Q ₄ Has A Very Small Two-dimensional Spherical Surface Of The Kelvin Angle

J. Bolton, G. R. Jensen, M. Rigoli and L. M. Woodward [1] perfectly clas-sify minimal constant curved two-dimensional spheres in the complex pro-jective space CP" through the Veronese sequence. The hyperquadric Qn is a complex submanifolds of CPn+1, and its holomorphic sectional curvature is not constant. The famous mathematician, S. S. Chern studied the geomet-ric properties of Qn. In recent years, many scholars studied the geometry of submanifolds of Qn. Recently, totally real minimal two-dimensional spheres with constant curvature in Qn are also completely determined. It is proved that minimal two-dimensional spheres with constant Kahler angle in Q2, neither holomorphic nor anti-holomorphic, is totally real. In this paper, we consider minimal immersion, neither holomorphic nor anti-holomorphic, of minimal two-dimensional spheres into the hyperquadric Q4. It is prove that the immer-sion is totally real if its Kahler angle is constant, moreover, we have some ge-ometric properties of the fundamental collineations of minimal minimal two-dimensional spheres in Q4, We obtain the following results.Theorem 3.1.1 Let F:S2 â†' Q4 be a linearly full conformal minimal immersion with constant Kahler angle θ ∈ (0, Ï€). Then F is totally real.Theorem 3.2.1 Let F:S2 â†' Q4 be a linearly full conformal minimal im-mersion with constant Kahler angle θ ∈ (0, Ï€), Ï„x =Ï„y = 0 and r≠ 0(defined by(3.7)). Then we have(1) (?)F:S2 â†' Q4 is anti-holomorphic conformal minimal immersion, the Gauss curvature of the induced metric K=2, the length of the second funda-mental||B||=2;(2) (?)F:S2 â†'Q4 is holomorphic conformal minimal immersion, the Gauss curvature of the induced metric K=2, the length of the second fundamental ||B||=2.