Department of Mathematics, University of California at Berkeley, and Mathematical Sciences Research Institute, Berkeley, CA 94720.
Proc Natl Acad Sci U S A. 1985 Apr;82(8):2217-9. doi: 10.1073/pnas.82.8.2217.
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).
设 G(k, n) 为 C(k) 在 C(n) 中的 Grassmann 流形，分别为维数 k 和 n 的复空间，或者说，它是 P(k-1) 在 P(n-1) 中的所有射影空间的流形，因此 G(1, n) 本身就是复射影空间 P(n-1)。我们研究二维球面 S(2) 到 G(k, n) 的调和映射。k = 1 的情况已经被几位作者研究过[例如，Din, A. M. & Zakrzewski, W. J. (1980) Nucl. Phys. B 174, 397-406; Eells, J. & Wood, J. C. (1983) Adv. Math. 49, 217-263; 和 Wolfson, J. G. Trans. Am. Math. Soc., in press]。Ramanathan [Ramanathan, J. (1984) J. Differ. Geom. 19, 207-219]研究了 S(2) --> G(2, 4) 的调和映射。我们将描述所有 S(2) --> G(2, n) 的调和映射。该方法基于几种几何构造，这些构造从给定的调和映射到新的调和映射，其中图像射影空间通过“基本共线”联系起来。关键结果是一些基本共线的退化，这是一个全局结果，这是由于域流形是 S(2)。该方法扩展到 G(k, n)。
