Harmonic maps of S into a complex Grassmann manifold.
Let G(k, n) be the Grassmann manifold of all C(k) in C(n), the complex spaces of dimensions k and n, respectively, or, what is the same, the manifold of all projective spaces P(k-1) in P(n-1), so that G(1, n) is the complex projective space P(n-1) itself. We study harmonic maps of the two-dimensional sphere S(2) 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 S(2) --> G(2, 4) have been studied by Ramanathan [Ramanathan, J. (1984) J. Differ. Geom. 19, 207-219]. We shall describe all harmonic maps S(2) --> 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 S(2). The method extends to G(k, n).
Department of Mathematics, University of California at Berkeley, and Mathematical Sciences Research Institute, Berkeley, CA 94720.
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