Grassmann Manifold G(2,8) And Complex Structures On R～8

In this paper, we use Clifford algebra Cl₈ to study the complex structures on R⁸, unit sphere S⁶ and S⁴. We consider Grassmann manifold G(2,8) as a submanifold of the Clifford algebra Cl₈. Thus we construct homeomorphism between Grassmann manifold G(2,8) and the set of orthogonal complex structures on R⁸ by the isomorphism between the Clifford algebra Cl₈ and the matrix algebra R(16) .We show that G(2,8) is a totally geodesic submanifold of SO(8). Restricting the homeomorphism on the fibres of fibre bundles Ï€ : (7(2,8) → S⁶ and Ï„ : CP³ → S⁴ respectively, we get the set of complex structures on the tangent space of S⁶ and S⁴ respectively.