Geodesic Lines Of The Complex Quadric Q~n

The complex quadric Qn is a complex hypersurface in(n+1)-dimensional com-plex projective space CPn+1,and its Riemannian metric is induced from the Fubini-Study metric on CPn+1.In this paper,we study geodesic lines of Qn.For any geodesic line γ(t)of Qn,there is a curve γ(t)∈Q that satisfies π(γ(t))=γ(t),we obtain parametrization of γ(t)by giving a explicit parametrization of curve γ(t),where t is the arc length parameter,Q is a submanifold of real codimensional 2 in unit sphere S2n+3 and π is the natural projection from Q to Qn.