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 $K\leq 2$ (or $K\geq 2$), 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 $K\leq 1$ (or $K\geq 1$), then $K=1$. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in $G(2,4)$ must be $U(4)$-equivalent.
Volume 131, 2021
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