spaces

Keywords: Gauss maps; twistor bundles; harmonic morphisms; harmonic maps; harmonic submersion; Grassmann bundle; Gauss section; almost complex structures

Citations: Zbl 0575.00016

Cited in: Zbl 0952.53031

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]