Rigidity of Conformal Minimal Immersions of Constant Curvature from $$S^2$$S2 to $$Q_4$$Q4

Geometry of conformal minimal two-spheres immersed in $$G(2,6;\mathbb {R})$$ is studied in this paper by harmonic maps. We construct a nonhomogeneous constant curved minimal two-sphere in $$G(2,6;\mathbb {R})$$, by establishing a classification theorem of linearly full conformal minimal immersions of constant curvature from $$S^2$$ to $$G(2,6;\mathbb {R})$$ identified with the complex hyperquadric $$Q_{4}$$, which illustrates minimal two-spheres of constant curvature in $$Q_{4}$$ are in general not congruent.