Curve theory in S²× S² with applications to ruled surfaces in S³

The Riemannian product S²× S² of two unit spheres admits several important geometric interpretations. It can be identified both with the complex quadric Q² and with the space L(S³) of oriented geodesics in the unit $3$-sphere S³. In this paper, we first develop a unified framework for the geometry of curves in S²× S². We introduce projection-regular curves in S²× S² and their curvature data, and prove a congruence theorem for these curves. As an application of this framework, regarding S²× S² as the complex quadric Q², we classify Kähler magnetic geodesics associated with the product Kähler--Einstein structure. For closed generic projection-regular curves, we establish a sharp Fenchel-type inequality. We then identify S²× S² with the space L(S³) of oriented geodesics in the unit $3$-sphere S³ and apply our curve theory to ruled surfaces in S³. In particular, we prove that every nonsingular closed generic projection-regular ruled surface attaining equality in the Fenchel-type inequality is an embedded torus.