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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

58E20 Harmonic maps, etc.
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
58A15 Exterior differential systems (Cartan theory)
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