Harmonic maps of S2 into a complex Grassmann manifold

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Let G(k, n) be the Grassmann manifold of all Ck in Cn, the complex spaces of dimensions k and n, respectively, or, what is the same, the manifold of all projective spaces Pk-1 in Pn-1, so that G(1, n) is the complex projective space Pn-1 itself. We study harmonic maps of the two-dimensional sphere S2 into G(k, n). The case k = 1 has been the subject of investigation by several authors [see, for example, Din, A. M. & Zakrzewski, W. J. (1980) Nucl. Phys. B 174, 397-406; Eells, J. & Wood, J. C. (1983) Adv. Math. 49, 217-263; and Wolfson, J. G. Trans. Am. Math. Soc., in press]. The harmonic maps S2 → G(2, 4) have been studied by Ramanathan [Ramanathan, J. (1984) J. Differ. Geom. 19, 207-219]. We shall describe all harmonic maps S2 → G(2, n). The method is based on several geometrical constructions, which lead from a given harmonic map to new harmonic maps, in which the image projective spaces are related by “fundamental collineations.” The key result is the degeneracy of some fundamental collineations, which is a global consequence, following from the fact that the domain manifold is S2. The method extends to G(k, n).

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