Minimal two-spheres in G(2, 4)

Abstract

In this paper, we mainly study the geometry of conformal minimal immersions of two-spheres in a complex Grassmann manifold G(2,4). At first, we give a precise description of any non-±holomorphic harmonic 2-sphere in G(2,4) with the linearly full holomorphic maps ψ0: S2 → ℂP3 (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ: S2 → G(2, 4) with constant curvature has constant Kaahler angle. Furthermore, ϕ is either V1(3) + V3(3), which is totally geodesic, with constant Gauss curvature 2/5 and constant Kaahler angle given by t = 3/2 or V3(3) + V2(3), which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V0(3), V1(3), V2(3), V3(3): S2 → ℂP3 is the Veronese sequence.