On holomorphic curves in a complex Grassmann manifold G(2, n)

Abstract

Let s : S2 → G(2, n) be a linearly full totally unramified non-degenerate holomorphic curve in a complex Grassmann manifold G(2, n), and let K(s) be its Gaussian curvature. It is proved that \({K(s) = \frac{4}{n-2}}\) if K(s) satisfies \({K(s) \geq \frac{4}{n-2}}\) or \({K(s) \leq \frac{4}{n-2} }\) everywhere on S2. In particular, \({K(s) = \frac{4}{n-2}}\) if K(s) is constant.