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Harmonic maps of $$S^ 2$$ into a complex Grassmann manifold. (English) Zbl 0601.58023
The authors study harmonic maps from the Riemann sphere $$S^ 2$$ to the complex Grassmann manifold $$G_{2,n}$$ of complex 2-planes in $$C^ n$$. Two processes are described called ”crossing” and ”turning” which, when applied repeatedly, change a harmonic map $$\phi$$ : $$S^ 2\to G_{2,n}$$ into a harmonic map $$\phi_ 0: S^ 2\to G_{2,n}$$ describable in terms of holomorphic maps. These processes depend for their success on the automatic vanishing of holomorphic differentials on $$S^ 2$$. The original harmonic map can then be obtained from $$\phi_ 0$$ by inverse procedures which involve choosing arbitrary holomorphic sections of $${\mathbb{C}}P^ 1$$ bundles. The methods are those of moving frames; full details appear in part II of this paper [Preprint; per revr.]. For an alternative treatment and some further results see work of F. E. Burstall and the reviewer [J. Differ. Geom. 23, 255-297 (1986; Zbl 0588.58018)].
{Reviewer’s remark: In the statement of Theorem 6 it seems that there should be a third possibility (called a mixed pair in Burstall and the reviewer’s paper [op. cit., cf. Theorem 3.3]) where the orthogonal harmonic points in the double harmonic flag $$w_ 0$$ are, respectively, a holomorpic and an antiholomorphic map h,g: $$S^ 2\to P_{n-1}$$ into complex projective space with $$\partial h\perp g$$ (mixed pairs are always harmonic).}

##### MSC:
 58E20 Harmonic maps, etc. 14M15 Grassmannians, Schubert varieties, flag manifolds
##### Keywords:
harmonic maps; complex Grassmann manifold
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