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Harmonic maps from $$S^ 2$$ to $$G_{2,4}$$. (English) Zbl 0546.58019
This article gives a classification of harmonic maps of $${\mathbb{C}}P^ 1$$ into $$G_{2,4}$$, the Grassmannian of 2-planes in $${\mathbb{C}}^ 4$$, by holomorphic maps of $${\mathbb{C}}P^ 1$$ into $${\mathbb{C}}P^ 3$$ and certain antiholomorphic rank one sub-bundles of holomorphic bundles over $${\mathbb{C}}P^ 1$$. Let $$\phi:{\mathbb{C}}P^ 1\to G_{2,4}$$ be a smooth map. Denoting by E, the pull back by $$\phi$$ of the tautological subbundle of $$G_{2,4}\times {\mathbb{C}}^ 4$$, we identify $$\phi^{- 1}T^{(1,0)}G_{2,4}$$ with $$Hom(E,E^{\perp})$$ and then the (1,0) part of the derivative of $$\phi$$ is given by $$dz\otimes\beta_ z+d\bar z\otimes\beta_{\bar z}$$ with $$\beta_ z,\beta_{\bar z}$$ local sections of $$Hom(E,E^{\perp}).$$
Now we state Ramanathan’s results as follows: Let $$h_{k-1},h_ k:{\mathbb{C}}P^ 1\to {\mathbb{C}}P^ 3$$, 0$$\leq k\leq 3$$, be consecutive legs of the Frénet frame of some holomorphic curve $$h_ 0:{\mathbb{C}}P^ 1\to {\mathbb{C}}P^ 3$$. If H is the subbundle of $${\mathbb{C}}P^ 1\times {\mathbb{C}}^ 4$$ given by $$(h_{k-1}\oplus h_ k)^{\perp}$$ equipped with a suitable holomorphic structure, then any rank one antiholomorphic subbundle of H induces a map $$\Phi_ 1:{\mathbb{C}}P^ 1\to {\mathbb{C}}P^ 3$$ such that $$\phi =\Phi_ 1\oplus h_{k-1}:{\mathbb{C}}P^ 1\to G_{2,4}$$ is a harmonic map with rank $$\beta {}_ z<2$$ and $$\beta_ z\not\equiv 0$$. Further, any such harmonic map arises in this way, indeed we take $$\Phi_ 1=\ker \beta_ z$$ almost everywhere and $$h_{k-1}=\phi\cap \Phi_ 1^{\perp}$$. Lastly it is shown that for any non-(anti)holomorphic harmonic $$\phi:{\mathbb{C}}P^ 1\to G_{2,4}$$ either $$\phi$$ or $$\phi^{\perp}$$ has rk $$\beta {}_ z<2$$ so that these constructions account for essentially all harmonic maps from $${\mathbb{C}}P^ 1$$ into $$G_{2,4}.$$
The main ingredients in the proof are the results of J. Eells and J. C. Wood [Adv. Math. 49, 217-263 (1983; Zbl 0528.58007)] which characterise any harmonic map of $${\mathbb{C}}P^ 1$$ into $${\mathbb{C}}P^ 3$$ as a leg of the Frénet frame of some holomorphic map of $${\mathbb{C}}P^ 1$$ into $${\mathbb{C}}P^ 3$$ and a conservation law which states that $$\beta_ z\circ\beta^*_{\bar z}$$ is a nilpotent endomorphism on each fibre of E.
Reviewer’s remark: Since the publication of this article, the reviewer and several others have considerably extended the results obtained therein.
Reviewer: F.E.Burstall

##### MSC:
 58E20 Harmonic maps, etc. 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 58A15 Exterior differential systems (Cartan theory)
##### Keywords:
complex Grassmannians; 2-sphere; harmonic maps; holomorphic map
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