Abstract
In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent.
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Xu, X., Jiao, X. Holomorphic two-spheres in complex Grassmann manifold G(2, 4). Proc Math Sci 118, 381–388 (2008). https://doi.org/10.1007/s12044-008-0030-8
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DOI: https://doi.org/10.1007/s12044-008-0030-8